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+{"documenter":{"julia_version":"1.9.3","generation_timestamp":"2023-10-08T15:31:19","documenter_version":"1.1.0"}} \ No newline at end of file diff --git a/previews/PR178/anisotropy.html b/previews/PR178/anisotropy.html new file mode 100644 index 000000000..61113dedb --- /dev/null +++ b/previews/PR178/anisotropy.html @@ -0,0 +1,40 @@ + +Interaction Strength Renormalization · Sunny documentation
GitHub

Interaction Strength Renormalization

A unique feature of Sunny is its support for building classical models where each quantum spin is represented as a full $N$-level system, rather than just an expected dipole. This formalism enables more accurate modeling of quantum spin Hamiltonians that include, e.g., a single-ion anisotropy, or a biquadratic coupling between sites.

Local operators

A quantum spin-$S$ state has $N = 2S + 1$ levels. Each local spin operator $\hat{S}^{\{x,y,z\}}$ is faithfully represented as an $N×N$ matrix. Access these matrices using spin_matrices for a given label $S$. For example, spin_matrices(1/2) returns the Pauli matrices divided by 2.

When $S > 1/2$, it is possible to construct multipole moments beyond the spin-dipole. For example,

XXX TODO XXX

S = spin_matrices(2)

The Stevens operators $\hat{\mathcal{O}}_{k,q}$ are polynomials of the spin operators, and are accessed using stevens_matrices. With these building blocks, a single-ion anisotropy is defined using set_onsite_coupling!. For example:

# An easy axis anisotropy in the z-direction
+S = spin_operators(sys, i)
+set_onsite_coupling!(sys, -D*S[3]^3, i)
+
+# The unique quartic single-ion anisotropy for a site with cubic point group
+# symmetry
+O = stevens_operators(sys, i)
+set_onsite_coupling!(sys, O[4,0] + 5*O[4,4], i)
+
+# An equivalent expression of this quartic anisotropy, up to a constant shift
+set_onsite_coupling!(sys, 20*(S[1]^4 + S[2]^4 + S[3]^4), i)

Renormalization procedure for :dipole mode

There are two allowed modes for a System. The mode :SUN models each spin as an SU(N) coherent state (i.e., as a set of $N$ complex amplitudes), and is the most variationally accurate. The mode :dipole constrains the SU(N) coherent-state dynamics to the space of pure dipoles. In either mode, Sunny encourages specifying single-ion anisotropies as $N×N$ matrices. In :dipole mode, Sunny will automatically renormalize the anisotropy operator to achieve maximal consistency with :SUN mode. This procedure was derived in D. Dahlbom et al., [arXiv:2304.03874]. Here, we summarize the final results.

The starting point is a quantum operator $\hat{\mathcal{H}}_{\mathrm{local}}$ giving the single-ion anisotropy for one site. It can be expanded in Stevens operators,

\[\hat{\mathcal H}_{\mathrm{local}} = \sum_{k, q} A_{k,q} \hat{\mathcal{O}}_{k,q}.\]

See the documentation of print_stevens_expansion for some explicit examples of this expansion.

The traditional classical limit of a quantum spin Hamiltonian, which yields the Landau-Lifshitz dynamics, can be derived by taking the formal $S \to\infty$ limit, such that each spin operator $\hat{\mathbf{S}}$ is replaced by its dipole expectation value $\mathbf{s}$. Correspondingly, the Stevens operators $\hat{\mathcal{O}}_{k,q}$ become polynomials $\mathcal{O}_{k,q}(\mathbf{s})$ in the classical dipole. With this traditional approach, one would arrive at the bare expected energy,

\[H_{\mathrm{bare}}(\mathbf{s}) = \sum_{k, q} A_{k,q} \mathcal{O}_{k,q}(\mathbf{s}).\]

In a real magnetic compound, however, $S$ may not be very large, and one can achieve a better approximation by avoiding the $S \to\infty$ limit. The strategy is to begin with the full dynamics of SU(N) coherent states, and then constrain it to the space of dipoles $\mathbf{s}$. Doing so will again yield the Landau-Lifshitz dynamics, but now involving the renormalized expected energy,

\[H_{\mathrm{renormalized}}(\mathbf{s}) = \sum_{k, q} c_k A_{k,q} \mathcal{O}_{k,q}(\mathbf{s}).\]

The $k$-dependent renormalization factors are

\[\begin{align*} +c_2 &= 1-\frac{1}{2}S^{-1} \\ +c_4 &= 1-3S^{-1}+\frac{11}{4}S^{-2}-\frac{3}{4}S^{-3} \\ +c_6 &= 1-\frac{15}{2}S^{-1}+\frac{85}{4}S^{-2}-\frac{225}{8}S^{-3}+\frac{137}{8}S^{-4}-\frac{15}{4}S^{-5}. +\end{align*}\]

Sunny will use $H_{\mathrm{renormalized}}(\mathbf{s})$ in its classical dynamics of dipoles. Because of this renormalization, Sunny is more variationally accurate than traditional codes like SpinW.

How and when to disable renormalization?

Although we generally recommend the above renormalization procedure, there are circumstances where it is not desirable. Examples include reproducing a model-system study, or describing a micromagnetic system for which the $S\to\infty$ limit is quantitatively realized. To get symbolic operators in the large-$S$ limit, use spin_matrices or stevens_matrices with the argument Inf. Sunny will not perform any renormalization on anisotropy operators constructed through these primitives.

Note that Sunny will also renormalize scalar biquadratic exchange interactions by default. Disable this renormalization by setting large_S = true in the call to set_exchange!.

Stevens operators

The Stevens operators $\hat{\mathcal{O}}_{k,q}$ are defined as polynomials of angular momentum operators $\hat{S}_{\{x,y,z\}}$ in some spin-$S$ representation.

Using

\[\begin{align*} +X &= \mathbf{\hat{S}} \cdot \mathbf{\hat{S}} = S (S+1) \\ +\hat{S}_\pm &= \hat{S}_x \pm i \hat{S}_y \\ +\phi_+ &= \frac{1}{4},\quad \phi_- = \frac{1}{4 i}, +\end{align*}\]

the relevant Stevens operators are defined as,

\[\begin{align*} +\hat{\mathcal{O}}_{0,0} & = 1 \\ +\\ +\hat{\mathcal{O}}_{2,\pm2} & =\phi_{\pm}(\hat{S}_{+}^{2}\pm \hat{S}_{-}^{2})+\mathrm{h.c.}\\ +\hat{\mathcal{O}}_{2,\pm1} & =\phi_{\pm}(\hat{S}_{+}\pm \hat{S}_{-})\hat{S}_{z}+\mathrm{h.c.}\\ +\hat{\mathcal{O}}_{2,0} & =3\hat{S}_{z}^{2}-X\\ +\\ +\hat{\mathcal{O}}_{4,\pm4} & =\phi_{\pm}(\hat{S}_{+}^{4}\pm \hat{S}_{-}^{4})+\mathrm{h.c.}\\ +\hat{\mathcal{O}}_{4,\pm3} & =\phi_{\pm}(\hat{S}_{+}^{3}\pm \hat{S}_{-}^{3})\hat{S}_{z}+\mathrm{h.c.}\\ +\hat{\mathcal{O}}_{4,\pm2} & =\phi_{\pm}(\hat{S}_{+}^{2}\pm \hat{S}_{-}^{2})(7\hat{S}_{z}^{2}-(X+5))+\mathrm{h.c.}\\ +\hat{\mathcal{O}}_{4,\pm1} & =\phi_{\pm}(\hat{S}_{+}\pm \hat{S}_{-})(7\hat{S}_{z}^{3}-(3X+1)\hat{S}_{z})+\mathrm{h.c.}\\ +\hat{\mathcal{O}}_{4,0} & =35\hat{S}_{z}^{4}-(30X-25)\hat{S}_{z}^{2}+(3X^{2}-6X)\\ +\\ +\hat{\mathcal{O}}_{6,\pm6} & =\phi_{\pm}(\hat{S}_{+}^{6}\pm \hat{S}_{-}^{6})+\mathrm{h.c.}\\ +\hat{\mathcal{O}}_{6,\pm5} & =\phi_{\pm}(\hat{S}_{+}^{5}\pm \hat{S}_{-}^{5})\hat{S}_{z}+\mathrm{h.c.}\\ +\hat{\mathcal{O}}_{6,\pm4} & =\phi_{\pm}(\hat{S}_{+}^{4}\pm \hat{S}_{-}^{4})(11\hat{S}_{z}^{2}-X-38)+\mathrm{h.c.}\\ +\hat{\mathcal{O}}_{6,\pm3} & =\phi_{\pm}(\hat{S}_{+}^{3}\pm \hat{S}_{-}^{3})(11\hat{S}_{z}^{3}-(3X+59)\hat{S}_{z})+\mathrm{h.c.}\\ +\hat{\mathcal{O}}_{6,\pm2} & =\phi_{\pm}(\hat{S}_{+}^{2}\pm \hat{S}_{-}^{2})(33\hat{S}_{z}^{4}-(18X+123)\hat{S}_{z}^{2}+X^{2}+10X+102)+\mathrm{h.c.}\\ +\hat{\mathcal{O}}_{6,\pm1} & =\phi_{\pm}(\hat{S}_{+}\pm \hat{S}_{-})(33\hat{S}_{z}^{5}-(30X-15)\hat{S}_{z}^{3}+(5X^{2}-10X+12)\hat{S}_{z})+\mathrm{h.c.}\\ +\hat{\mathcal{O}}_{6,0} & =231\hat{S}_{z}^{6}-(315X-735)\hat{S}_{z}^{4}+(105X^{2}-525X+294)\hat{S}_{z}^{2}-5X^{3}+40X^{2}-60X +\end{align*}\]

Stevens operators $\hat{\mathcal{O}}_{k,q}$ for odd $k$ are disallowed from the single-ion anisotropy under the assumption of time-reversal symmetry. Computer-generated tables of Stevens operators with larger k are available from C. Rudowicz and C. Y. Chung, J. Phys.: Condens. Matter 16, 5825 (2004).

For each $k$ value, the collection of operators $\{\hat{\mathcal{O}}_{k,q'}\}$ for $q' = -k, \dots, k$ is an irreducible representation of the group of rotations O(3). In particular, a physical rotation will transform $\hat{\mathcal{O}}_{k,q}$ into a linear combination of $\hat{\mathcal{O}}_{k,q'}$ where $q'$ varies but $k$ remains fixed.

In taking the large-$S$ limit, each dipole operator is replaced by its expectation value $\mathbf{s} = \langle \hat{\mathbf{S}} \rangle$, and only leading-order terms are retained. The operator $\hat{\mathcal{O}}_{k,q}$ becomes a homogeneous polynomial $O_{k,q}(\mathbf{s})$ of order $k$ in the spin components. One can see these polynomials by constructing stevens_matrices with the argument S = Inf. Due to the normalization constraint, each dipole can be expressed in polar angles, $(\theta, \phi)$. Then the Stevens functions $O_{k,q}(\mathbf{s})$ correspond to the spherical harmonic functions $Y_l^m(\theta, \phi)$ where $l=k$ and $m=q$, and modulo $k$ and $q$-dependent rescaling factors.

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// minisearch requires a unique for each object + return x; +}); + +// list below is the lunr 2.1.3 list minus the intersect with names(Base) +// (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) +// ideally we'd just filter the original list but it's not available as a variable +const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", +]); + +let index = new minisearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with search results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + boost: { title: 100 }, + fuzzy: 2, + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + } + + return word ?? null; + }, + tokenize: (string) => string.split(/[\s\-\.]+/), + }, +}); + +index.addAll(data); + +let filters = [...new Set(data.map((x) => x.category))]; +var modal_filters = make_modal_body_filters(filters); +var filter_results = []; + +$(document).on("keyup", ".documenter-search-input", function (event) { + // Adding a debounce to prevent disruptions from super-speed typing! + debounce(() => update_search(filter_results), 300); +}); + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + $(this).removeClass("search-filter-selected"); + } else { + $(this).addClass("search-filter-selected"); + } + + // Adding a debounce to prevent disruptions from crazy clicking! + debounce(() => get_filters(), 300); +}); + +/** + * A debounce function, takes a function and an optional timeout in milliseconds + * + * @function callback + * @param {number} timeout + */ +function debounce(callback, timeout = 300) { + clearTimeout(timer); + timer = setTimeout(callback, timeout); +} + +/** + * Make/Update the search component + * + * @param {string[]} selected_filters + */ +function update_search(selected_filters = []) { + let initial_search_body = ` +

Type something to get started!
+ `; + + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + results = index.search(querystring, { + filter: (result) => { + // Filtering results + if (selected_filters.length === 0) { + return result.score >= 1; + } else { + return ( + result.score >= 1 && selected_filters.includes(result.category) + ); + } + }, + }); + + let search_result_container = ``; + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + results.forEach(function (result) { + if (result.location) { + // Checking for duplication of results for the same page + if (!links.includes(result.location)) { + search_results += make_search_result(result, querystring); + count++; + } + + links.push(result.location); + } + }); + + let result_count = `
${count} result(s)
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + filter_results = []; + modal_filters = make_modal_body_filters(filters, filter_results); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(initial_search_body); + } +} + +/** + * Make the modal filter html + * + * @param {string[]} filters + * @param {string[]} selected_filters + * @returns string + */ +function make_modal_body_filters(filters, selected_filters = []) { + let str = ``; + + filters.forEach((val) => { + if (selected_filters.includes(val)) { + str += `${val}`; + } else { + str += `${val}`; + } + }); + + let filter_html = ` +
+ Filters: + ${str} +
+ `; + + return filter_html; +} + +/** + * Make the result component given a minisearch result data object and the value of the search input as queryString. + * To view the result object structure, refer: https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ +function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`\\b${querystring}\\b`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`\\b${querystring}\\b`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${result.title}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; +} + +/** + * Get selected filters, remake the filter html and lastly update the search modal + */ +function get_filters() { + let ele = $(".search-filters .search-filter-selected").get(); + filter_results = ele.map((x) => $(x).text().toLowerCase()); + modal_filters = make_modal_body_filters(filters, filter_results); + update_search(filter_results); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+`; + +let initial_search_body = ` +
Type something to get started!
+`; + +let search_modal_footer = ` +
+ + Ctrl + + / to search + + esc to close +
+`; + +$(document.body).append( + ` + + ` +); + +document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); +}); + +document.querySelector(".close-search-modal").addEventListener("click", () => { + closeModal(); +}); + +$(document).on("click", ".search-result-link", function () { + closeModal(); +}); + +document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; +}); + +// Functions to open and close a modal +function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); +} + +function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); +} + +document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $( + "" + ); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}); + +}) diff --git a/previews/PR178/assets/logo.png b/previews/PR178/assets/logo.png new file mode 100644 index 0000000000000000000000000000000000000000..803f96e15b790a9c303c5edb9ed85e40d2bc6183 GIT binary patch literal 80589 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zN{mHnKhdT!S6wI-GaltVDl{p#Fph{>O4Ooeo;Vh+lr%mz7{cBwo&wQFKP1EyCR5pu znYic*5`f{MKt6x@MzTn!hW)4HvC^~QdT$7Trh%wlX5R`|1nsys7h^Ghpx#Sxf8ST# zcO~(?d{xx|`8jsy4{sdp-+E34vWJA|4Bx&l4t+6FkasnDw2Guiwj5L1MEH*JE1a0A zdB`3k#T8-uoq1}Agm~4A`|gFiPG8|kttemUx!?Xxi7xTJH2dg@V{iocg}qQrvJj&Q zqmHBf>T2OXPcAc zN%89eX`T?RYxb8*M9GnSV+w!gf%x~iA6}=Qo~Qrbk@QEH?)S^$pWK1}{kp%GX#QV$ JUx -D*S[3]^2, 1)#hide\n", + "sys" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "## Finding a ground state" + ], + "metadata": {} + }, + { + "cell_type": "markdown", + "source": [ + "Sunny uses the [Langevin dynamics of SU(_N_) coherent\n", + "states](https://arxiv.org/abs/2209.01265) to sample spin configurations\n", + "from the thermal equlibrium. One first constructs a\n", + "`Langevin` integrator. This requires a time step, temperature, and a\n", + "phenomenological damping parameter $λ$ that sets the coupling to the thermal\n", + "bath." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "Δt = 0.05/D # Should be inversely proportional to the largest energy scale\n", + " # in the system. For FeI2, this is the easy-axis anisotropy,\n", + " # `D = 2.165` (meV). The prefactor 0.05 is relatively small,\n", + " # and achieves high accuracy.\n", + "kT = 0.2 # Temperature of the thermal bath (meV).\n", + "λ = 0.1 # This value is typically good for Monte Carlo sampling,\n", + " # independent of system details.\n", + "\n", + "langevin = Langevin(Δt; kT, λ);" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Langevin dynamics can be used to search for a magnetically ordered state. For\n", + "this, the temperature `kT` must be below the ordering temperature, but large\n", + "enough that the dynamical sampling procedure can overcome local energy\n", + "barriers and eliminate defects." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "randomize_spins!(sys)\n", + "for _ in 1:20_000\n", + " step!(sys, langevin)\n", + "end" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Although thermal fluctuations are present, the correct antiferromagnetic order\n", + "(2 up, 2 down) is apparent." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "plot_spins(sys; color=[s[3] for s in sys.dipoles])" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "For other systems, it can be much harder to find the magnetic ordering in an\n", + "unbiased way, and more complicated sampling procedures may be necessary." + ], + "metadata": {} + }, + { + "cell_type": "markdown", + "source": [ + "## Calculating Thermal-Averaged Correlations $\\langle S^{\\alpha\\beta}(𝐪,ω)\\rangle$\n", + "\n", + "Our aim is to study the classical spin dynamics for states sampled in thermal\n", + "equilibrium. To minimize finite size effects, and achieve sufficient momentum\n", + "space resolution, we should significantly enlarge the system volume. The\n", + "function `resize_supercell` takes new dimensions as multiples of the\n", + "unit cell lattice vectors." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "sys_large = resize_supercell(sys, (16,16,4)) # 16x16x4 copies of the original unit cell\n", + "plot_spins(sys_large; color=[s[3] for s in sys_large.dipoles])" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Now we will re-thermalize the system to a configuration just above the\n", + "ordering temperature. Sunny expects energies in meV by default, so\n", + "we use `meV_per_K` to convert from kelvin." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "kT = 3.5 * meV_per_K # 3.5K ≈ 0.30 meV\n", + "langevin.kT = kT\n", + "for _ in 1:10_000\n", + " step!(sys_large, langevin)\n", + "end" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "The next step is to collect correlation data $S^{\\alpha\\beta}$. This will\n", + "involve sampling spin configurations from thermal equilibrium, and then\n", + "integrating the [Hamiltonian dynamics of SU(_N_) coherent\n", + "states](https://arxiv.org/abs/2204.07563) to collect Fourier-space information\n", + "about normal modes. Quantization of these modes yields the magnons, and the\n", + "associated dynamical spin-spin correlations can be compared with neutron\n", + "scattering intensities $S^{\\alpha\\beta}(q,\\omega)$. Because this a\n", + "real-space calculation, data is only available for discrete $q$ modes (the\n", + "resolution scales like inverse system size).\n", + "\n", + "To store the correlation data, we initialize a `SampledCorrelations` object by\n", + "calling `dynamical_correlations`. It requires three keyword arguments:\n", + "an integration step size, a target number of ωs to retain, and a maximum\n", + "energy ω to resolve. For the time step, twice the value used for the Langevin\n", + "integrator is usually a good choice." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "sc = dynamical_correlations(sys_large; Δt=2Δt, nω=120, ωmax=7.5)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "The function `add_sample!` will collect data by running a dynamical\n", + "trajectory starting from the current system configuration." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "add_sample!(sc, sys_large) # Accumulate the sample into `sc`" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "To collect additional data, it is required to re-sample the spin configuration\n", + "from the thermal distribution. For efficiency, the dynamics should be run long\n", + "enough that consecutive samples are uncorrelated." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "for _ in 1:2\n", + " for _ in 1:1000 # Enough steps to decorrelate spins\n", + " step!(sys_large, langevin)\n", + " end\n", + " add_sample!(sc, sys_large) # Accumulate the sample into `sc`\n", + "end" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Now, `sc` has more samples included:" + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "sc" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "## Computing Scattering Intensities" + ], + "metadata": {} + }, + { + "cell_type": "markdown", + "source": [ + "With the thermally-averaged correlation data $\\langle S^{\\alpha\\beta}(q,\\omega)\\rangle$\n", + "in hand, we now need to specify how to extract a scattering intensity from this information.\n", + "This is done by constructing an `intensity_formula`.\n", + "By way of example, we will use a formula which computes the trace of the structure\n", + "factor and applies a classical-to-quantum temperature-dependent rescaling `kT`." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "formula = intensity_formula(sc, :trace; kT)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Recall that $\\langle S^{\\alpha\\beta}(q,\\omega)\\rangle$ is only available at certain discrete\n", + "$q$ values, due to the finite lattice size.\n", + "There are two basic approaches to handling this discreteness.\n", + "The first approach is to interpolate between the available\n", + "data using `intensities_interpolated`. For example, we can plot single-$q$ slices\n", + "at (0,0,0) and (π,π,π) using this method:" + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "qs = [[0, 0, 0], [0.5, 0.5, 0.5]]\n", + "is = intensities_interpolated(sc, qs, formula; interpolation = :round)\n", + "\n", + "ωs = available_energies(sc)\n", + "fig = lines(ωs, is[1,:]; axis=(xlabel=\"meV\", ylabel=\"Intensity\"), label=\"(0,0,0)\")\n", + "lines!(ωs, is[2,:]; label=\"(π,π,π)\")\n", + "axislegend()\n", + "fig" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "The resolution in energy can be improved by increasing `nω` (and decreasing `Δt`),\n", + "and the general accuracy can be improved by collecting additional samples from the thermal\n", + "equilibrium.\n", + "\n", + "For real calculations, one often wants to apply further corrections and more\n", + "accurate formulas. Here, we apply `FormFactor` corrections appropriate\n", + "for `Fe2` magnetic ions, and a dipole polarization correction `:perp`." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "formfactors = [FormFactor(\"Fe2\"; g_lande=3/2)]\n", + "new_formula = intensity_formula(sc, :perp; kT, formfactors = formfactors)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Frequently, one wants to extract energy intensities along lines that connect\n", + "special wave vectors--a so-called \"spaghetti plot\". The function\n", + "`reciprocal_space_path` creates an appropriate horizontal axis for\n", + "this plot by linearly sampling between provided $q$-points with a given\n", + "sample density. The number of sample points between two wavevectors `q1` and\n", + "`q2` is given by `dist*density` where `dist = norm(cryst.recipvecs * (q1 -\n", + "q2))` is measured in the global frame." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "points = [[0, 0, 0], # List of wave vectors that define a path\n", + " [1, 0, 0],\n", + " [0, 1, 0],\n", + " [1/2, 0, 0],\n", + " [0, 1, 0],\n", + " [0, 0, 0]]\n", + "density = 40\n", + "path, xticks = reciprocal_space_path(cryst, points, density);" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Again using `intensities_interpolated`, we can evaluate the (interpolated) intensity\n", + "at each point on the `path`.\n", + "Since scattering intensities are only available at a certain discrete $(Q,\\omega)$\n", + "points, the intensity on the path can be calculated by interpolating between these\n", + "discrete points:" + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "is_interpolated = intensities_interpolated(sc, path, new_formula;\n", + " interpolation = :linear, # Interpolate between available wave vectors\n", + ");\n", + "# Add artificial broadening\n", + "is_interpolated_broadened = broaden_energy(sc, is, (ω, ω₀)->lorentzian(ω-ω₀, 0.05));" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "The second approach to handle the discreteness of the data is to bin the intensity\n", + "at the discrete points into the bins of a histogram.\n", + "First, the five sub-histograms are set up using `reciprocal_space_path_bins` in\n", + "analogy to `reciprocal_space_path`." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "cut_width = 0.3\n", + "density = 15\n", + "paramsList, markers, ranges = reciprocal_space_path_bins(sc,points,density,cut_width);" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Then, the intensity data is computed using `intensities_binned` for each sub-histogram:" + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "total_bins = ranges[end][end]\n", + "energy_bins = paramsList[1].numbins[4]\n", + "is_binned = zeros(Float64,total_bins,energy_bins)\n", + "integrated_kernel = integrated_lorentzian(0.05) # Lorentzian broadening\n", + "for k in eachindex(paramsList)\n", + " bin_data, counts = intensities_binned(sc,paramsList[k], new_formula;\n", + " integrated_kernel = integrated_kernel\n", + " )\n", + " is_binned[ranges[k],:] = bin_data[:,1,1,:] ./ counts[:,1,1,:]\n", + "end" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "The graph produced by interpolating (top) is similar to the one produced by binning (bottom):" + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "fig = Figure()\n", + "ax_top = Axis(fig[1,1],ylabel = \"meV\",xticklabelrotation=π/8,xticklabelsize=12;xticks)\n", + "ax_bottom = Axis(fig[2,1],ylabel = \"meV\",xticks = (markers, string.(points)),xticklabelrotation=π/8,xticklabelsize=12)\n", + "\n", + "heatmap!(ax_top,1:size(is_interpolated,1), ωs, is_interpolated;\n", + " colorrange=(0.0,0.07),\n", + ")\n", + "\n", + "heatmap!(ax_bottom,1:size(is_binned,1), ωs, is_binned;\n", + " colorrange=(0.0,0.05),\n", + ")\n", + "\n", + "fig" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Note that we have clipped the colors in order to make the higher-energy\n", + "excitations more visible." + ], + "metadata": {} + }, + { + "cell_type": "markdown", + "source": [ + "# Unconventional RLU Systems and Constant Energy Cuts" + ], + "metadata": {} + }, + { + "cell_type": "markdown", + "source": [ + "Often it is useful to plot cuts across multiple wave vectors but at a single\n", + "energy. We'll pick an energy," + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "ωidx = 60\n", + "target_ω = ωs[ωidx]" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "and take a constant-energy cut at that energy.\n", + "The most straightforward way is to make a plot whose axes are aligned with\n", + "the conventional reciprocal lattice of the crystal.\n", + "This is accomplished using `unit_resolution_binning_parameters`:" + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "params = unit_resolution_binning_parameters(sc)\n", + "params.binstart[1:2] .= -1 # Expand plot range slightly\n", + "\n", + "# Set energy integration range\n", + "omega_width = 0.3\n", + "params.binstart[4] = target_ω - (omega_width/2)\n", + "params.binend[4] = target_ω # `binend` should be inside (e.g. at the center) of the range\n", + "params.binwidth[4] = omega_width\n", + "\n", + "integrate_axes!(params, axes = 3) # Integrate out z direction entirely\n", + "\n", + "params" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "In each of the following plots, black dashed lines represent (direct) lattice vectors.\n", + "Since these plots are in reciprocal space, direct lattice vectors are represented\n", + "as covectors (i.e. coordinate grids) instead of as arrows." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "is, counts = intensities_binned(sc,params,new_formula)\n", + "\n", + "fig = Figure()\n", + "ax = Axis(fig[1,1];\n", + " title=\"Δω=0.3 meV (Binned)\", aspect=true,\n", + " xlabel = \"[H, 0, 0]\",\n", + " ylabel = \"[0, K, 0]\"\n", + ")\n", + "bcs = axes_bincenters(params)\n", + "hm = heatmap!(ax,bcs[1],bcs[2],is[:,:,1,1] ./ counts[:,:,1,1])\n", + "function add_lines!(ax,params)#hide\n", + " bes = Sunny.axes_binedges(params)#hide\n", + " hrange = range(-2,2,length=17)#hide\n", + " linesegments!(ax,[(Point2f(params.covectors[1,1:3] ⋅ [h,-10,0],params.covectors[2,1:3] ⋅ [h,-10,0]),Point2f(params.covectors[1,1:3] ⋅ [h,10,0],params.covectors[2,1:3] ⋅ [h,10,0])) for h = hrange],linestyle=:dash,color=:black)#hide\n", + " krange = range(-2,2,length=17)#hide\n", + " linesegments!(ax,[(Point2f(params.covectors[1,1:3] ⋅ [-10,k,0],params.covectors[2,1:3] ⋅ [-10,k,0]),Point2f(params.covectors[1,1:3] ⋅ [10,k,0],params.covectors[2,1:3] ⋅ [10,k,0])) for k = krange],linestyle=:dash,color=:black)#hide\n", + " xlims!(ax,bes[1][1],bes[1][end])#hide\n", + " ylims!(ax,bes[2][1],bes[2][end])#hide\n", + "end#hide\n", + "add_lines!(ax,params)\n", + "Colorbar(fig[1,2], hm);\n", + "fig" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "In the above plot, the dashed-line (direct) lattice vectors are clearly orthogonal.\n", + "However, we know that in real space, the lattice vectors $a$ and $b$ are *not* orthogonal, but rather\n", + "point along the edges of a hexagon (see lower left corner):\n", + "\n", + "\n", + "

\n", + "\n", + "\n", + "Thus, plotting the direct lattice vectors as orthogonal (even in reciprocal space) is somewhat misleading.\n", + "Worse yet, the `[H,0,0]` by `[0,K,0]` plot apparently loses the 6-fold symmetry of the crystal!\n", + "Lastly, if one works out the components of the real-space metric with respect to the axes of the plot,\n", + "one finds that there are non-zero off-diagonal entries," + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "latvecs = sys.crystal.latvecs\n", + "metric = latvecs' * I(3) * latvecs" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "so real-space rotations and angles map into reciprocal space rotations angles in a complicated way.\n", + "\n", + "To resolve these important issues, we want to use axes which are orthogonal (i.e. they diagonalize\n", + "the metric and solve all of the problems just mentioned). The canonical choice is to use\n", + "the combination $\\frac{1}{2}a + b$ of lattice vectors (equiv. $a^* - \\frac{1}{2}b^*$), which is orthogonal to $a$:" + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "(latvecs * [1/2,1,0]) ⋅ (latvecs * [1,0,0]) == 0" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "This new vector $\\frac{1}{2}a+b$ is visibly orthogonal to $a$ in real space:" + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "f = Figure()#hide\n", + "ax = Axis(f[1,1])#hide\n", + "arrows!(ax,[Point2f(0,0),Point2f(latvecs[1:2,1] ./ 2)],[Vec2f(latvecs[1:2,1] ./ 2), Vec2f(latvecs[1:2,2])],arrowcolor = :blue,arrowsize = 30.,linewidth = 5.,linecolor = :blue)#hide\n", + "arrows!(ax,[Point2f(0,0)],[Vec2f(latvecs[1:2,:] * [1/2,1,0])],arrowcolor = :red,arrowsize = 30.,linewidth = 5.,linecolor = :red, linestyle = :dash)#hide\n", + "scatter!(ax,[Point2f(latvecs[1:2,:] * [a,b,0]) for a in -1:1, b in -1:1][:],color = :black)#hide\n", + "annotations!(ax,[\"0\",\"0+b\",\"0+a\", \"a/2\", \"b\"],[Point2f(0,-0.3),Point2f(latvecs[1:2,2]) .- Vec2f(0,0.3),Point2f(latvecs[1:2,1]) .- Vec2f(0,0.3),Point2f(latvecs[1:2,1] ./ 4) .- Vec2f(0,0.3),Point2f(latvecs[1:2,1] ./ 2) .+ Vec2f(latvecs[1:2,2] ./ 2) .+ Vec2f(0.3,0.3)],color=[:black,:black,:black,:blue,:blue])#hide\n", + "f#hide" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "To use \"projection onto the new vector\" as a histogram axis, only a single change is needed to the binning parameters.\n", + "The second covector (previously $b$) must be swapped out for $\\frac{1}{2}a + b$ (recall that reciprocal space covectors, such\n", + "as those used in `BinningParameters` correspond to direct space vectors)." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "params.covectors[2,1:3] = [1/2,1,0] # [1/2,1,0] times [a;b;c] is (a/2 + b)\n", + "params#hide" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "The second axis of the histogram now agrees with what is conventionally labelled as `[H,-H/2,0]`." + ], + "metadata": {} + }, + { + "cell_type": "markdown", + "source": [ + "!!! warning \"Length of the new vector\"\n", + "\n", + " Note that, although $\\frac{1}{2}a+b$ is orthogonal to $a$, it is not the same length as $a$.\n", + " Instead, it is `sqrt(3/4)` times as long. Note the unsymmetrical axes labels in the plots that\n", + " follow as a direct result of this!" + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "# Zoom out horizontal axis\n", + "params.binstart[1], params.binend[1] = -2, 2\n", + "\n", + "# Adjust vertical axis bounds to account for\n", + "# length of a/2 + b\n", + "params.binstart[2], params.binend[2] = -2 * sqrt(3/4), 2 * sqrt(3/4)\n", + "\n", + "# Re-compute in the new coordinate system\n", + "is, counts = intensities_binned(sc,params,new_formula)\n", + "\n", + "fig = Figure(; resolution=(1200,500))#hide\n", + "ax_right = Axis(fig[1,3];#hide\n", + " title=\"ω≈$(round(target_ω, digits=2)) meV with Δω=0.3 meV (Binned)\", aspect=true,#hide\n", + " xlabel = \"[H, -1/2H, 0]\"#hide\n", + ")#hide\n", + "bcs = axes_bincenters(params)#hide\n", + "hm_right = heatmap!(ax_right,bcs[1],bcs[2],is[:,:,1,1] ./ counts[:,:,1,1])#hide\n", + "add_lines!(ax_right,params)\n", + "Colorbar(fig[1,4], hm_right);#hide" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "For comparison purposes, we will make the same plot using\n", + "`intensities_interpolated` to emulate zero-width bins.\n", + "This time, it's more convenient to think in terms of reciprocal vectors $a^*$ and $b^*$.\n", + "Now, our coordinate transformation consists of\n", + "establishing a new, orthogonal basis\n", + "to specify our wave vectors: $a^* - \\frac{1}{2}b^*$, $b^*$ and $c^*$.\n", + "Writing this in matrix form allows us to sample a rectilinear grid of wave vectors in this frame.\n", + "Finally, we'll convert these back into the original RLU system for input into Sunny." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "# New basis matrix\n", + "A = [1 0 0\n", + " -1/2 1 0\n", + " 0 0 1]\n", + "\n", + "# Define our grid of wave vectors\n", + "npoints = 60\n", + "as = range(-2, 2, npoints)\n", + "bs = range(-3/√3, 3/√3, npoints)\n", + "qs_ortho = [[a, b, 0] for a in as, b in bs]\n", + "\n", + "# Convert to original RLU system for input to Sunny\n", + "qs = [A * q for q in qs_ortho]\n", + "\n", + "# Use interpolation to get intensities\n", + "is = intensities_interpolated(sc, qs, new_formula; interpolation=:linear)\n", + "\n", + "ax_left = Axis(fig[1,2];#hide\n", + " title=\"ω≈$(round(ωs[ωidx], digits=2)) meV (Interpolated)\", aspect=true,#hide\n", + " xlabel = \"[H, -1/2H, 0]\", ylabel = \"[0, K, 0]\"#hide\n", + ")#hide\n", + "hm_left = heatmap!(ax_left, as, bs, is[:,:,ωidx])#hide\n", + "add_lines!(ax_left,params)\n", + "Colorbar(fig[1,1], hm_left);#hide\n", + "fig" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Now, not only are the dashed-line lattice vectors no longer misleadingly orthogonal,\n", + "but the six-fold symmetry has been restored as well!\n", + "Further, the metric has been diagonalized:" + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "metric = (latvecs * inv(A'))' * I(3) * (latvecs * inv(A'))" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Finally, we note that instantaneous structure factor data, $𝒮(𝐪)$, can be\n", + "obtained from a dynamic structure factor with\n", + "`instant_intensities_interpolated`. Here we'll reuse the grid of wave\n", + "vectors we generated above." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "is_static = instant_intensities_interpolated(sc, qs, new_formula; interpolation = :linear)\n", + "\n", + "hm = heatmap(as, bs, is_static;\n", + " axis=(\n", + " title=\"Instantaneous Structure Factor\",\n", + " xlabel = \"[H, -1/2H, 0]\",\n", + " ylabel = \"[0, K, 0]\",\n", + " aspect=true\n", + " )\n", + ")\n", + "Colorbar(hm.figure[1,2], hm.plot)\n", + "hm" + ], + "metadata": {}, + "execution_count": null + } + ], + "nbformat_minor": 3, + "metadata": { + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.9.3" + }, + "kernelspec": { + "name": "julia-1.9", + "display_name": "Julia 1.9.3", + "language": "julia" + } + }, + "nbformat": 4 +} diff --git a/previews/PR178/assets/notebooks/fei2_tutorial.ipynb b/previews/PR178/assets/notebooks/fei2_tutorial.ipynb new file mode 100644 index 000000000..8fbbdc3c0 --- /dev/null +++ b/previews/PR178/assets/notebooks/fei2_tutorial.ipynb @@ -0,0 +1,770 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "source": [ + "# Case Study: FeI₂\n", + "\n", + "FeI₂ is an effective spin-1 material with strong single-ion anisotropy.\n", + "Quadrupolar fluctuations give rise to a single-ion bound state that cannot be\n", + "described by a dipole-only model. This tutorial illustrates how to use the\n", + "linear spin wave theory of SU(3) coherent states (i.e. 2-flavor bosons) to\n", + "model the magnetic behavior in FeI₂. The original study was performed in [Bai\n", + "et al., Nature Physics 17, 467–472\n", + "(2021)](https://doi.org/10.1038/s41567-020-01110-1).\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "The Fe atoms are arranged in stacked triangular layers. The effective spin\n", + "Hamiltonian takes the form,\n", + "\n", + "$$\n", + "\\mathcal{H}=\\sum_{(i,j)} 𝐒_i ⋅ J_{ij} 𝐒_j - D\\sum_i \\left(S_i^z\\right)^2,\n", + "$$\n", + "\n", + "where the set of exchange matrices $J_{ij}$ between bonded sites $(i,j)$\n", + "includes competing ferromagnetic and antiferromagnetic interactions. This\n", + "model also includes a strong easy axis anisotropy, $D > 0$.\n", + "\n", + "We will formulate this Hamiltonian in Sunny and then calculate its dynamic\n", + "structure factor." + ], + "metadata": {} + }, + { + "cell_type": "markdown", + "source": [ + "## Get Julia and Sunny\n", + "\n", + "Sunny is implemented in Julia. This is a relatively new programming language\n", + "that allows for interactive development (like Python or Matlab) while also\n", + "providing high numerical efficiency (like C++ or Fortran). New Julia users may\n", + "wish to take a look at our [Getting Started with\n", + "Julia](https://github.com/SunnySuite/Sunny.jl/wiki/Getting-started-with-Julia)\n", + "guide. Sunny requires Julia 1.9 or later.\n", + "\n", + "From the Julia prompt, load `Sunny`. For plotting, one can choose either\n", + "`GLMakie` (a pop-up window) or `WGLMakie` (inline plots for a Jupyter notebook\n", + "or VSCode)." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "using Sunny, WGLMakie" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "If these packages are not yet installed, Julia should offer to install them\n", + "using its built-in package management system. If old versions are installed,\n", + "you may need to update them to run this tutorial." + ], + "metadata": {} + }, + { + "cell_type": "markdown", + "source": [ + "## Crystals\n", + "\n", + "A `Crystal` describes the crystallographic unit cell and will usually\n", + "be loaded from a `.cif` file. Here, we instead build a crystal by listing all\n", + "atoms and their types." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "a = b = 4.05012 # Lattice constants for triangular lattice\n", + "c = 6.75214 # Spacing in the z-direction\n", + "\n", + "latvecs = lattice_vectors(a, b, c, 90, 90, 120) # A 3x3 matrix of lattice vectors that\n", + " # define the conventional unit cell\n", + "positions = [[0, 0, 0], [1/3, 2/3, 1/4], [2/3, 1/3, 3/4]] # Positions of atoms in fractions\n", + " # of lattice vectors\n", + "types = [\"Fe\", \"I\", \"I\"]\n", + "FeI2 = Crystal(latvecs, positions; types)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Observe that Sunny inferred the space group, 'P -3 m 1' (164) and labeled the\n", + "atoms according to their point group symmetries." + ], + "metadata": {} + }, + { + "cell_type": "markdown", + "source": [ + "Only the Fe atoms are magnetic, so we discard the I ions using\n", + "`subcrystal`." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "cryst = subcrystal(FeI2, \"Fe\")" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Observe that `cryst` retains the spacegroup symmetry of the full FeI₂ crystal.\n", + "This information will be used, for example, to propagate exchange interactions\n", + "between symmetry-equivalent bonds.\n", + "\n", + "In a running Julia environment, the crystal can be viewed interactively using\n", + "`view_crystal`." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "view_crystal(cryst, 8.0)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "## Symmetry analysis\n", + "\n", + "The command `print_symmetry_table` provides a list of all the\n", + "symmetry-allowed interactions up to a cutoff distance." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "print_symmetry_table(cryst, 8.0)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "The allowed $g$-tensor is expressed as a 3×3 matrix in the free coefficients\n", + "`A`, `B`, ... The allowed single-ion anisotropy is expressed as a linear\n", + "combination of Stevens operators. The latter correspond to polynomials of the\n", + "spin operators, as we will describe below.\n", + "\n", + "The allowed exchange interactions are given as a 3×3 matrix for representative\n", + "bonds. The notation `Bond(i, j, n)` indicates a bond between atom indices `i`\n", + "and `j`, with cell offset `n`. In the general case, it will be necessary to\n", + "associate atom indices with their positions in the unit cell; these can be\n", + "viewed with `display(cryst)`. Note that the order of the pair $(i, j)$ is\n", + "significant if the exchange tensor contains antisymmetric\n", + "Dzyaloshinskii–Moriya (DM) interactions.\n", + "\n", + "In the case of FeI₂, `Bond(1, 1, [1,0,0])` is one of the 6 nearest-neighbor\n", + "Fe-Fe bonds on a triangular lattice layer, and `Bond(1, 1, [0,0,1])` is an\n", + "Fe-Fe bond between layers." + ], + "metadata": {} + }, + { + "cell_type": "markdown", + "source": [ + "## Building a spin System" + ], + "metadata": {} + }, + { + "cell_type": "markdown", + "source": [ + "In constructing a spin `System`, we must provide several additional\n", + "details about the spins." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "sys = System(cryst, (4, 4, 4), [SpinInfo(1, S=1, g=2)], :SUN, seed=2)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "This system includes $4×4×4$ unit cells, i.e. 64 Fe atoms, each with spin $S=1$\n", + "and a $g$-factor of 2. Quantum mechanically, spin $S=1$ involves a\n", + "superposition of $2S+1=3$ distinct angular momentum states. In `:SUN` mode,\n", + "this superposition will be modeled explicitly using the formalism of SU(3)\n", + "coherent states, which captures both dipolar and quadrupolar fluctuations. For\n", + "the more traditional dipole dynamics, use `:dipole` mode instead." + ], + "metadata": {} + }, + { + "cell_type": "markdown", + "source": [ + "Next we will use `set_exchange!` to assign interaction to bonds. Sunny\n", + "will automatically propagate each interaction to all symmetry-equivalent bonds\n", + "in the unit cell. The FeI₂ interactions below follow [Bai et\n", + "al](https://doi.org/10.1038/s41567-020-01110-1)." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "J1pm = -0.236\n", + "J1pmpm = -0.161\n", + "J1zpm = -0.261\n", + "J2pm = 0.026\n", + "J3pm = 0.166\n", + "J′0pm = 0.037\n", + "J′1pm = 0.013\n", + "J′2apm = 0.068\n", + "\n", + "J1zz = -0.236\n", + "J2zz = 0.113\n", + "J3zz = 0.211\n", + "J′0zz = -0.036\n", + "J′1zz = 0.051\n", + "J′2azz = 0.073\n", + "\n", + "J1xx = J1pm + J1pmpm\n", + "J1yy = J1pm - J1pmpm\n", + "J1yz = J1zpm\n", + "\n", + "set_exchange!(sys, [J1xx 0.0 0.0;\n", + " 0.0 J1yy J1yz;\n", + " 0.0 J1yz J1zz], Bond(1,1,[1,0,0]))\n", + "set_exchange!(sys, [J2pm 0.0 0.0;\n", + " 0.0 J2pm 0.0;\n", + " 0.0 0.0 J2zz], Bond(1,1,[1,2,0]))\n", + "set_exchange!(sys, [J3pm 0.0 0.0;\n", + " 0.0 J3pm 0.0;\n", + " 0.0 0.0 J3zz], Bond(1,1,[2,0,0]))\n", + "set_exchange!(sys, [J′0pm 0.0 0.0;\n", + " 0.0 J′0pm 0.0;\n", + " 0.0 0.0 J′0zz], Bond(1,1,[0,0,1]))\n", + "set_exchange!(sys, [J′1pm 0.0 0.0;\n", + " 0.0 J′1pm 0.0;\n", + " 0.0 0.0 J′1zz], Bond(1,1,[1,0,1]))\n", + "set_exchange!(sys, [J′2apm 0.0 0.0;\n", + " 0.0 J′2apm 0.0;\n", + " 0.0 0.0 J′2azz], Bond(1,1,[1,2,1]))" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "The function `set_onsite_coupling!` assigns a single-ion anisotropy.\n", + "The argument can be constructed using `spin_matrices` or\n", + "`stevens_matrices`. Here we use Julia's anonymous function syntax to\n", + "assign an easy-axis anisotropy along the direction $\\hat{z}$." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "D = 2.165\n", + "set_onsite_coupling!(sys, S -> -D*S[3]^2, 1)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "# Calculating structure factor intensities" + ], + "metadata": {} + }, + { + "cell_type": "markdown", + "source": [ + "In the remainder of this tutorial, we will examine Sunny's tools for\n", + "calculating the dynamical structure factor using a [multi-boson\n", + "generalization](https://arxiv.org/abs/1307.7731) of linear spin wave theory\n", + "(LSWT). This theory describes non-interacting quasi-particle excitations that\n", + "hybridize dipolar and quadrupolar modes." + ], + "metadata": {} + }, + { + "cell_type": "markdown", + "source": [ + "## Finding the ground state" + ], + "metadata": {} + }, + { + "cell_type": "markdown", + "source": [ + "Begin with a random configuration and use `minimize_energy!` to find a\n", + "configuration of the SU(3) coherent states (i.e. spin dipoles and quadrupoles)\n", + "that locally minimizes energy." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "randomize_spins!(sys)\n", + "minimize_energy!(sys)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "A positive number above indicates that the procedure has converged to a local\n", + "energy minimum. The configuration, however, may still have defects. This can\n", + "be checked by visualizing the spins, colored according to their\n", + "$z$-components." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "plot_spins(sys; color=[s[3] for s in sys.dipoles])" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "A different understanding of the magnetic ordering can be obtained by moving\n", + "to Fourier space. The 'instant' structure factor $𝒮(𝐪)$ is an experimental\n", + "observable. To investigate $𝒮(𝐪)$ as true 3D data, Sunny provides\n", + "`instant_correlations` and related functions. Here, however, we will\n", + "use `print_wrapped_intensities`, which gives average intensities for\n", + "the individual Bravais sublattices (in effect, all wavevectors are wrapped to\n", + "the first Brillouin zone)." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "print_wrapped_intensities(sys)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "The result will likely be approximately consistent with the known zero-field\n", + "energy-minimizing magnetic structure of FeI₂, which is single-$Q$ (two-up,\n", + "two-down antiferromagnetic order). Mathematically, spontaneous symmetry\n", + "breaking should select one of $±Q = [0, -1/4, 1/4]$, $[1/4, 0, 1/4]$, or\n", + "$[-1/4,1/4,1/4]$, associated with the three-fold rotational symmetry of the\n", + "crystal spacegroup. In nature, however, one will frequently encounter\n", + "competing \"domains\" associated with the three possible orientations of the\n", + "ground state." + ], + "metadata": {} + }, + { + "cell_type": "markdown", + "source": [ + "If the desired ground state is already known, as with FeI₂, it could be\n", + "entered by hand using `set_dipole!`. Alternatively, in the case of\n", + "FeI₂, we could repeatedly employ the above randomization and minimization\n", + "procedure until a defect-free configuration is found. Some systems will have\n", + "more complicated ground states, which can be much more challenging to find.\n", + "For this, Sunny provides experimental support for powerful simulated annealing\n", + "via [parallel tempering](https://en.wikipedia.org/wiki/Parallel_tempering),\n", + "but that is outside the scope of this tutorial." + ], + "metadata": {} + }, + { + "cell_type": "markdown", + "source": [ + "Here, let's break the three-fold symmetry of FeI₂ by hand. Given one or more\n", + "desired $Q$ modes, Sunny can suggest a magnetic supercell with appropriate\n", + "periodicity. Let's arbitrarily select one of the three possible ordering\n", + "wavevectors, $Q = [0, -1/4, 1/4]$. Sunny suggests a corresponding magnetic\n", + "supercell in units of the crystal lattice vectors." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "suggest_magnetic_supercell([[0, -1/4, 1/4]])" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "The system returned by `reshape_supercell` is smaller, and is sheared\n", + "relative to the original system. This makes it much easier to find the global\n", + "energy minimum." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "sys_min = reshape_supercell(sys, [1 0 0; 0 2 1; 0 -2 1])\n", + "randomize_spins!(sys_min)\n", + "minimize_energy!(sys_min);" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Plot the system again, now including \"ghost\" spins out to 12Å" + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "plot_spins(sys_min; color=[s[3] for s in sys_min.dipoles], ghost_radius=12)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "## Linear spin wave theory\n", + "\n", + "Now that we have found the ground state for a magnetic supercell, we can\n", + "immediately proceed to perform zero-temperature calculations using linear spin\n", + "wave theory. We begin by instantiating a `SpinWaveTheory` type using the\n", + "supercell." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "swt = SpinWaveTheory(sys_min)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Select a sequence of wavevectors that will define a piecewise linear\n", + "interpolation in reciprocal lattice units (RLU)." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "q_points = [[0,0,0], [1,0,0], [0,1,0], [1/2,0,0], [0,1,0], [0,0,0]];" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "The function `reciprocal_space_path` will linearly sample a `path`\n", + "between the provided $q$-points with a given `density`. The `xticks` return\n", + "value provides labels for use in plotting." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "density = 50\n", + "path, xticks = reciprocal_space_path(cryst, q_points, density);" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "The `dispersion` function defines the quasiparticle excitation\n", + "energies $ω_i(𝐪)$ for each point $𝐪$ along the reciprocal space path." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "disp = dispersion(swt, path);" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "In addition to the band energies $ω_i(𝐪)$, Sunny can calculate the inelastic\n", + "neutron scattering intensity $I_i(𝐪)$ for each band $i$ according to an\n", + "`intensity_formula`. We choose to apply a polarization correction\n", + "$(1 - 𝐪⊗𝐪)$ by setting the mode argument to `:perp`. Selecting\n", + "`delta_function_kernel` specifies that we want the energy and intensity of\n", + "each band individually." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "formula = intensity_formula(swt, :perp; kernel=delta_function_kernel)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "The function `intensities_bands` uses linear spin wave theory to\n", + "calculate both the dispersion and intensity data for the provided path." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "disp, intensity = intensities_bands(swt, path, formula);" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "These can be plotted in GLMakie." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "fig = Figure()\n", + "ax = Axis(fig[1,1]; xlabel=\"Momentum (r.l.u.)\", ylabel=\"Energy (meV)\", xticks, xticklabelrotation=π/6)\n", + "ylims!(ax, 0.0, 7.5)\n", + "xlims!(ax, 1, size(disp, 1))\n", + "colorrange = extrema(intensity)\n", + "for i in axes(disp)[2]\n", + " lines!(ax, 1:length(disp[:,i]), disp[:,i]; color=intensity[:,i], colorrange)\n", + "end\n", + "fig" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "To make comparisons with inelastic neutron scattering (INS) data, it is\n", + "helpful to employ an empirical broadening kernel, e.g., a\n", + "`lorentzian`." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "γ = 0.15 # width in meV\n", + "broadened_formula = intensity_formula(swt, :perp; kernel=lorentzian(γ))" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "The `intensities_broadened` function requires an energy range in\n", + "addition to the $𝐪$-space path." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "energies = collect(0:0.01:10) # 0 < ω < 10 (meV).\n", + "is1 = intensities_broadened(swt, path, energies, broadened_formula);" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "A real FeI₂ sample will exhibit competing magnetic domains associated with\n", + "spontaneous symmetry breaking of the 6-fold rotational symmetry of the\n", + "triangular lattice. Note that the wavevectors $𝐪$ and $-𝐪$ are equivalent in\n", + "the structure factor, which leaves three distinct domain orientations, which\n", + "are related by 120° rotations about the $ẑ$-axis. Rather than rotating the\n", + "spin configuration directly, on can rotate the $𝐪$-space path. Below, we use\n", + "`rotation_in_rlu` to average the intensities over all three possible\n", + "orientations." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "R = rotation_in_rlu(cryst, [0, 0, 1], 2π/3)\n", + "is2 = intensities_broadened(swt, [R*q for q in path], energies, broadened_formula)\n", + "is3 = intensities_broadened(swt, [R*R*q for q in path], energies, broadened_formula)\n", + "is_averaged = (is1 + is2 + is3) / 3\n", + "\n", + "fig = Figure()\n", + "ax = Axis(fig[1,1]; xlabel=\"Momentum (r.l.u.)\", ylabel=\"Energy (meV)\", xticks, xticklabelrotation=π/6)\n", + "heatmap!(ax, 1:size(is_averaged, 1), energies, is_averaged)\n", + "fig" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "This result can be directly compared to experimental neutron scattering data\n", + "from [Bai et al.](https://doi.org/10.1038/s41567-020-01110-1)\n", + "\n", + "\n", + "\n", + "\n", + "(The publication figure accidentally used a non-standard coordinate system to\n", + "label the wave vectors.)\n", + "\n", + "To get this agreement, the use of SU(3) coherent states is essential. In other\n", + "words, we needed a theory of multi-flavored bosons. The lower band has large\n", + "quadrupolar character, and arises from the strong easy-axis anisotropy of\n", + "FeI₂. By setting `mode = :SUN`, the calculation captures this coupled\n", + "dipole-quadrupole dynamics.\n", + "\n", + "An interesting exercise is to repeat the same study, but using `mode =\n", + ":dipole` instead of `:SUN`. That alternative choice would constrain the\n", + "coherent state dynamics to the space of dipoles only." + ], + "metadata": {} + }, + { + "cell_type": "markdown", + "source": [ + "The full dynamical spin structure factor (DSSF) can be retrieved as a $3×3$\n", + "matrix with the `dssf` function, for a given path of $𝐪$-vectors." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "disp, is = dssf(swt, path);" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "The first output `disp` is identical to that obtained from `dispersion`. The\n", + "second output `is` contains a list of $3×3$ matrix of intensities. For\n", + "example, `is[q,n][2,3]` yields the $(ŷ,ẑ)$ component of the structure factor\n", + "intensity for `nth` mode at the `q`th wavevector in the `path`." + ], + "metadata": {} + }, + { + "cell_type": "markdown", + "source": [ + "## What's next?\n", + "\n", + "The multi-boson linear spin wave theory, applied above, can be understood as\n", + "the quantization of a certain generalization of the Landau-Lifshitz spin\n", + "dynamics. Rather than dipoles, this dynamics takes places on the space of\n", + "[SU(_N_) coherent states](https://arxiv.org/abs/2106.14125).\n", + "\n", + "The full SU(_N_) coherent state dynamics, with appropriate quantum correction\n", + "factors, can be useful to model finite temperature scattering data. In\n", + "particular, it captures certain anharmonic effects due to thermal\n", + "fluctuations. This is the subject of our FeI₂ at Finite Temperature\n", + "tutorial.\n", + "\n", + "The classical dynamics is also a good starting point to study non-equilibrium\n", + "phenomena. Empirical noise and damping terms can be used to model [coupling to\n", + "a thermal bath](https://arxiv.org/abs/2209.01265). This yields a Langevin\n", + "dynamics of SU(_N_) coherent states. Our CP² Skyrmion Quench\n", + "tutorial shows how this dynamics gives rise to the formation of novel\n", + "topological defects in a temperature quench.\n", + "\n", + "Relative to LSWT calculations, it can take much more time to estimate\n", + "$\\mathcal{S}(𝐪,ω)$ intensities using classical dynamics simulation. See the\n", + "[SunnyTutorials\n", + "notebooks](https://nbviewer.org/github/SunnySuite/SunnyTutorials/tree/main/Tutorials/)\n", + "for examples of \"production-scale\" simulations." + ], + "metadata": {} + } + ], + "nbformat_minor": 3, + "metadata": { + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.9.3" + }, + "kernelspec": { + "name": "julia-1.9", + "display_name": "Julia 1.9.3", + "language": "julia" + } + }, + "nbformat": 4 +} diff --git a/previews/PR178/assets/notebooks/ising2d.ipynb b/previews/PR178/assets/notebooks/ising2d.ipynb new file mode 100644 index 000000000..3d9a59ddd --- /dev/null +++ b/previews/PR178/assets/notebooks/ising2d.ipynb @@ -0,0 +1,179 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "source": [ + "# Classical Ising model\n", + "\n", + "This tutorial illustrates simulation of the classical 2D Ising model." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "using Sunny, WGLMakie" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Sunny expects a 3D `Crystal` unit cell. To model a square lattice, we\n", + "create an orthogonal unit cell where the $z$-spacing is distinct from the $x$\n", + "and $y$ spacing." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "a = 1\n", + "latvecs = lattice_vectors(a,a,10a,90,90,90)\n", + "crystal = Crystal(latvecs, [[0,0,0]])" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Create a `System` of spins with linear size `L` in the $x$ and $y$\n", + "directions, and only one layer in the $z$ direction. The option `:dipole`\n", + "means that the system will store Heisenberg spins, as opposed to SU($N$)\n", + "coherent states. Polarize the initial spin configuration using\n", + "`polarize_spins!`. Following the Ising convention, we will restrict\n", + "these spins to the $z$-axis and give them magnitude $S=1$.\n", + "\n", + "By default, Sunny uses physical units, e.g. magnetic field in tesla. Here we\n", + "specify an alternative `Units` system, so that the Zeeman coupling\n", + "between the spin dipole $𝐬$ and an external field $𝐁$ has the dimensionless\n", + "form $-𝐁⋅𝐬$." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "L = 128\n", + "sys = System(crystal, (L,L,1), [SpinInfo(1, S=1, g=1)], :dipole, units=Units.theory, seed=0)\n", + "polarize_spins!(sys, (0,0,1))" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Use `set_exchange!` to include a ferromagnetic Heisenberg interaction\n", + "along nearest-neighbor bonds. The `Bond` below connects two spins\n", + "displaced by one lattice constant in the $x$-direction. This interaction will\n", + "be propagated to all nearest-neighbors bonds in the system, consistent with\n", + "the symmetries of the square lattice." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "set_exchange!(sys, -1.0, Bond(1,1,(1,0,0)))" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "If an external field is desired, it can be set using\n", + "`set_external_field!`." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "B = 0\n", + "set_external_field!(sys, (0,0,B))" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "The critical temperature for the Ising model is known analytically." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "Tc = 2/log(1+√2)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Use a `LocalSampler` to perform `nsweeps` Monte Carlo sweeps. A sweep\n", + "consists of, on average, one trial update per spin in the system. Each\n", + "proposed update is accepted or rejected according to the Metropolis acceptance\n", + "probability. As its name suggests, the `propose_flip` function will\n", + "only propose pure spin flips, $𝐬 \\rightarrow -𝐬$." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "nsweeps = 4000\n", + "sampler = LocalSampler(kT=Tc, propose=propose_flip)\n", + "for i in 1:nsweeps\n", + " step!(sys, sampler)\n", + "end" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Plot the Ising spins by extracting the $z$-component of the dipoles" + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "heatmap(reshape([s.z for s in sys.dipoles], (L,L)))" + ], + "metadata": {}, + "execution_count": null + } + ], + "nbformat_minor": 3, + "metadata": { + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.9.3" + }, + "kernelspec": { + "name": "julia-1.9", + "display_name": "Julia 1.9.3", + "language": "julia" + } + }, + "nbformat": 4 +} diff --git a/previews/PR178/assets/notebooks/out_of_equilibrium.ipynb b/previews/PR178/assets/notebooks/out_of_equilibrium.ipynb new file mode 100644 index 000000000..9574a45a8 --- /dev/null +++ b/previews/PR178/assets/notebooks/out_of_equilibrium.ipynb @@ -0,0 +1,278 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "source": [ + "# CP² Skyrmion Quench\n", + "\n", + "This example demonstrates Sunny's ability to simulate the out-of-equilibrium\n", + "dynamics of generalized spin systems. We will implement the model Hamiltonian\n", + "of [Zhang et al., Nature Communications **14**, 3626\n", + "(2023)](https://www.nature.com/articles/s41467-023-39232-8), which supports a\n", + "novel type of topological defect, a CP² skyrmion, that involves both the\n", + "dipolar and quadrupolar parts of a quantum spin.\n", + "\n", + "Beginning from an initial high-temperature state, a disordered gas of CP²\n", + "skyrmions can be formed by rapidly quenching to low temperature. To model the\n", + "coupled dynamics of dipoles and quadrupoles, Sunny uses a recently developed\n", + "generalization of the Landau-Lifshitz spin dynamics, [Dahlbom et al., Phys.\n", + "Rev. B **106**, 235154 (2022)](https://doi.org/10.1103/PhysRevB.106.235154)." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "using Sunny, WGLMakie" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "The Hamiltonian we will implement,\n", + "$$\n", + "\\mathcal{H} = \\sum_{\\langle i,j \\rangle} J_{ij}( \\hat{S}_i^x \\hat{S}_j^x + \\hat{S}_i^y \\hat{S}_j^y + \\Delta\\hat{S}_i^z \\hat{S}_j^z) - h\\sum_{i}\\hat{S}_i^z + D\\sum_{i}(\\hat{S}_i^z)^2\n", + "$$\n", + "contains competing ferromagnetic nearest-neightbor and antiferromagnetic\n", + "next-nearest-neighbor exchange terms on a triangular lattice. Both exchanges\n", + "exhibit anisotropy on the z-term. Additionally, there is an external magnetic\n", + "field, $h$, and easy-plane single-ion anisotropy, $D > 0$. We begin by\n", + "implementing the `Crystal`." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "lat_vecs = lattice_vectors(1.0, 1.0, 2.0, 90, 90, 120)\n", + "basis_vecs = [[0,0,0]]\n", + "cryst = Crystal(lat_vecs, basis_vecs)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "The crystal is then used to create a spin `System`. All parameters in\n", + "this model system are dimensionless, so we select \"theory\" units and set the\n", + "g-factor to one." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "L = 40\n", + "dims = (L, L, 1)\n", + "sys = System(cryst, dims, [SpinInfo(1, S=1, g=1)], :SUN; seed=101, units=Units.theory)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "We proceed to implement each term of the Hamiltonian, selecting our parameters\n", + "so that the system occupies a region of the phase diagram that supports\n", + "skyrmions. The exchange interactions are set as follows." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "J1 = -1 # Nearest-neighbor ferromagnetic\n", + "J2 = (2.0/(1+√5)) # Tune competing exchange to set skyrmion scale length\n", + "Δ = 2.6 # Exchange anisotropy\n", + "\n", + "ex1 = J1 * [1.0 0.0 0.0;\n", + " 0.0 1.0 0.0;\n", + " 0.0 0.0 Δ]\n", + "ex2 = J2 * [1.0 0.0 0.0;\n", + " 0.0 1.0 0.0;\n", + " 0.0 0.0 Δ]\n", + "set_exchange!(sys, ex1, Bond(1, 1, [1, 0, 0]))\n", + "set_exchange!(sys, ex2, Bond(1, 1, [1, 2, 0]))" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Next we add the external field," + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "h = 15.5\n", + "field = set_external_field!(sys, [0.0 0.0 h])" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "and finally an easy-plane single-ion anisotropy," + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "D = 19.0\n", + "set_onsite_coupling!(sys, S -> D*S[3]^2, 1)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Initialize system to an infinite temperature (fully randomized) initial\n", + "condition." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "randomize_spins!(sys)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "We are now ready to simulate the quenching process using a generalized\n", + "`Langevin` spin dynamics. If we were working with spin dipoles only,\n", + "then `Langevin` dynamics would be the usual Landau-Lifshitz spin dynamics,\n", + "augmented with damping and noise terms. In the present study, we are instead\n", + "working with quantum spin-1 (an ($N=3$)-level system that includes both\n", + "dipoles and quadrupoles). Here, `Langevin` captures the coupled\n", + "dipole-quadrupole dynamics using the formalism of SU($N$) coherent states.\n", + "\n", + "Selecting `kT = 0` in the Langevin dynamics will effective disable the noise\n", + "term. Then the parameter `λ` effectively determines the damping time-scale." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "Δt = 0.2/D # Integration time step (inverse meV). Typically this will be\n", + " # inversely proportional to the largest energy scale in the\n", + " # system. We can use a fairly large time-step here because\n", + " # accuracy isn't critical.\n", + "kT = 0 # Target equilibrium temperature (meV)\n", + "λ = 0.1 # Magnitude of coupling to thermal bath (dimensionless)\n", + "integrator = Langevin(Δt; kT, λ)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Finally we run the dynamics. We will record the state of the system at three\n", + "different times during the quenching process by copying the `coherents` field\n", + "of the `System`." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "τs = [4., 16, 256] # Times to record snapshots\n", + "frames = [] # Empty array to store snapshots\n", + "for i in eachindex(τs)\n", + " dur = i == 1 ? τs[1] : τs[i] - τs[i-1] # Determine the length of time to simulate\n", + " numsteps = round(Int, dur/Δt)\n", + " for _ in 1:numsteps # Perform the integration\n", + " step!(sys, integrator)\n", + " end\n", + " push!(frames, copy(sys.coherents)) # Save a snapshot spin configuration\n", + "end" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "To visualize the state of the system contained in each snapshot, we will\n", + "calculate and plot the skyrmion density on each plaquette of our lattice. The\n", + "function `plot_triangular_plaquettes` is not part of the core Sunny package,\n", + "but rather something you could define yourself. We are using the definition in\n", + "`plotting2d.jl` from the Sunny [`examples/extra`\n", + "directory](https://github.com/SunnySuite/Sunny.jl/tree/main/examples/extra)." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "include(pkgdir(Sunny, \"examples\", \"extra\", \"plotting2d.jl\"))\n", + "\n", + "function sun_berry_curvature(z₁, z₂, z₃)\n", + " z₁, z₂, z₃ = normalize.((z₁, z₂, z₃))\n", + " n₁ = z₁ ⋅ z₂\n", + " n₂ = z₂ ⋅ z₃\n", + " n₃ = z₃ ⋅ z₁\n", + " return angle(n₁ * n₂ * n₃)\n", + "end\n", + "\n", + "plot_triangular_plaquettes(sun_berry_curvature, frames; resolution=(1800,600),\n", + " offset_spacing=10, texts = [\"\\tt = \"*string(τ) for τ in τs], text_offset = (0.0, 6.0)\n", + ")" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "The times are given in $\\hbar/|J_1|$. The white\n", + "background corresponds to a quantum paramagnetic state, where the local spin\n", + "exhibits a strong quadrupole moment and little or no dipole moment. Observe\n", + "that the process has generated a number of well-formed skyrmions of both\n", + "positive (red) and negative (blue) charge in addition to a number of other\n", + "metastable spin configurations. A full-sized version of this figure is\n", + "available in [Dahlbom et al.](https://doi.org/10.1103/PhysRevB.106.235154)." + ], + "metadata": {} + } + ], + "nbformat_minor": 3, + "metadata": { + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.9.3" + }, + "kernelspec": { + "name": "julia-1.9", + "display_name": "Julia 1.9.3", + "language": "julia" + } + }, + "nbformat": 4 +} diff --git a/previews/PR178/assets/notebooks/powder_averaging.ipynb b/previews/PR178/assets/notebooks/powder_averaging.ipynb new file mode 100644 index 000000000..0f84c42b8 --- /dev/null +++ b/previews/PR178/assets/notebooks/powder_averaging.ipynb @@ -0,0 +1,264 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "source": [ + "# Powder Averaged CoRh₂O₄\n", + "\n", + "This tutorial illustrates the calculation of the powder-averaged structure\n", + "factor by performing an orientational average. We consider a simple model of\n", + "the diamond-cubic crystal CoRh₂O₄, with parameters extracted from [Ge et al.,\n", + "Phys. Rev. B 96, 064413](https://doi.org/10.1103/PhysRevB.96.064413)." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "using Sunny, WGLMakie" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Construct a diamond `Crystal` in the conventional (non-primitive)\n", + "cubic unit cell. Sunny will populate all eight symmetry-equivalent sites when\n", + "given the international spacegroup number 227 (\"Fd-3m\") and the appropriate\n", + "setting. For this spacegroup, there are two conventional translations of the\n", + "unit cell, and it is necessary to disambiguate through the `setting` keyword\n", + "argument. (On your own: what happens if `setting` is omitted?)" + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "a = 8.5031 # (Å)\n", + "latvecs = lattice_vectors(a, a, a, 90, 90, 90)\n", + "cryst = Crystal(latvecs, [[0,0,0]], 227, setting=\"1\")" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "In a running Julia environment, the crystal can be viewed interactively using\n", + "`view_crystal`." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "view_crystal(cryst, 8.0)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Construct a `System` with an antiferromagnetic nearest neighbor\n", + "interaction `J`. Because the diamond crystal is bipartite, the ground state\n", + "will have unfrustrated Néel order. Selecting `latsize=(1,1,1)` is sufficient\n", + "because the ground state is periodic over each cubic unit cell. By passing an\n", + "explicit `seed`, the system's random number generator will give repeatable\n", + "results." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "latsize = (2, 2, 2)\n", + "seed = 0\n", + "S = 3/2\n", + "J = 7.5413*meV_per_K # (~ 0.65 meV)\n", + "sys = System(cryst, latsize, [SpinInfo(1; S, g=2)], :dipole; seed=0)\n", + "set_exchange!(sys, J, Bond(1, 3, [0,0,0]))" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "In the ground state, each spin is exactly anti-aligned with its 4\n", + "nearest-neighbors. Because every bond contributes an energy of $-JS^2$, the\n", + "energy per site is $-2JS^2$. In this calculation, a factor of 1/2 avoids\n", + "double-counting the bonds. Due to lack of frustration, direct energy\n", + "minimization is successful in finding the ground state." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "randomize_spins!(sys)\n", + "minimize_energy!(sys)\n", + "\n", + "@assert energy_per_site(sys) ≈ -2J*S^2" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Plotting the spins confirms the expected Néel order. Note that the overall,\n", + "global rotation of dipoles is arbitrary." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "s0 = sys.dipoles[1,1,1,1]\n", + "plot_spins(sys; color=[s'*s0 for s in sys.dipoles])" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "For numerical efficiency, it will be helpful to work with the smallest\n", + "possible magnetic supercell. Here, it is the primitive unit cell, which\n", + "contains just two sites. The variable `shape` below defines the primitive\n", + "lattice vectors `cryst.prim_latvecs` in units of the conventional lattice\n", + "vectors. This result is used as input to `reshape_supercell`. The\n", + "energy per site remains the same, which verifies that the magnetic supercell\n", + "is valid." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "shape = cryst.latvecs \\ cryst.prim_latvecs\n", + "sys_prim = reshape_supercell(sys, shape)\n", + "@assert energy_per_site(sys_prim) ≈ -2J*S^2" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Now estimate $𝒮(𝐪,ω)$ with `SpinWaveTheory` and an\n", + "`intensity_formula`. The mode `:perp` contracts with a dipole factor\n", + "to return the unpolarized intensity. The formula also employs\n", + "`lorentzian` broadening. The isotropic `FormFactor` for\n", + "Cobalt(2+) dampens intensities at large $𝐪$." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "swt = SpinWaveTheory(sys_prim)\n", + "η = 0.4 # (meV)\n", + "kernel = lorentzian(η)\n", + "formfactors = [FormFactor(\"Co2\")]\n", + "formula = intensity_formula(swt, :perp; kernel, formfactors)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "For the \"single crystal\" result, we may use `reciprocal_space_path` to\n", + "construct a path that connects high-symmetry points in reciprocal space. The\n", + "`intensities_broadened` function collects intensities along this path\n", + "for the given set of energy values." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "qpoints = [[0.0, 0.0, 0.0], [0.5, 0.0, 0.0], [0.5, 0.5, 0.0], [0.0, 0.0, 0.0]]\n", + "path, xticks = reciprocal_space_path(cryst, qpoints, 50)\n", + "energies = collect(0:0.01:6)\n", + "is = intensities_broadened(swt, path, energies, formula)\n", + "\n", + "fig = Figure()\n", + "ax = Axis(fig[1,1]; aspect=1.4, ylabel=\"ω (meV)\", xlabel=\"𝐪 (r.l.u.)\",\n", + " xticks, xticklabelrotation=π/10)\n", + "heatmap!(ax, 1:size(is, 1), energies, is, colormap=:gnuplot2)\n", + "fig" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "A powder measurement effectively involves an average over all possible crystal\n", + "orientations. We use the function `reciprocal_space_shell` to sample\n", + "`n` wavevectors on a sphere of a given radius (inverse angstroms), and then\n", + "calculate the spherically-averaged intensity." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "radii = 0.01:0.02:3 # (1/Å)\n", + "output = zeros(Float64, length(radii), length(energies))\n", + "for (i, radius) in enumerate(radii)\n", + " n = 300\n", + " qs = reciprocal_space_shell(cryst, radius, n)\n", + " is = intensities_broadened(swt, qs, energies, formula)\n", + " output[i, :] = sum(is, dims=1) / size(is, 1)\n", + "end\n", + "\n", + "fig = Figure()\n", + "ax = Axis(fig[1,1]; xlabel=\"Q (Å⁻¹)\", ylabel=\"ω (meV)\")\n", + "heatmap!(ax, radii, energies, output, colormap=:gnuplot2)\n", + "fig" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "This result can be compared to experimental neutron scattering data\n", + "from Fig. 5 of [Ge et al.](https://doi.org/10.1103/PhysRevB.96.064413)\n", + "\n", + "" + ], + "metadata": {} + } + ], + "nbformat_minor": 3, + "metadata": { + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.9.3" + }, + "kernelspec": { + "name": "julia-1.9", + "display_name": "Julia 1.9.3", + "language": "julia" + } + }, + "nbformat": 4 +} diff --git a/previews/PR178/assets/paraviewcolormap.png b/previews/PR178/assets/paraviewcolormap.png new file mode 100644 index 0000000000000000000000000000000000000000..1d1073abe36ddf9a3a8ebfdf6c61f1277291bf22 GIT binary patch literal 27585 zcmbUJWmH^k&@GBWaJOI~xCeLF;0_6Hfgr)%f(CcD;K3b&I|O%kcXy|O?z4EmbI#rS zj(z{!jKK)Ky4UJQW>w9qXBJ^9O44Y^ACRG-pwMJxB-Egwpk1J#-bo_91Flq{T~-1A zK|85Qi$hh8e>?(Sz?zFGia|mBjX`-af&*S7*~@4VL0M_YN{D@R zH#loY`u2Vx39{jID0K)Q$QKkF*tI%HfZ&CE8T60Hm{!TUPun~yb8*-|N+oejSR@id-`BegKBtz})(SyrF-)q(oDMQXW4vM(ME_K{V-$Zf 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z3UPLJ$cDIbb#Xb#NKRf*@`>6=b*o8PY-B#8VEoeDdoxqr-gB5$U$He^ac&*DyJyH# zevULN7k5N+t6cOZhR2nN34j=Y_L%z4cm8B+%cfh1;4PBOG!~(1X7=J16K|U~%qF(s zYKWGg10R*=^72C!7FJdY^77d^Id3~j<*MPJBGlaXjJo}SI+gLog62;*jFG$&A3J7d zU%*XmgL`z4xwQmP3G0U7cMH}?x9HVFp(UpSnUj~jt_XfQqYZPCV2&Nt*{ zhxZxK7!I5@$So($D`3Jo*b(51p%x3xCNYPSZf(k})u#V1wNg=0=`aJxV05CzaS+P$SoiJQhv`pVS{~|l z{=uhsG_(U@_B_0(5WFKo)<~PDPo6R_@oB7MvM*>p>fY+70Gd0odzcy&io9n;?zK5l zjA=}?x65c$`rh(1HUJTELfXziij}S|d*y16aii1KBI=r03H<^8wuI?c{oN(0V5q8| zCuU<+$2S2w=jB^unsU6MZ~Qaeh^o=YK4Em3+l3pFMB_e`Y6P+@xt-HB02ppNG^J4W z2XlbUJEt^GR--wvuXHIhZyK3Z*hdzhSFKeYRNa(O@zOmF7}x=Zj#TdU4p18JuznbW zc)ObIe(`p -D*S[3]^2, 1)#hide +sys + +Δt = 0.05/D # Should be inversely proportional to the largest energy scale + # in the system. For FeI2, this is the easy-axis anisotropy, + # `D = 2.165` (meV). The prefactor 0.05 is relatively small, + # and achieves high accuracy. +kT = 0.2 # Temperature of the thermal bath (meV). +λ = 0.1 # This value is typically good for Monte Carlo sampling, + # independent of system details. + +langevin = Langevin(Δt; kT, λ); + +randomize_spins!(sys) +for _ in 1:20_000 + step!(sys, langevin) +end + +plot_spins(sys; color=[s[3] for s in sys.dipoles]) + +sys_large = resize_supercell(sys, (16,16,4)) # 16x16x4 copies of the original unit cell +plot_spins(sys_large; color=[s[3] for s in sys_large.dipoles]) + +kT = 3.5 * meV_per_K # 3.5K ≈ 0.30 meV +langevin.kT = kT +for _ in 1:10_000 + step!(sys_large, langevin) +end + +sc = dynamical_correlations(sys_large; Δt=2Δt, nω=120, ωmax=7.5) + +add_sample!(sc, sys_large) # Accumulate the sample into `sc` + +for _ in 1:2 + for _ in 1:1000 # Enough steps to decorrelate spins + step!(sys_large, langevin) + end + add_sample!(sc, sys_large) # Accumulate the sample into `sc` +end + +sc + +formula = intensity_formula(sc, :trace; kT) + +qs = [[0, 0, 0], [0.5, 0.5, 0.5]] +is = intensities_interpolated(sc, qs, formula; interpolation = :round) + +ωs = available_energies(sc) +fig = lines(ωs, is[1,:]; axis=(xlabel="meV", ylabel="Intensity"), label="(0,0,0)") +lines!(ωs, is[2,:]; label="(π,π,π)") +axislegend() +fig + +formfactors = [FormFactor("Fe2"; g_lande=3/2)] +new_formula = intensity_formula(sc, :perp; kT, formfactors = formfactors) + +points = [[0, 0, 0], # List of wave vectors that define a path + [1, 0, 0], + [0, 1, 0], + [1/2, 0, 0], + [0, 1, 0], + [0, 0, 0]] +density = 40 +path, xticks = reciprocal_space_path(cryst, points, density); + +is_interpolated = intensities_interpolated(sc, path, new_formula; + interpolation = :linear, # Interpolate between available wave vectors +); +# Add artificial broadening +is_interpolated_broadened = broaden_energy(sc, is, (ω, ω₀)->lorentzian(ω-ω₀, 0.05)); + +cut_width = 0.3 +density = 15 +paramsList, markers, ranges = reciprocal_space_path_bins(sc,points,density,cut_width); + +total_bins = ranges[end][end] +energy_bins = paramsList[1].numbins[4] +is_binned = zeros(Float64,total_bins,energy_bins) +integrated_kernel = integrated_lorentzian(0.05) # Lorentzian broadening +for k in eachindex(paramsList) + bin_data, counts = intensities_binned(sc,paramsList[k], new_formula; + integrated_kernel = integrated_kernel + ) + is_binned[ranges[k],:] = bin_data[:,1,1,:] ./ counts[:,1,1,:] +end + +fig = Figure() +ax_top = Axis(fig[1,1],ylabel = "meV",xticklabelrotation=π/8,xticklabelsize=12;xticks) +ax_bottom = Axis(fig[2,1],ylabel = "meV",xticks = (markers, string.(points)),xticklabelrotation=π/8,xticklabelsize=12) + +heatmap!(ax_top,1:size(is_interpolated,1), ωs, is_interpolated; + colorrange=(0.0,0.07), +) + +heatmap!(ax_bottom,1:size(is_binned,1), ωs, is_binned; + colorrange=(0.0,0.05), +) + +fig + +ωidx = 60 +target_ω = ωs[ωidx] + +params = unit_resolution_binning_parameters(sc) +params.binstart[1:2] .= -1 # Expand plot range slightly + +# Set energy integration range +omega_width = 0.3 +params.binstart[4] = target_ω - (omega_width/2) +params.binend[4] = target_ω # `binend` should be inside (e.g. at the center) of the range +params.binwidth[4] = omega_width + +integrate_axes!(params, axes = 3) # Integrate out z direction entirely + +params + +is, counts = intensities_binned(sc,params,new_formula) + +fig = Figure() +ax = Axis(fig[1,1]; + title="Δω=0.3 meV (Binned)", aspect=true, + xlabel = "[H, 0, 0]", + ylabel = "[0, K, 0]" +) +bcs = axes_bincenters(params) +hm = heatmap!(ax,bcs[1],bcs[2],is[:,:,1,1] ./ counts[:,:,1,1]) +function add_lines!(ax,params)#hide + bes = Sunny.axes_binedges(params)#hide + hrange = range(-2,2,length=17)#hide + linesegments!(ax,[(Point2f(params.covectors[1,1:3] ⋅ [h,-10,0],params.covectors[2,1:3] ⋅ [h,-10,0]),Point2f(params.covectors[1,1:3] ⋅ [h,10,0],params.covectors[2,1:3] ⋅ [h,10,0])) for h = hrange],linestyle=:dash,color=:black)#hide + krange = range(-2,2,length=17)#hide + linesegments!(ax,[(Point2f(params.covectors[1,1:3] ⋅ [-10,k,0],params.covectors[2,1:3] ⋅ [-10,k,0]),Point2f(params.covectors[1,1:3] ⋅ [10,k,0],params.covectors[2,1:3] ⋅ [10,k,0])) for k = krange],linestyle=:dash,color=:black)#hide + xlims!(ax,bes[1][1],bes[1][end])#hide + ylims!(ax,bes[2][1],bes[2][end])#hide +end#hide +add_lines!(ax,params) +Colorbar(fig[1,2], hm); +fig + +latvecs = sys.crystal.latvecs +metric = latvecs' * I(3) * latvecs + +(latvecs * [1/2,1,0]) ⋅ (latvecs * [1,0,0]) == 0 + +f = Figure()#hide +ax = Axis(f[1,1])#hide +arrows!(ax,[Point2f(0,0),Point2f(latvecs[1:2,1] ./ 2)],[Vec2f(latvecs[1:2,1] ./ 2), Vec2f(latvecs[1:2,2])],arrowcolor = :blue,arrowsize = 30.,linewidth = 5.,linecolor = :blue)#hide +arrows!(ax,[Point2f(0,0)],[Vec2f(latvecs[1:2,:] * [1/2,1,0])],arrowcolor = :red,arrowsize = 30.,linewidth = 5.,linecolor = :red, linestyle = :dash)#hide +scatter!(ax,[Point2f(latvecs[1:2,:] * [a,b,0]) for a in -1:1, b in -1:1][:],color = :black)#hide +annotations!(ax,["0","0+b","0+a", "a/2", "b"],[Point2f(0,-0.3),Point2f(latvecs[1:2,2]) .- Vec2f(0,0.3),Point2f(latvecs[1:2,1]) .- Vec2f(0,0.3),Point2f(latvecs[1:2,1] ./ 4) .- Vec2f(0,0.3),Point2f(latvecs[1:2,1] ./ 2) .+ Vec2f(latvecs[1:2,2] ./ 2) .+ Vec2f(0.3,0.3)],color=[:black,:black,:black,:blue,:blue])#hide +f#hide + +params.covectors[2,1:3] = [1/2,1,0] # [1/2,1,0] times [a;b;c] is (a/2 + b) +params#hide + +# Zoom out horizontal axis +params.binstart[1], params.binend[1] = -2, 2 + +# Adjust vertical axis bounds to account for +# length of a/2 + b +params.binstart[2], params.binend[2] = -2 * sqrt(3/4), 2 * sqrt(3/4) + +# Re-compute in the new coordinate system +is, counts = intensities_binned(sc,params,new_formula) + +fig = Figure(; resolution=(1200,500))#hide +ax_right = Axis(fig[1,3];#hide + title="ω≈$(round(target_ω, digits=2)) meV with Δω=0.3 meV (Binned)", aspect=true,#hide + xlabel = "[H, -1/2H, 0]"#hide +)#hide +bcs = axes_bincenters(params)#hide +hm_right = heatmap!(ax_right,bcs[1],bcs[2],is[:,:,1,1] ./ counts[:,:,1,1])#hide +add_lines!(ax_right,params) +Colorbar(fig[1,4], hm_right);#hide + +# New basis matrix +A = [1 0 0 + -1/2 1 0 + 0 0 1] + +# Define our grid of wave vectors +npoints = 60 +as = range(-2, 2, npoints) +bs = range(-3/√3, 3/√3, npoints) +qs_ortho = [[a, b, 0] for a in as, b in bs] + +# Convert to original RLU system for input to Sunny +qs = [A * q for q in qs_ortho] + +# Use interpolation to get intensities +is = intensities_interpolated(sc, qs, new_formula; interpolation=:linear) + +ax_left = Axis(fig[1,2];#hide + title="ω≈$(round(ωs[ωidx], digits=2)) meV (Interpolated)", aspect=true,#hide + xlabel = "[H, -1/2H, 0]", ylabel = "[0, K, 0]"#hide +)#hide +hm_left = heatmap!(ax_left, as, bs, is[:,:,ωidx])#hide +add_lines!(ax_left,params) +Colorbar(fig[1,1], hm_left);#hide +fig + +metric = (latvecs * inv(A'))' * I(3) * (latvecs * inv(A')) + +is_static = instant_intensities_interpolated(sc, qs, new_formula; interpolation = :linear) + +hm = heatmap(as, bs, is_static; + axis=( + title="Instantaneous Structure Factor", + xlabel = "[H, -1/2H, 0]", + ylabel = "[0, K, 0]", + aspect=true + ) +) +Colorbar(hm.figure[1,2], hm.plot) +hm diff --git a/previews/PR178/assets/scripts/fei2_tutorial.jl b/previews/PR178/assets/scripts/fei2_tutorial.jl new file mode 100644 index 000000000..2ac71832d --- /dev/null +++ b/previews/PR178/assets/scripts/fei2_tutorial.jl @@ -0,0 +1,117 @@ +using Sunny, GLMakie + +a = b = 4.05012 # Lattice constants for triangular lattice +c = 6.75214 # Spacing in the z-direction + +latvecs = lattice_vectors(a, b, c, 90, 90, 120) # A 3x3 matrix of lattice vectors that + # define the conventional unit cell +positions = [[0, 0, 0], [1/3, 2/3, 1/4], [2/3, 1/3, 3/4]] # Positions of atoms in fractions + # of lattice vectors +types = ["Fe", "I", "I"] +FeI2 = Crystal(latvecs, positions; types) + +cryst = subcrystal(FeI2, "Fe") + +view_crystal(cryst, 8.0) + +print_symmetry_table(cryst, 8.0) + +sys = System(cryst, (4, 4, 4), [SpinInfo(1, S=1, g=2)], :SUN, seed=2) + +J1pm = -0.236 +J1pmpm = -0.161 +J1zpm = -0.261 +J2pm = 0.026 +J3pm = 0.166 +J′0pm = 0.037 +J′1pm = 0.013 +J′2apm = 0.068 + +J1zz = -0.236 +J2zz = 0.113 +J3zz = 0.211 +J′0zz = -0.036 +J′1zz = 0.051 +J′2azz = 0.073 + +J1xx = J1pm + J1pmpm +J1yy = J1pm - J1pmpm +J1yz = J1zpm + +set_exchange!(sys, [J1xx 0.0 0.0; + 0.0 J1yy J1yz; + 0.0 J1yz J1zz], Bond(1,1,[1,0,0])) +set_exchange!(sys, [J2pm 0.0 0.0; + 0.0 J2pm 0.0; + 0.0 0.0 J2zz], Bond(1,1,[1,2,0])) +set_exchange!(sys, [J3pm 0.0 0.0; + 0.0 J3pm 0.0; + 0.0 0.0 J3zz], Bond(1,1,[2,0,0])) +set_exchange!(sys, [J′0pm 0.0 0.0; + 0.0 J′0pm 0.0; + 0.0 0.0 J′0zz], Bond(1,1,[0,0,1])) +set_exchange!(sys, [J′1pm 0.0 0.0; + 0.0 J′1pm 0.0; + 0.0 0.0 J′1zz], Bond(1,1,[1,0,1])) +set_exchange!(sys, [J′2apm 0.0 0.0; + 0.0 J′2apm 0.0; + 0.0 0.0 J′2azz], Bond(1,1,[1,2,1])) + +D = 2.165 +set_onsite_coupling!(sys, S -> -D*S[3]^2, 1) + +randomize_spins!(sys) +minimize_energy!(sys) + +plot_spins(sys; color=[s[3] for s in sys.dipoles]) + +print_wrapped_intensities(sys) + +suggest_magnetic_supercell([[0, -1/4, 1/4]]) + +sys_min = reshape_supercell(sys, [1 0 0; 0 2 1; 0 -2 1]) +randomize_spins!(sys_min) +minimize_energy!(sys_min); + +plot_spins(sys_min; color=[s[3] for s in sys_min.dipoles], ghost_radius=12) + +swt = SpinWaveTheory(sys_min) + +q_points = [[0,0,0], [1,0,0], [0,1,0], [1/2,0,0], [0,1,0], [0,0,0]]; + +density = 50 +path, xticks = reciprocal_space_path(cryst, q_points, density); + +disp = dispersion(swt, path); + +formula = intensity_formula(swt, :perp; kernel=delta_function_kernel) + +disp, intensity = intensities_bands(swt, path, formula); + +fig = Figure() +ax = Axis(fig[1,1]; xlabel="Momentum (r.l.u.)", ylabel="Energy (meV)", xticks, xticklabelrotation=π/6) +ylims!(ax, 0.0, 7.5) +xlims!(ax, 1, size(disp, 1)) +colorrange = extrema(intensity) +for i in axes(disp)[2] + lines!(ax, 1:length(disp[:,i]), disp[:,i]; color=intensity[:,i], colorrange) +end +fig + +γ = 0.15 # width in meV +broadened_formula = intensity_formula(swt, :perp; kernel=lorentzian(γ)) + +energies = collect(0:0.01:10) # 0 < ω < 10 (meV). +is1 = intensities_broadened(swt, path, energies, broadened_formula); + +R = rotation_in_rlu(cryst, [0, 0, 1], 2π/3) +is2 = intensities_broadened(swt, [R*q for q in path], energies, broadened_formula) +is3 = intensities_broadened(swt, [R*R*q for q in path], energies, broadened_formula) +is_averaged = (is1 + is2 + is3) / 3 + +fig = Figure() +ax = Axis(fig[1,1]; xlabel="Momentum (r.l.u.)", ylabel="Energy (meV)", xticks, xticklabelrotation=π/6) +heatmap!(ax, 1:size(is_averaged, 1), energies, is_averaged) +fig + +disp, is = dssf(swt, path); diff --git a/previews/PR178/assets/scripts/ising2d.jl b/previews/PR178/assets/scripts/ising2d.jl new file mode 100644 index 000000000..6ab158eb6 --- /dev/null +++ b/previews/PR178/assets/scripts/ising2d.jl @@ -0,0 +1,24 @@ +using Sunny, GLMakie + +a = 1 +latvecs = lattice_vectors(a,a,10a,90,90,90) +crystal = Crystal(latvecs, [[0,0,0]]) + +L = 128 +sys = System(crystal, (L,L,1), [SpinInfo(1, S=1, g=1)], :dipole, units=Units.theory, seed=0) +polarize_spins!(sys, (0,0,1)) + +set_exchange!(sys, -1.0, Bond(1,1,(1,0,0))) + +B = 0 +set_external_field!(sys, (0,0,B)) + +Tc = 2/log(1+√2) + +nsweeps = 4000 +sampler = LocalSampler(kT=Tc, propose=propose_flip) +for i in 1:nsweeps + step!(sys, sampler) +end + +heatmap(reshape([s.z for s in sys.dipoles], (L,L))) diff --git a/previews/PR178/assets/scripts/out_of_equilibrium.jl b/previews/PR178/assets/scripts/out_of_equilibrium.jl new file mode 100644 index 000000000..a20a0c078 --- /dev/null +++ b/previews/PR178/assets/scripts/out_of_equilibrium.jl @@ -0,0 +1,63 @@ +using Sunny, GLMakie + +lat_vecs = lattice_vectors(1.0, 1.0, 2.0, 90, 90, 120) +basis_vecs = [[0,0,0]] +cryst = Crystal(lat_vecs, basis_vecs) + +L = 40 +dims = (L, L, 1) +sys = System(cryst, dims, [SpinInfo(1, S=1, g=1)], :SUN; seed=101, units=Units.theory) + +J1 = -1 # Nearest-neighbor ferromagnetic +J2 = (2.0/(1+√5)) # Tune competing exchange to set skyrmion scale length +Δ = 2.6 # Exchange anisotropy + +ex1 = J1 * [1.0 0.0 0.0; + 0.0 1.0 0.0; + 0.0 0.0 Δ] +ex2 = J2 * [1.0 0.0 0.0; + 0.0 1.0 0.0; + 0.0 0.0 Δ] +set_exchange!(sys, ex1, Bond(1, 1, [1, 0, 0])) +set_exchange!(sys, ex2, Bond(1, 1, [1, 2, 0])) + +h = 15.5 +field = set_external_field!(sys, [0.0 0.0 h]) + +D = 19.0 +set_onsite_coupling!(sys, S -> D*S[3]^2, 1) + +randomize_spins!(sys) + +Δt = 0.2/D # Integration time step (inverse meV). Typically this will be + # inversely proportional to the largest energy scale in the + # system. We can use a fairly large time-step here because + # accuracy isn't critical. +kT = 0 # Target equilibrium temperature (meV) +λ = 0.1 # Magnitude of coupling to thermal bath (dimensionless) +integrator = Langevin(Δt; kT, λ) + +τs = [4., 16, 256] # Times to record snapshots +frames = [] # Empty array to store snapshots +for i in eachindex(τs) + dur = i == 1 ? τs[1] : τs[i] - τs[i-1] # Determine the length of time to simulate + numsteps = round(Int, dur/Δt) + for _ in 1:numsteps # Perform the integration + step!(sys, integrator) + end + push!(frames, copy(sys.coherents)) # Save a snapshot spin configuration +end + +include(pkgdir(Sunny, "examples", "extra", "plotting2d.jl")) + +function sun_berry_curvature(z₁, z₂, z₃) + z₁, z₂, z₃ = normalize.((z₁, z₂, z₃)) + n₁ = z₁ ⋅ z₂ + n₂ = z₂ ⋅ z₃ + n₃ = z₃ ⋅ z₁ + return angle(n₁ * n₂ * n₃) +end + +plot_triangular_plaquettes(sun_berry_curvature, frames; resolution=(1800,600), + offset_spacing=10, texts = ["\tt = "*string(τ) for τ in τs], text_offset = (0.0, 6.0) +) diff --git a/previews/PR178/assets/scripts/powder_averaging.jl b/previews/PR178/assets/scripts/powder_averaging.jl new file mode 100644 index 000000000..5e18d9af5 --- /dev/null +++ b/previews/PR178/assets/scripts/powder_averaging.jl @@ -0,0 +1,57 @@ +using Sunny, GLMakie + +a = 8.5031 # (Å) +latvecs = lattice_vectors(a, a, a, 90, 90, 90) +cryst = Crystal(latvecs, [[0,0,0]], 227, setting="1") + +view_crystal(cryst, 8.0) + +latsize = (2, 2, 2) +seed = 0 +S = 3/2 +J = 7.5413*meV_per_K # (~ 0.65 meV) +sys = System(cryst, latsize, [SpinInfo(1; S, g=2)], :dipole; seed=0) +set_exchange!(sys, J, Bond(1, 3, [0,0,0])) + +randomize_spins!(sys) +minimize_energy!(sys) + +@assert energy_per_site(sys) ≈ -2J*S^2 + +s0 = sys.dipoles[1,1,1,1] +plot_spins(sys; color=[s'*s0 for s in sys.dipoles]) + +shape = cryst.latvecs \ cryst.prim_latvecs +sys_prim = reshape_supercell(sys, shape) +@assert energy_per_site(sys_prim) ≈ -2J*S^2 + +swt = SpinWaveTheory(sys_prim) +η = 0.4 # (meV) +kernel = lorentzian(η) +formfactors = [FormFactor("Co2")] +formula = intensity_formula(swt, :perp; kernel, formfactors) + +qpoints = [[0.0, 0.0, 0.0], [0.5, 0.0, 0.0], [0.5, 0.5, 0.0], [0.0, 0.0, 0.0]] +path, xticks = reciprocal_space_path(cryst, qpoints, 50) +energies = collect(0:0.01:6) +is = intensities_broadened(swt, path, energies, formula) + +fig = Figure() +ax = Axis(fig[1,1]; aspect=1.4, ylabel="ω (meV)", xlabel="𝐪 (r.l.u.)", + xticks, xticklabelrotation=π/10) +heatmap!(ax, 1:size(is, 1), energies, is, colormap=:gnuplot2) +fig + +radii = 0.01:0.02:3 # (1/Å) +output = zeros(Float64, length(radii), length(energies)) +for (i, radius) in enumerate(radii) + n = 300 + qs = reciprocal_space_shell(cryst, radius, n) + is = intensities_broadened(swt, qs, energies, formula) + output[i, :] = sum(is, dims=1) / size(is, 1) +end + +fig = Figure() +ax = Axis(fig[1,1]; xlabel="Q (Å⁻¹)", ylabel="ω (meV)") +heatmap!(ax, radii, energies, output, colormap=:gnuplot2) +fig diff --git 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.button.is-black.is-outlined:focus,html.theme--documenter-dark .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--documenter-dark .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-black.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:hover,html.theme--documenter-dark .button.is-light.is-hovered{background-color:#e5eaec;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus,html.theme--documenter-dark .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus:not(:active),html.theme--documenter-dark .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(236,240,241,0.25)}html.theme--documenter-dark .button.is-light:active,html.theme--documenter-dark .button.is-light.is-active{background-color:#dde4e6;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:#ecf0f1;box-shadow:none}html.theme--documenter-dark .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined{background-color:transparent;border-color:#ecf0f1;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-outlined.is-focused{background-color:#ecf0f1;border-color:#ecf0f1;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-outlined{background-color:transparent;border-color:#ecf0f1;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-dark,html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:hover,html.theme--documenter-dark .content kbd.button:hover,html.theme--documenter-dark .button.is-dark.is-hovered,html.theme--documenter-dark .content kbd.button.is-hovered{background-color:#232829;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus,html.theme--documenter-dark .content kbd.button:focus,html.theme--documenter-dark .button.is-dark.is-focused,html.theme--documenter-dark .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus:not(:active),html.theme--documenter-dark .content kbd.button:focus:not(:active),html.theme--documenter-dark .button.is-dark.is-focused:not(:active),html.theme--documenter-dark .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(40,47,47,0.25)}html.theme--documenter-dark .button.is-dark:active,html.theme--documenter-dark .content kbd.button:active,html.theme--documenter-dark .button.is-dark.is-active,html.theme--documenter-dark .content kbd.button.is-active{background-color:#1d2122;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark[disabled],html.theme--documenter-dark .content kbd.button[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark,fieldset[disabled] html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:#282f2f;box-shadow:none}html.theme--documenter-dark .button.is-dark.is-inverted,html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted:hover,html.theme--documenter-dark .content kbd.button.is-inverted:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-dark.is-inverted[disabled],html.theme--documenter-dark .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-loading::after,html.theme--documenter-dark .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined,html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-outlined.is-focused{background-color:#282f2f;border-color:#282f2f;color:#fff}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-primary,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink{background-color:#375a7f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:hover,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:hover,html.theme--documenter-dark .button.is-primary.is-hovered,html.theme--documenter-dark .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#335476;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:focus,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:focus,html.theme--documenter-dark .button.is-primary.is-focused,html.theme--documenter-dark .docstring>section>a.button.is-focused.docs-sourcelink{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:focus:not(:active),html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:focus:not(:active),html.theme--documenter-dark .button.is-primary.is-focused:not(:active),html.theme--documenter-dark .docstring>section>a.button.is-focused.docs-sourcelink:not(:active){box-shadow:0 0 0 0.125em rgba(55,90,127,0.25)}html.theme--documenter-dark .button.is-primary:active,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:active,html.theme--documenter-dark .button.is-primary.is-active,html.theme--documenter-dark .docstring>section>a.button.is-active.docs-sourcelink{background-color:#2f4d6d;border-color:transparent;color:#fff}html.theme--documenter-dark 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.docstring>section>a.button.is-inverted.docs-sourcelink[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-primary.is-inverted,fieldset[disabled] html.theme--documenter-dark .docstring>section>a.button.is-inverted.docs-sourcelink{background-color:#fff;border-color:transparent;box-shadow:none;color:#375a7f}html.theme--documenter-dark .button.is-primary.is-loading::after,html.theme--documenter-dark .docstring>section>a.button.is-loading.docs-sourcelink::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-primary.is-outlined,html.theme--documenter-dark .docstring>section>a.button.is-outlined.docs-sourcelink{background-color:transparent;border-color:#375a7f;color:#375a7f}html.theme--documenter-dark .button.is-primary.is-outlined:hover,html.theme--documenter-dark .docstring>section>a.button.is-outlined.docs-sourcelink:hover,html.theme--documenter-dark .button.is-primary.is-outlined.is-hovered,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-hovered.docs-sourcelink,html.theme--documenter-dark .button.is-primary.is-outlined:focus,html.theme--documenter-dark .docstring>section>a.button.is-outlined.docs-sourcelink:focus,html.theme--documenter-dark .button.is-primary.is-outlined.is-focused,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-focused.docs-sourcelink{background-color:#375a7f;border-color:#375a7f;color:#fff}html.theme--documenter-dark .button.is-primary.is-outlined.is-loading::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink::after{border-color:transparent transparent #375a7f #375a7f !important}html.theme--documenter-dark .button.is-primary.is-outlined.is-loading:hover::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink:hover::after,html.theme--documenter-dark .button.is-primary.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.is-hovered.docs-sourcelink::after,html.theme--documenter-dark .button.is-primary.is-outlined.is-loading:focus::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink:focus::after,html.theme--documenter-dark .button.is-primary.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.is-focused.docs-sourcelink::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-primary.is-outlined[disabled],html.theme--documenter-dark .docstring>section>a.button.is-outlined.docs-sourcelink[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-primary.is-outlined,fieldset[disabled] html.theme--documenter-dark .docstring>section>a.button.is-outlined.docs-sourcelink{background-color:transparent;border-color:#375a7f;box-shadow:none;color:#375a7f}html.theme--documenter-dark 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.button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons 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h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#f2f2f2}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#f2f2f2}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--documenter-dark .icon.is-small,html.theme--documenter-dark #documenter 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#documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--documenter-dark .image.is-fullwidth,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--documenter-dark .image.is-square img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--documenter-dark .image.is-square .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--documenter-dark .image.is-1by1 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--documenter-dark .image.is-1by1 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--documenter-dark .image.is-5by4 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--documenter-dark .image.is-5by4 .has-ratio,html.theme--documenter-dark #documenter 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+ Maintainer: @highlightjs/core-team + Website: https://highlightjs.org/ + License: see project LICENSE + Touched: 2021 +*/pre code.hljs{display:block;overflow-x:auto;padding:1em}code.hljs{padding:3px 5px}.hljs{background:#F3F3F3;color:#444}.hljs-comment{color:#697070}.hljs-tag,.hljs-punctuation{color:#444a}.hljs-tag .hljs-name,.hljs-tag .hljs-attr{color:#444}.hljs-keyword,.hljs-attribute,.hljs-selector-tag,.hljs-meta .hljs-keyword,.hljs-doctag,.hljs-name{font-weight:bold}.hljs-type,.hljs-string,.hljs-number,.hljs-selector-id,.hljs-selector-class,.hljs-quote,.hljs-template-tag,.hljs-deletion{color:#880000}.hljs-title,.hljs-section{color:#880000;font-weight:bold}.hljs-regexp,.hljs-symbol,.hljs-variable,.hljs-template-variable,.hljs-link,.hljs-selector-attr,.hljs-operator,.hljs-selector-pseudo{color:#ab5656}.hljs-literal{color:#695}.hljs-built_in,.hljs-bullet,.hljs-code,.hljs-addition{color:#397300}.hljs-meta{color:#1f7199}.hljs-meta .hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold} diff --git a/previews/PR178/assets/themeswap.js b/previews/PR178/assets/themeswap.js new file mode 100644 index 000000000..9f5eebe6a --- /dev/null +++ b/previews/PR178/assets/themeswap.js @@ -0,0 +1,84 @@ +// Small function to quickly swap out themes. Gets put into the tag.. +function set_theme_from_local_storage() { + // Initialize the theme to null, which means default + var theme = null; + // If the browser supports the localstorage and is not disabled then try to get the + // documenter theme + if (window.localStorage != null) { + // Get the user-picked theme from localStorage. May be `null`, which means the default + // theme. + theme = window.localStorage.getItem("documenter-theme"); + } + // Check if the users preference is for dark color scheme + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; + // Initialize a few variables for the loop: + // + // - active: will contain the index of the theme that should be active. Note that there + // is no guarantee that localStorage contains sane values. If `active` stays `null` + // we either could not find the theme or it is the default (primary) theme anyway. + // Either way, we then need to stick to the primary theme. + // + // - disabled: style sheets that should be disabled (i.e. all the theme style sheets + // that are not the currently active theme) + var active = null; + var disabled = []; + var primaryLightTheme = null; + var primaryDarkTheme = null; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // To distinguish the default (primary) theme, it needs to have the data-theme-primary + // attribute set. + if (ss.ownerNode.getAttribute("data-theme-primary") !== null) { + primaryLightTheme = themename; + } + // Check if the theme is primary dark theme so that we could store its name in darkTheme + if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) { + primaryDarkTheme = themename; + } + // If we find a matching theme (and it's not the default), we'll set active to non-null + if (themename === theme) active = i; + // Store the style sheets of inactive themes so that we could disable them + if (themename !== theme) disabled.push(ss); + } + var activeTheme = null; + if (active !== null) { + // If we did find an active theme, we'll (1) add the theme--$(theme) class to + document.getElementsByTagName("html")[0].className = "theme--" + theme; + activeTheme = theme; + } else { + // If we did _not_ find an active theme, then we need to fall back to the primary theme + // which can either be dark or light, depending on the user's OS preference. + var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme; + // In case it somehow happens that the relevant primary theme was not found in the + // preceding loop, we abort without doing anything. + if (activeTheme === null) { + console.error("Unable to determine primary theme."); + return; + } + // When switching to the primary light theme, then we must not have a class name + // for the tag. That's only for non-primary or the primary dark theme. + if (darkPreference) { + document.getElementsByTagName("html")[0].className = + "theme--" + activeTheme; + } else { + document.getElementsByTagName("html")[0].className = ""; + } + } + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // we'll disable all the stylesheets, except for the active one + ss.disabled = !(themename == activeTheme); + } +} +set_theme_from_local_storage(); diff --git a/previews/PR178/assets/warner.js b/previews/PR178/assets/warner.js new file mode 100644 index 000000000..3f6f5d008 --- /dev/null +++ b/previews/PR178/assets/warner.js @@ -0,0 +1,52 @@ +function maybeAddWarning() { + // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE + // in siteinfo.js. + // If either of these are undefined something went horribly wrong, so we abort. + if ( + window.DOCUMENTER_NEWEST === undefined || + window.DOCUMENTER_CURRENT_VERSION === undefined || + window.DOCUMENTER_STABLE === undefined + ) { + return; + } + + // Current version is not a version number, so we can't tell if it's the newest version. Abort. + if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) { + return; + } + + // Current version is newest version, so no need to add a warning. + if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) { + return; + } + + // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs. + if (document.body.querySelector('meta[name="robots"]') === null) { + const meta = document.createElement("meta"); + meta.name = "robots"; + meta.content = "noindex"; + + document.getElementsByTagName("head")[0].appendChild(meta); + } + + const div = document.createElement("div"); + div.classList.add("outdated-warning-overlay"); + const closer = document.createElement("button"); + closer.classList.add("outdated-warning-closer", "delete"); + closer.addEventListener("click", function () { + document.body.removeChild(div); + }); + const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE; + div.innerHTML = + 'This documentation is not for the latest stable release, but for either the development version or an older release.
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    GitHub

    Download this example as Jupyter notebook or Julia script.

    Case Study: FeI₂

    FeI₂ is an effective spin-1 material with strong single-ion anisotropy. Quadrupolar fluctuations give rise to a single-ion bound state that cannot be described by a dipole-only model. This tutorial illustrates how to use the linear spin wave theory of SU(3) coherent states (i.e. 2-flavor bosons) to model the magnetic behavior in FeI₂. The original study was performed in Bai et al., Nature Physics 17, 467–472 (2021).

    The Fe atoms are arranged in stacked triangular layers. The effective spin Hamiltonian takes the form,

    \[\mathcal{H}=\sum_{(i,j)} 𝐒_i ⋅ J_{ij} 𝐒_j - D\sum_i \left(S_i^z\right)^2,\]

    where the set of exchange matrices $J_{ij}$ between bonded sites $(i,j)$ includes competing ferromagnetic and antiferromagnetic interactions. This model also includes a strong easy axis anisotropy, $D > 0$.

    We will formulate this Hamiltonian in Sunny and then calculate its dynamic structure factor.

    Get Julia and Sunny

    Sunny is implemented in Julia. This is a relatively new programming language that allows for interactive development (like Python or Matlab) while also providing high numerical efficiency (like C++ or Fortran). New Julia users may wish to take a look at our Getting Started with Julia guide. Sunny requires Julia 1.9 or later.

    From the Julia prompt, load Sunny. For plotting, one can choose either GLMakie (a pop-up window) or WGLMakie (inline plots for a Jupyter notebook or VSCode).

    using Sunny, GLMakie

    If these packages are not yet installed, Julia should offer to install them using its built-in package management system. If old versions are installed, you may need to update them to run this tutorial.

    Crystals

    A Crystal describes the crystallographic unit cell and will usually be loaded from a .cif file. Here, we instead build a crystal by listing all atoms and their types.

    a = b = 4.05012  # Lattice constants for triangular lattice
+c = 6.75214      # Spacing in the z-direction
+
+latvecs = lattice_vectors(a, b, c, 90, 90, 120) # A 3x3 matrix of lattice vectors that
+                                                # define the conventional unit cell
+positions = [[0, 0, 0], [1/3, 2/3, 1/4], [2/3, 1/3, 3/4]]  # Positions of atoms in fractions
+                                                           # of lattice vectors
+types = ["Fe", "I", "I"]
+FeI2 = Crystal(latvecs, positions; types)
    Crystal
+HM symbol 'P -3 m 1' (164)
+Lattice params a=4.05, b=4.05, c=6.752, α=90°, β=90°, γ=120°
+Cell volume 95.92
+Type 'Fe', Wyckoff 1a (point group '-3m.'):
+   1. [0, 0, 0]
+Type 'I', Wyckoff 2d (point group '3m.'):
+   2. [1/3, 2/3, 1/4]
+   3. [2/3, 1/3, 3/4]
+

    Observe that Sunny inferred the space group, 'P -3 m 1' (164) and labeled the atoms according to their point group symmetries.

    Only the Fe atoms are magnetic, so we discard the I ions using subcrystal.

    cryst = subcrystal(FeI2, "Fe")
    Crystal
+HM symbol 'P -3 m 1' (164)
+Lattice params a=4.05, b=4.05, c=6.752, α=90°, β=90°, γ=120°
+Cell volume 95.92
+Type 'Fe', Wyckoff 1a (point group '-3m.'):
+   1. [0, 0, 0]
+

    Observe that cryst retains the spacegroup symmetry of the full FeI₂ crystal. This information will be used, for example, to propagate exchange interactions between symmetry-equivalent bonds.

    In a running Julia environment, the crystal can be viewed interactively using view_crystal.

    view_crystal(cryst, 8.0)
    Example block output

    Symmetry analysis

    The command print_symmetry_table provides a list of all the symmetry-allowed interactions up to a cutoff distance.

    print_symmetry_table(cryst, 8.0)
    Atom 1
+Type 'Fe', position [0, 0, 0], multiplicity 1
+Allowed g-tensor: [A 0 0
+                   0 A 0
+                   0 0 B]
+Allowed anisotropy in Stevens operators:
+    c₁*𝒪[2,0] +
+    c₂*𝒪[4,-3] + c₃*𝒪[4,0] +
+    c₄*𝒪[6,-3] + c₅*𝒪[6,0] + c₆*𝒪[6,6]
+
+Bond(1, 1, [1, 0, 0])
+Distance 4.05012, coordination 6
+Connects 'Fe' at [0, 0, 0] to 'Fe' at [1, 0, 0]
+Allowed exchange matrix:[A 0 0
+                         0 B D
+                         0 D C]
+
+Bond(1, 1, [0, 0, 1])
+Distance 6.75214, coordination 2
+Connects 'Fe' at [0, 0, 0] to 'Fe' at [0, 0, 1]
+Allowed exchange matrix:[A 0 0
+                         0 A 0
+                         0 0 B]
+
+Bond(1, 1, [1, 2, 0])
+Distance 7.01501361675086, coordination 6
+Connects 'Fe' at [0, 0, 0] to 'Fe' at [1, 2, 0]
+Allowed exchange matrix:[A 0 0
+                         0 B D
+                         0 D C]
+
+Bond(1, 1, [1, 0, 1])
+Distance 7.8736818956572, coordination 12
+Connects 'Fe' at [0, 0, 0] to 'Fe' at [1, 0, 1]
+Allowed exchange matrix:[A F E
+                         F B D
+                         E D C]

    The allowed $g$-tensor is expressed as a 3×3 matrix in the free coefficients A, B, ... The allowed single-ion anisotropy is expressed as a linear combination of Stevens operators. The latter correspond to polynomials of the spin operators, as we will describe below.

    The allowed exchange interactions are given as a 3×3 matrix for representative bonds. The notation Bond(i, j, n) indicates a bond between atom indices i and j, with cell offset n. In the general case, it will be necessary to associate atom indices with their positions in the unit cell; these can be viewed with display(cryst). Note that the order of the pair $(i, j)$ is significant if the exchange tensor contains antisymmetric Dzyaloshinskii–Moriya (DM) interactions.

    In the case of FeI₂, Bond(1, 1, [1,0,0]) is one of the 6 nearest-neighbor Fe-Fe bonds on a triangular lattice layer, and Bond(1, 1, [0,0,1]) is an Fe-Fe bond between layers.

    Building a spin System

    In constructing a spin System, we must provide several additional details about the spins.

    sys = System(cryst, (4, 4, 4), [SpinInfo(1, S=1, g=2)], :SUN, seed=2)
    System [SU(3)]
+Lattice: (4, 4, 4)×1
+

    This system includes $4×4×4$ unit cells, i.e. 64 Fe atoms, each with spin $S=1$ and a $g$-factor of 2. Quantum mechanically, spin $S=1$ involves a superposition of $2S+1=3$ distinct angular momentum states. In :SUN mode, this superposition will be modeled explicitly using the formalism of SU(3) coherent states, which captures both dipolar and quadrupolar fluctuations. For the more traditional dipole dynamics, use :dipole mode instead.

    Next we will use set_exchange! to assign interaction to bonds. Sunny will automatically propagate each interaction to all symmetry-equivalent bonds in the unit cell. The FeI₂ interactions below follow Bai et al.

    J1pm   = -0.236
+J1pmpm = -0.161
+J1zpm  = -0.261
+J2pm   = 0.026
+J3pm   = 0.166
+J′0pm  = 0.037
+J′1pm  = 0.013
+J′2apm = 0.068
+
+J1zz   = -0.236
+J2zz   = 0.113
+J3zz   = 0.211
+J′0zz  = -0.036
+J′1zz  = 0.051
+J′2azz = 0.073
+
+J1xx = J1pm + J1pmpm
+J1yy = J1pm - J1pmpm
+J1yz = J1zpm
+
+set_exchange!(sys, [J1xx   0.0    0.0;
+                    0.0    J1yy   J1yz;
+                    0.0    J1yz   J1zz], Bond(1,1,[1,0,0]))
+set_exchange!(sys, [J2pm   0.0    0.0;
+                    0.0    J2pm   0.0;
+                    0.0    0.0    J2zz], Bond(1,1,[1,2,0]))
+set_exchange!(sys, [J3pm   0.0    0.0;
+                    0.0    J3pm   0.0;
+                    0.0    0.0    J3zz], Bond(1,1,[2,0,0]))
+set_exchange!(sys, [J′0pm  0.0    0.0;
+                    0.0    J′0pm  0.0;
+                    0.0    0.0    J′0zz], Bond(1,1,[0,0,1]))
+set_exchange!(sys, [J′1pm  0.0    0.0;
+                    0.0    J′1pm  0.0;
+                    0.0    0.0    J′1zz], Bond(1,1,[1,0,1]))
+set_exchange!(sys, [J′2apm 0.0    0.0;
+                    0.0    J′2apm 0.0;
+                    0.0    0.0    J′2azz], Bond(1,1,[1,2,1]))

    The function set_onsite_coupling! assigns a single-ion anisotropy. The argument can be constructed using spin_matrices or stevens_matrices. Here we use Julia's anonymous function syntax to assign an easy-axis anisotropy along the direction $\hat{z}$.

    D = 2.165
+set_onsite_coupling!(sys, S -> -D*S[3]^2, 1)

    Calculating structure factor intensities

    In the remainder of this tutorial, we will examine Sunny's tools for calculating the dynamical structure factor using a multi-boson generalization of linear spin wave theory (LSWT). This theory describes non-interacting quasi-particle excitations that hybridize dipolar and quadrupolar modes.

    Finding the ground state

    Begin with a random configuration and use minimize_energy! to find a configuration of the SU(3) coherent states (i.e. spin dipoles and quadrupoles) that locally minimizes energy.

    randomize_spins!(sys)
+minimize_energy!(sys)
    50

    A positive number above indicates that the procedure has converged to a local energy minimum. The configuration, however, may still have defects. This can be checked by visualizing the spins, colored according to their $z$-components.

    plot_spins(sys; color=[s[3] for s in sys.dipoles])
    Example block output

    A different understanding of the magnetic ordering can be obtained by moving to Fourier space. The 'instant' structure factor $𝒮(𝐪)$ is an experimental observable. To investigate $𝒮(𝐪)$ as true 3D data, Sunny provides instant_correlations and related functions. Here, however, we will use print_wrapped_intensities, which gives average intensities for the individual Bravais sublattices (in effect, all wavevectors are wrapped to the first Brillouin zone).

    print_wrapped_intensities(sys)
    Dominant wavevectors for spin sublattices:
+
+    [-1/4, 1/4, 1/4]       37.33% weight
+    [1/4, -1/4, -1/4]      37.33%
+    [-1/4, 1/4, 0]          1.60%
+    [1/4, -1/4, 0]          1.60%
+    [-1/4, 1/4, 1/2]        1.60%
+    [1/4, -1/4, 1/2]        1.60%
+    [1/4, -1/4, 1/4]        1.59%
+    [-1/4, 1/4, -1/4]       1.59%
+    [0, -1/4, 1/4]          0.77%
+    [0, 1/4, -1/4]          0.77%
+    [1/4, 0, 1/4]           0.77%
+    ...                     ...

    The result will likely be approximately consistent with the known zero-field energy-minimizing magnetic structure of FeI₂, which is single-$Q$ (two-up, two-down antiferromagnetic order). Mathematically, spontaneous symmetry breaking should select one of $±Q = [0, -1/4, 1/4]$, $[1/4, 0, 1/4]$, or $[-1/4,1/4,1/4]$, associated with the three-fold rotational symmetry of the crystal spacegroup. In nature, however, one will frequently encounter competing "domains" associated with the three possible orientations of the ground state.

    If the desired ground state is already known, as with FeI₂, it could be entered by hand using set_dipole!. Alternatively, in the case of FeI₂, we could repeatedly employ the above randomization and minimization procedure until a defect-free configuration is found. Some systems will have more complicated ground states, which can be much more challenging to find. For this, Sunny provides experimental support for powerful simulated annealing via parallel tempering, but that is outside the scope of this tutorial.

    Here, let's break the three-fold symmetry of FeI₂ by hand. Given one or more desired $Q$ modes, Sunny can suggest a magnetic supercell with appropriate periodicity. Let's arbitrarily select one of the three possible ordering wavevectors, $Q = [0, -1/4, 1/4]$. Sunny suggests a corresponding magnetic supercell in units of the crystal lattice vectors.

    suggest_magnetic_supercell([[0, -1/4, 1/4]])
    Suggested magnetic supercell in multiples of lattice vectors:
+
+    [1 0 0; 0 2 1; 0 -2 1]
+
+for wavevectors [[0, -1/4, 1/4]].

    The system returned by reshape_supercell is smaller, and is sheared relative to the original system. This makes it much easier to find the global energy minimum.

    sys_min = reshape_supercell(sys, [1 0 0; 0 2 1; 0 -2 1])
+randomize_spins!(sys_min)
+minimize_energy!(sys_min);

    Plot the system again, now including "ghost" spins out to 12Å

    plot_spins(sys_min; color=[s[3] for s in sys_min.dipoles], ghost_radius=12)
    Example block output

    Linear spin wave theory

    Now that we have found the ground state for a magnetic supercell, we can immediately proceed to perform zero-temperature calculations using linear spin wave theory. We begin by instantiating a SpinWaveTheory type using the supercell.

    swt = SpinWaveTheory(sys_min)
    SpinWaveTheory [Dipole correlations]
+Atoms in magnetic supercell: 4
+

    Select a sequence of wavevectors that will define a piecewise linear interpolation in reciprocal lattice units (RLU).

    q_points = [[0,0,0], [1,0,0], [0,1,0], [1/2,0,0], [0,1,0], [0,0,0]];

    The function reciprocal_space_path will linearly sample a path between the provided $q$-points with a given density. The xticks return value provides labels for use in plotting.

    density = 50
+path, xticks = reciprocal_space_path(cryst, q_points, density);

    The dispersion function defines the quasiparticle excitation energies $ω_i(𝐪)$ for each point $𝐪$ along the reciprocal space path.

    disp = dispersion(swt, path);

    In addition to the band energies $ω_i(𝐪)$, Sunny can calculate the inelastic neutron scattering intensity $I_i(𝐪)$ for each band $i$ according to an intensity_formula. We choose to apply a polarization correction $(1 - 𝐪⊗𝐪)$ by setting the mode argument to :perp. Selecting delta_function_kernel specifies that we want the energy and intensity of each band individually.

    formula = intensity_formula(swt, :perp; kernel=delta_function_kernel)
    Quantum Scattering Intensity Formula
+At any Q and for each band ωᵢ = εᵢ(Q), with S = S(Q,ωᵢ):
+
+  Intensity(Q,ω) = ∑ᵢ δ(ω-ωᵢ) ∑_ij (I - Q⊗Q){i,j} S{i,j}
+                              
+                              (i,j = Sx,Sy,Sz)
+
+BandStructure information (ωᵢ and intensity) reported for each band
+

    The function intensities_bands uses linear spin wave theory to calculate both the dispersion and intensity data for the provided path.

    disp, intensity = intensities_bands(swt, path, formula);

    These can be plotted in GLMakie.

    fig = Figure()
+ax = Axis(fig[1,1]; xlabel="Momentum (r.l.u.)", ylabel="Energy (meV)", xticks, xticklabelrotation=π/6)
+ylims!(ax, 0.0, 7.5)
+xlims!(ax, 1, size(disp, 1))
+colorrange = extrema(intensity)
+for i in axes(disp)[2]
+    lines!(ax, 1:length(disp[:,i]), disp[:,i]; color=intensity[:,i], colorrange)
+end
+fig
    Example block output

    To make comparisons with inelastic neutron scattering (INS) data, it is helpful to employ an empirical broadening kernel, e.g., a lorentzian.

    γ = 0.15 # width in meV
+broadened_formula = intensity_formula(swt, :perp; kernel=lorentzian(γ))
    Quantum Scattering Intensity Formula
+At any (Q,ω), with S = S(Q,ωᵢ):
+
+  Intensity(Q,ω) = ∑ᵢ Kernel(ω-ωᵢ) ∑_ij (I - Q⊗Q){i,j} S{i,j}
+                                   
+                                   (i,j = Sx,Sy,Sz)
+
+Intensity(ω) reported
+

    The intensities_broadened function requires an energy range in addition to the $𝐪$-space path.

    energies = collect(0:0.01:10)  # 0 < ω < 10 (meV).
+is1 = intensities_broadened(swt, path, energies, broadened_formula);

    A real FeI₂ sample will exhibit competing magnetic domains associated with spontaneous symmetry breaking of the 6-fold rotational symmetry of the triangular lattice. Note that the wavevectors $𝐪$ and $-𝐪$ are equivalent in the structure factor, which leaves three distinct domain orientations, which are related by 120° rotations about the $ẑ$-axis. Rather than rotating the spin configuration directly, on can rotate the $𝐪$-space path. Below, we use rotation_in_rlu to average the intensities over all three possible orientations.

    R = rotation_in_rlu(cryst, [0, 0, 1], 2π/3)
+is2 = intensities_broadened(swt, [R*q for q in path], energies, broadened_formula)
+is3 = intensities_broadened(swt, [R*R*q for q in path], energies, broadened_formula)
+is_averaged = (is1 + is2 + is3) / 3
+
+fig = Figure()
+ax = Axis(fig[1,1]; xlabel="Momentum (r.l.u.)", ylabel="Energy (meV)", xticks, xticklabelrotation=π/6)
+heatmap!(ax, 1:size(is_averaged, 1), energies, is_averaged)
+fig
    Example block output

    This result can be directly compared to experimental neutron scattering data from Bai et al.

    (The publication figure accidentally used a non-standard coordinate system to label the wave vectors.)

    To get this agreement, the use of SU(3) coherent states is essential. In other words, we needed a theory of multi-flavored bosons. The lower band has large quadrupolar character, and arises from the strong easy-axis anisotropy of FeI₂. By setting mode = :SUN, the calculation captures this coupled dipole-quadrupole dynamics.

    An interesting exercise is to repeat the same study, but using mode = :dipole instead of :SUN. That alternative choice would constrain the coherent state dynamics to the space of dipoles only.

    The full dynamical spin structure factor (DSSF) can be retrieved as a $3×3$ matrix with the dssf function, for a given path of $𝐪$-vectors.

    disp, is = dssf(swt, path);

    The first output disp is identical to that obtained from dispersion. The second output is contains a list of $3×3$ matrix of intensities. For example, is[q,n][2,3] yields the $(ŷ,ẑ)$ component of the structure factor intensity for nth mode at the qth wavevector in the path.

    What's next?

    The multi-boson linear spin wave theory, applied above, can be understood as the quantization of a certain generalization of the Landau-Lifshitz spin dynamics. Rather than dipoles, this dynamics takes places on the space of SU(N) coherent states.

    The full SU(N) coherent state dynamics, with appropriate quantum correction factors, can be useful to model finite temperature scattering data. In particular, it captures certain anharmonic effects due to thermal fluctuations. This is the subject of our FeI₂ at Finite Temperature tutorial.

    The classical dynamics is also a good starting point to study non-equilibrium phenomena. Empirical noise and damping terms can be used to model coupling to a thermal bath. This yields a Langevin dynamics of SU(N) coherent states. Our CP² Skyrmion Quench tutorial shows how this dynamics gives rise to the formation of novel topological defects in a temperature quench.

    Relative to LSWT calculations, it can take much more time to estimate $\mathcal{S}(𝐪,ω)$ intensities using classical dynamics simulation. See the SunnyTutorials notebooks for examples of "production-scale" simulations.

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    Classical Ising model

    This tutorial illustrates simulation of the classical 2D Ising model.

    using Sunny, GLMakie

    Sunny expects a 3D Crystal unit cell. To model a square lattice, we create an orthogonal unit cell where the $z$-spacing is distinct from the $x$ and $y$ spacing.

    a = 1
+latvecs = lattice_vectors(a,a,10a,90,90,90)
+crystal = Crystal(latvecs, [[0,0,0]])
    Crystal
+HM symbol 'P 4/m m m' (123)
+Lattice params a=1, b=1, c=10, α=90°, β=90°, γ=90°
+Cell volume 10
+Wyckoff 1a (point group '4/mmm'):
+   1. [0, 0, 0]
+

    Create a System of spins with linear size L in the $x$ and $y$ directions, and only one layer in the $z$ direction. The option :dipole means that the system will store Heisenberg spins, as opposed to SU($N$) coherent states. Polarize the initial spin configuration using polarize_spins!. Following the Ising convention, we will restrict these spins to the $z$-axis and give them magnitude $S=1$.

    By default, Sunny uses physical units, e.g. magnetic field in tesla. Here we specify an alternative Units system, so that the Zeeman coupling between the spin dipole $𝐬$ and an external field $𝐁$ has the dimensionless form $-𝐁⋅𝐬$.

    L = 128
+sys = System(crystal, (L,L,1), [SpinInfo(1, S=1, g=1)], :dipole, units=Units.theory, seed=0)
+polarize_spins!(sys, (0,0,1))

    Use set_exchange! to include a ferromagnetic Heisenberg interaction along nearest-neighbor bonds. The Bond below connects two spins displaced by one lattice constant in the $x$-direction. This interaction will be propagated to all nearest-neighbors bonds in the system, consistent with the symmetries of the square lattice.

    set_exchange!(sys, -1.0, Bond(1,1,(1,0,0)))

    If an external field is desired, it can be set using set_external_field!.

    B = 0
+set_external_field!(sys, (0,0,B))

    The critical temperature for the Ising model is known analytically.

    Tc = 2/log(1+√2)
    2.269185314213022

    Use a LocalSampler to perform nsweeps Monte Carlo sweeps. A sweep consists of, on average, one trial update per spin in the system. Each proposed update is accepted or rejected according to the Metropolis acceptance probability. As its name suggests, the propose_flip function will only propose pure spin flips, $𝐬 \rightarrow -𝐬$.

    nsweeps = 4000
+sampler = LocalSampler(kT=Tc, propose=propose_flip)
+for i in 1:nsweeps
+    step!(sys, sampler)
+end

    Plot the Ising spins by extracting the $z$-component of the dipoles

    heatmap(reshape([s.z for s in sys.dipoles], (L,L)))
    Example block output
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b/previews/PR178/examples/out_of_equilibrium.html @@ -0,0 +1,54 @@ + +CP² Skyrmion Quench · Sunny documentation

      GitHub

      Download this example as Jupyter notebook or Julia script.

      CP² Skyrmion Quench

      This example demonstrates Sunny's ability to simulate the out-of-equilibrium dynamics of generalized spin systems. We will implement the model Hamiltonian of Zhang et al., Nature Communications 14, 3626 (2023), which supports a novel type of topological defect, a CP² skyrmion, that involves both the dipolar and quadrupolar parts of a quantum spin.

      Beginning from an initial high-temperature state, a disordered gas of CP² skyrmions can be formed by rapidly quenching to low temperature. To model the coupled dynamics of dipoles and quadrupoles, Sunny uses a recently developed generalization of the Landau-Lifshitz spin dynamics, Dahlbom et al., Phys. Rev. B 106, 235154 (2022).

      using Sunny, GLMakie

      The Hamiltonian we will implement,

      \[\mathcal{H} = \sum_{\langle i,j \rangle} J_{ij}( \hat{S}_i^x \hat{S}_j^x + \hat{S}_i^y \hat{S}_j^y + \Delta\hat{S}_i^z \hat{S}_j^z) - h\sum_{i}\hat{S}_i^z + D\sum_{i}(\hat{S}_i^z)^2\]

      contains competing ferromagnetic nearest-neightbor and antiferromagnetic next-nearest-neighbor exchange terms on a triangular lattice. Both exchanges exhibit anisotropy on the z-term. Additionally, there is an external magnetic field, $h$, and easy-plane single-ion anisotropy, $D > 0$. We begin by implementing the Crystal.

      lat_vecs = lattice_vectors(1.0, 1.0, 2.0, 90, 90, 120)
+basis_vecs = [[0,0,0]]
+cryst = Crystal(lat_vecs, basis_vecs)
      Crystal
+HM symbol 'P 6/m m m' (191)
+Lattice params a=1, b=1, c=2, α=90°, β=90°, γ=120°
+Cell volume 1.732
+Wyckoff 1a (point group '6/mmm'):
+   1. [0, 0, 0]
+

      The crystal is then used to create a spin System. All parameters in this model system are dimensionless, so we select "theory" units and set the g-factor to one.

      L = 40
+dims = (L, L, 1)
+sys = System(cryst, dims, [SpinInfo(1, S=1, g=1)], :SUN; seed=101, units=Units.theory)
      System [SU(3)]
+Lattice: (40, 40, 1)×1
+

      We proceed to implement each term of the Hamiltonian, selecting our parameters so that the system occupies a region of the phase diagram that supports skyrmions. The exchange interactions are set as follows.

      J1 = -1           # Nearest-neighbor ferromagnetic
+J2 = (2.0/(1+√5)) # Tune competing exchange to set skyrmion scale length
+Δ = 2.6           # Exchange anisotropy
+
+ex1 = J1 * [1.0 0.0 0.0;
+            0.0 1.0 0.0;
+            0.0 0.0 Δ]
+ex2 = J2 * [1.0 0.0 0.0;
+            0.0 1.0 0.0;
+            0.0 0.0 Δ]
+set_exchange!(sys, ex1, Bond(1, 1, [1, 0, 0]))
+set_exchange!(sys, ex2, Bond(1, 1, [1, 2, 0]))

      Next we add the external field,

      h = 15.5
+field = set_external_field!(sys, [0.0 0.0 h])

      and finally an easy-plane single-ion anisotropy,

      D = 19.0
+set_onsite_coupling!(sys, S -> D*S[3]^2, 1)

      Initialize system to an infinite temperature (fully randomized) initial condition.

      randomize_spins!(sys)

      We are now ready to simulate the quenching process using a generalized Langevin spin dynamics. If we were working with spin dipoles only, then Langevin dynamics would be the usual Landau-Lifshitz spin dynamics, augmented with damping and noise terms. In the present study, we are instead working with quantum spin-1 (an ($N=3$)-level system that includes both dipoles and quadrupoles). Here, Langevin captures the coupled dipole-quadrupole dynamics using the formalism of SU($N$) coherent states.

      Selecting kT = 0 in the Langevin dynamics will effective disable the noise term. Then the parameter λ effectively determines the damping time-scale.

      Δt = 0.2/D  # Integration time step (inverse meV). Typically this will be
+            # inversely proportional to the largest energy scale in the
+            # system. We can use a fairly large time-step here because
+            # accuracy isn't critical.
+kT = 0      # Target equilibrium temperature (meV)
+λ = 0.1     # Magnitude of coupling to thermal bath (dimensionless)
+integrator = Langevin(Δt; kT, λ)
      Langevin(0.010526315789473684, 0.1, 0.0)

      Finally we run the dynamics. We will record the state of the system at three different times during the quenching process by copying the coherents field of the System.

      τs = [4., 16, 256]  # Times to record snapshots
+frames = []         # Empty array to store snapshots
+for i in eachindex(τs)
+    dur = i == 1 ? τs[1] : τs[i] - τs[i-1] # Determine the length of time to simulate
+    numsteps = round(Int, dur/Δt)
+    for _ in 1:numsteps                    # Perform the integration
+        step!(sys, integrator)
+    end
+    push!(frames, copy(sys.coherents))     # Save a snapshot spin configuration
+end

      To visualize the state of the system contained in each snapshot, we will calculate and plot the skyrmion density on each plaquette of our lattice. The function plot_triangular_plaquettes is not part of the core Sunny package, but rather something you could define yourself. We are using the definition in plotting2d.jl from the Sunny examples/extra directory.

      include(pkgdir(Sunny, "examples", "extra", "plotting2d.jl"))
+
+function sun_berry_curvature(z₁, z₂, z₃)
+    z₁, z₂, z₃ = normalize.((z₁, z₂, z₃))
+    n₁ = z₁ ⋅ z₂
+    n₂ = z₂ ⋅ z₃
+    n₃ = z₃ ⋅ z₁
+    return angle(n₁ * n₂ * n₃)
+end
+
+plot_triangular_plaquettes(sun_berry_curvature, frames; resolution=(1800,600),
+    offset_spacing=10, texts = ["\tt = "*string(τ) for τ in τs], text_offset = (0.0, 6.0)
+)
      Example block output

      The times are given in $\hbar/|J_1|$. The white background corresponds to a quantum paramagnetic state, where the local spin exhibits a strong quadrupole moment and little or no dipole moment. Observe that the process has generated a number of well-formed skyrmions of both positive (red) and negative (blue) charge in addition to a number of other metastable spin configurations. A full-sized version of this figure is available in Dahlbom et al..

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z-w;&p$PWhWV`?-<21@?*kLpgwu!Dz@yc~?AMJI@~fE%wMSz*9Y1Jmh#k#Vv33&i5D zB_Nhy6$ScB9w&LF=F`9A4^t*jHeF6gH9xOA)nuKcAio_q6Ee$hcU#M7$ z8)7MPYNF^leDN2$ x#Q}kV;Hu(3ZxQ-`@R;}fw_Qg6>k!6%vyE-`0minq*ChPB9 +Powder Averaged CoRh₂O₄ · Sunny documentation

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      Download this example as Jupyter notebook or Julia script.

      Powder Averaged CoRh₂O₄

      This tutorial illustrates the calculation of the powder-averaged structure factor by performing an orientational average. We consider a simple model of the diamond-cubic crystal CoRh₂O₄, with parameters extracted from Ge et al., Phys. Rev. B 96, 064413.

      using Sunny, GLMakie

      Construct a diamond Crystal in the conventional (non-primitive) cubic unit cell. Sunny will populate all eight symmetry-equivalent sites when given the international spacegroup number 227 ("Fd-3m") and the appropriate setting. For this spacegroup, there are two conventional translations of the unit cell, and it is necessary to disambiguate through the setting keyword argument. (On your own: what happens if setting is omitted?)

      a = 8.5031 # (Å)
+latvecs = lattice_vectors(a, a, a, 90, 90, 90)
+cryst = Crystal(latvecs, [[0,0,0]], 227, setting="1")
      Crystal
+HM symbol 'F d -3 m' (227)
+Lattice params a=8.503, b=8.503, c=8.503, α=90°, β=90°, γ=90°
+Cell volume 614.8
+Wyckoff 8a (point group '-43m'):
+   1. [0, 0, 0]
+   2. [1/2, 1/2, 0]
+   3. [1/4, 1/4, 1/4]
+   4. [3/4, 3/4, 1/4]
+   5. [1/2, 0, 1/2]
+   6. [0, 1/2, 1/2]
+   7. [3/4, 1/4, 3/4]
+   8. [1/4, 3/4, 3/4]
+

      In a running Julia environment, the crystal can be viewed interactively using view_crystal.

      view_crystal(cryst, 8.0)
      Example block output

      Construct a System with an antiferromagnetic nearest neighbor interaction J. Because the diamond crystal is bipartite, the ground state will have unfrustrated Néel order. Selecting latsize=(1,1,1) is sufficient because the ground state is periodic over each cubic unit cell. By passing an explicit seed, the system's random number generator will give repeatable results.

      latsize = (2, 2, 2)
+seed = 0
+S = 3/2
+J = 7.5413*meV_per_K # (~ 0.65 meV)
+sys = System(cryst, latsize, [SpinInfo(1; S, g=2)], :dipole; seed=0)
+set_exchange!(sys, J, Bond(1, 3, [0,0,0]))

      In the ground state, each spin is exactly anti-aligned with its 4 nearest-neighbors. Because every bond contributes an energy of $-JS^2$, the energy per site is $-2JS^2$. In this calculation, a factor of 1/2 avoids double-counting the bonds. Due to lack of frustration, direct energy minimization is successful in finding the ground state.

      randomize_spins!(sys)
+minimize_energy!(sys)
+
+@assert energy_per_site(sys) ≈ -2J*S^2

      Plotting the spins confirms the expected Néel order. Note that the overall, global rotation of dipoles is arbitrary.

      s0 = sys.dipoles[1,1,1,1]
+plot_spins(sys; color=[s'*s0 for s in sys.dipoles])
      Example block output

      For numerical efficiency, it will be helpful to work with the smallest possible magnetic supercell. Here, it is the primitive unit cell, which contains just two sites. The variable shape below defines the primitive lattice vectors cryst.prim_latvecs in units of the conventional lattice vectors. This result is used as input to reshape_supercell. The energy per site remains the same, which verifies that the magnetic supercell is valid.

      shape = cryst.latvecs \ cryst.prim_latvecs
+sys_prim = reshape_supercell(sys, shape)
+@assert energy_per_site(sys_prim) ≈ -2J*S^2

      Now estimate $𝒮(𝐪,ω)$ with SpinWaveTheory and an intensity_formula. The mode :perp contracts with a dipole factor to return the unpolarized intensity. The formula also employs lorentzian broadening. The isotropic FormFactor for Cobalt(2+) dampens intensities at large $𝐪$.

      swt = SpinWaveTheory(sys_prim)
+η = 0.4 # (meV)
+kernel = lorentzian(η)
+formfactors = [FormFactor("Co2")]
+formula = intensity_formula(swt, :perp; kernel, formfactors)
      Quantum Scattering Intensity Formula
+At any (Q,ω), with S = S(Q,ωᵢ):
+
+  Intensity(Q,ω) = ∑ᵢ Kernel(ω-ωᵢ) ∑_ij (I - Q⊗Q){i,j} S{i,j}
+                                   
+                                   (i,j = Sx,Sy,Sz)
+
+Intensity(ω) reported
+

      For the "single crystal" result, we may use reciprocal_space_path to construct a path that connects high-symmetry points in reciprocal space. The intensities_broadened function collects intensities along this path for the given set of energy values.

      qpoints = [[0.0, 0.0, 0.0], [0.5, 0.0, 0.0], [0.5, 0.5, 0.0], [0.0, 0.0, 0.0]]
+path, xticks = reciprocal_space_path(cryst, qpoints, 50)
+energies = collect(0:0.01:6)
+is = intensities_broadened(swt, path, energies, formula)
+
+fig = Figure()
+ax = Axis(fig[1,1]; aspect=1.4, ylabel="ω (meV)", xlabel="𝐪 (r.l.u.)",
+          xticks, xticklabelrotation=π/10)
+heatmap!(ax, 1:size(is, 1), energies, is, colormap=:gnuplot2)
+fig
      Example block output

      A powder measurement effectively involves an average over all possible crystal orientations. We use the function reciprocal_space_shell to sample n wavevectors on a sphere of a given radius (inverse angstroms), and then calculate the spherically-averaged intensity.

      radii = 0.01:0.02:3 # (1/Å)
+output = zeros(Float64, length(radii), length(energies))
+for (i, radius) in enumerate(radii)
+    n = 300
+    qs = reciprocal_space_shell(cryst, radius, n)
+    is = intensities_broadened(swt, qs, energies, formula)
+    output[i, :] = sum(is, dims=1) / size(is, 1)
+end
+
+fig = Figure()
+ax = Axis(fig[1,1]; xlabel="Q (Å⁻¹)", ylabel="ω (meV)")
+heatmap!(ax, radii, energies, output, colormap=:gnuplot2)
+fig
      Example block output

      This result can be compared to experimental neutron scattering data from Fig. 5 of Ge et al.

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zRuCGml3k%Z)Y~S+|eCrufAml;^aK6js(_brwPiJBCnL0$s)|WL0(oLfL30OJS+A mjabMr<^TVS|4)i%>Ih$IRIOm~)FD3jN7-v(cvqix@&5p+?zWWx literal 0 HcmV?d00001 diff --git a/previews/PR178/examples/spinw/15_Ba3NbFe3Si2O14.html b/previews/PR178/examples/spinw/15_Ba3NbFe3Si2O14.html new file mode 100644 index 000000000..8fbc97a31 --- /dev/null +++ b/previews/PR178/examples/spinw/15_Ba3NbFe3Si2O14.html @@ -0,0 +1,48 @@ + +Ba₃NbFe₃Si₂O₁₄ · Sunny documentation

      GitHub

      Download this example as Jupyter notebook or Julia script.

      Ba₃NbFe₃Si₂O₁₄

      This is a Sunny port of SpinW Tutorial 15, authored by Toth et al. The goal is to calculate the linear spin wave theory spectrum for Ba₃NbFe₃Si₂O₁₄.

      Load packages

      using Sunny, GLMakie

      Build a Crystal for Ba₃NbFe₃Si₂O₁₄ using the crystal structure from Marty et al., Phys. Rev. Lett. 101, 247201 (2008).

      a = b = 8.539 # (Å)
+c = 5.2414
+latvecs = lattice_vectors(a, b, c, 90, 90, 120)
+types = ["Fe","Nb","Ba","Si","O","O","O"]
+positions = [[0.24964,0,0.5],[0,0,0],[0.56598,0,0],[2/3,1/3,0.5220],[2/3,1/3,0.2162],[0.5259,0.7024,0.3536],[0.7840,0.9002,0.7760]]
+langasite = Crystal(latvecs, positions, 150; types)
+crystal = subcrystal(langasite, "Fe")
+view_crystal(crystal, 7)
      Example block output

      Create a System with a lattice size of $(1,1,7)$. The magnetic structure of Ba₃NbFe₃Si₂O₁₄ was determined to have the ordering wavevector $𝐐=(0,0,1/7)$ and hence the magnetic unit cell has 7 sites.

      latsize = (1,1,7)
+S = 5/2
+seed = 5
+sys = System(crystal, latsize, [SpinInfo(1; S, g=2)], :dipole)
      System [Dipole mode]
+Lattice: (1, 1, 7)×3
+

      Set exchange interactions as parametrized in Loire et al., Phys. Rev. Lett. 106, 207201 (2011)

      J₁ = 0.85
+J₂ = 0.24
+J₃ = 0.053
+J₄ = 0.017
+J₅ = 0.24
+set_exchange!(sys, J₁, Bond(3, 2, [1,1,0]))
+set_exchange!(sys, J₄, Bond(1, 1, [0,0,1]))
+set_exchange!(sys, J₂, Bond(1, 3, [0,0,0]))

      The final two exchanges define the chirality of the magnetic structure. The crystal chirality, $\epsilon_T$, the chirality of each triangle, $ϵ_D$ and the sense of rotation of the spin helices along $c$, $ϵ_{H}$. The three chiralities are related by $ϵ_T=ϵ_D ϵ_H$. We now assign $J_3$ and $J_5$ according to the crystal chirality.

      ϵD = -1
+ϵH = +1
+ϵT = ϵD * ϵH
+
+if ϵT == -1
+    set_exchange!(sys, J₃, Bond(2, 3, [-1,-1,1]))
+    set_exchange!(sys, J₅, Bond(3, 2, [1,1,1]))
+elseif ϵT == 1
+    set_exchange!(sys, J₅, Bond(2, 3, [-1,-1,1]))
+    set_exchange!(sys, J₃, Bond(3, 2, [1,1,1]))
+else
+    throw("Provide a valid chirality")
+end

      Whilst Sunny provides tools to optimize the ground state automatically, in this case we already know the model ground state. Set the spiral magnetic order using set_spiral_order_on_sublattice!. It takes an ordering wavevector q, an axis of rotation for the spins axis, and the initial spin S0 for each sublattice.

      q = [0, 0, 1/7]
+axis = [0,0,1]
+set_spiral_order_on_sublattice!(sys, 1; q, axis, S0=[1, 0, 0])
+set_spiral_order_on_sublattice!(sys, 2; q, axis, S0=[-1/2, -sqrt(3)/2, 0])
+set_spiral_order_on_sublattice!(sys, 3; q, axis, S0=[-1/2, +sqrt(3)/2, 0])
+
+plot_spins(sys; color=[s[1] for s in sys.dipoles])
      Example block output

      Define a path in reciprocal space, $[0,1,-1+\xi]$ for $\xi = 0 \dots 3$.

      points_rlu = [[0,1,-1],[0,1,-1+1],[0,1,-1+2],[0,1,-1+3]];
+density = 100
+path, xticks = reciprocal_space_path(crystal, points_rlu, density);

      Calculate broadened intensities

      swt = SpinWaveTheory(sys)
+γ = 0.15 # width in meV
+broadened_formula = intensity_formula(swt, :perp; kernel=lorentzian(γ))
+energies = collect(0:0.01:6)  # 0 < ω < 6 (meV).
+is = intensities_broadened(swt, path, energies, broadened_formula);

      Plot

      fig = Figure()
+ax = Axis(fig[1,1]; xlabel="Momentum (r.l.u.)", ylabel="Energy (meV)", xticks, xticklabelrotation=π/6)
+heatmap!(ax, 1:size(is,1), energies, is, colorrange=(0,5))
+fig
      Example block output
      diff --git a/previews/PR178/index.html b/previews/PR178/index.html new file mode 100644 index 000000000..7f2bcf460 --- /dev/null +++ b/previews/PR178/index.html @@ -0,0 +1,2 @@ + +Overview · Sunny documentation
      GitHub

      Overview

      Sunny is a Julia package for modeling atomic-scale magnetism. It provides powerful tools to study equilibrium and non-equilibrium magnetic phenomena. In particular, it allows estimation of dynamical structure factor intensities, $\mathcal{S}(𝐪,ω)$, to support quantitative modeling of experimental scattering data.

      Features include:

      • Generalized spin dynamics using SU(N) coherent states.
      • Ability to specify a crystal from a .cif file or its spacegroup symmetry.
      • Interactive visualizations of the 3D crystals and magnetic ordering.
      • Symmetry analysis to classify allowed interaction terms, and to propagate them by symmetry.
      • Single-ion anisotropy at arbitrary order, which can be specified using Stevens operators or as a polynomial of spin operators.
      • Monte Carlo sampling of spin configurations in thermal equilibrium, and optimization tools.
      • Measurements of dynamical correlations. At low temperature, one can use linear spin wave theory and its multi-boson generalization. This generalizes to finite temperatures using the classical dynamics, which allows for strongly nonlinear effects.
      • Long-range dipole-dipole interactions accelerated with the fast Fourier transform (FFT).
      • Support for comparison with experimental data: form factor, dipole factor, temperature-dependent classical-to-quantum factors, intensity binning, etc.
      diff --git a/previews/PR178/library.html b/previews/PR178/library.html new file mode 100644 index 000000000..cea7c347a --- /dev/null +++ b/previews/PR178/library.html @@ -0,0 +1,123 @@ + +Library API · Sunny documentation
      GitHub

      Library API

      This page describes the public types and functions exported by Sunny. This documentation can be also be accessed using the Julia help system (enter ? at the Julia command prompt).

      Sunny.Site — Type
      (cell1, cell2, cell3, i) :: Site

      Four indices identifying a single site in a System. The first three indices select the lattice cell and the last selects the sublattice (i.e., the atom within the unit cell).

      This object can be used to index dipoles and coherents fields of a System. A Site is also required to specify inhomogeneous interactions via functions such as set_external_field_at! or set_exchange_at!.

      Note that the definition of a cell may change when a system is reshaped. In this case, it is convenient to construct the Site using position_to_site, which always takes a position in fractional coordinates of the original lattice vectors.

      source
      Sunny.Units — Constant
      Units.meV
+Units.theory

      The unit system is implicitly determined by the definition of two physical constants: the vacuum permeability $μ₀$ and the Bohr magneton $μ_B$. Temperatures are effectively measured in units of energy ($k_B = 1$) and time is effectively measured in units of inverse energy ($ħ = 1$). The default unit system, Units.meV, employs (meV, Å, tesla). Select alternatively Units.theory for a units system defined so that $μ₀ = μ_B = 1$.

      See also meV_per_K

      source
      Sunny.meV_per_K — Constant
      meV_per_K = 0.086173332621451774

      A physical constant. Useful for converting kelvin into the default energy units, meV.

      source
      Sunny.BinningParameters — Type
      BinningParameters(binstart,binend,binwidth;covectors = I(4))
+BinningParameters(binstart,binend;numbins,covectors = I(4))

      Describes a 4D parallelepided histogram in a format compatible with experimental Inelasitic Neutron Scattering data. See generate_mantid_script_from_binning_parameters to convert BinningParameters to a format understandable by the Mantid software, or load_nxs to load BinningParameters from a Mantid .nxs file.

      The coordinates of the histogram axes are specified by multiplication of (q,ω) with each row of the covectors matrix, with q given in [R.L.U.]. Since the default covectors matrix is the identity matrix, the default axes are (qx,qy,qz,ω) in absolute units.

      The convention for the binning scheme is that:

      • The left edge of the first bin starts at binstart
      • The bin width is binwidth
      • The last bin contains binend
      • There are no "partial bins;" the last bin may contain values greater than binend. C.f. count_bins.

      A value can be binned by computing its bin index:

      coords = covectors * value
+bin_ix = 1 .+ floor.(Int64,(coords .- binstart) ./ binwidth)
      source
      Sunny.Bond — Type
      Bond(i, j, n)

      Represents a bond between atom indices i and j. n is a vector of three integers specifying unit cell displacement in terms of lattice vectors.

      source
      Sunny.Crystal — Type

      An object describing a crystallographic unit cell and its space group symmetry. Constructors are as follows:

      Crystal(filename; symprec=1e-5)

      Reads the crystal from a .cif file located at the path filename. The optional parameter symprec controls the precision tolerance for spacegroup symmetries.

      Crystal(latvecs, positions; types=nothing, symprec=1e-5)

      Constructs a crystal from the complete list of atom positions positions, with coordinates (between 0 and 1) in units of lattice vectors latvecs. Spacegroup symmetry information is automatically inferred. The optional parameter types is a list of strings, one for each atom, and can be used to break symmetry-equivalence between atoms.

      Crystal(latvecs, positions, spacegroup_number; types=nothing, setting=nothing, symprec=1e-5)

      Builds a crystal by applying symmetry operators for a given international spacegroup number. For certain spacegroups, there are multiple possible unit cell settings; in this case, a warning message will be printed, and a list of crystals will be returned, one for every possible setting. Alternatively, the optional setting string will disambiguate between unit cell conventions.

      Currently, crystals built using only the spacegroup number will be missing some symmetry information. It is generally preferred to build a crystal from a .cif file or from the full specification of the unit cell.

      Examples

      # Read a Crystal from a .cif file
+Crystal("filename.cif")
+
+# Build an FCC crystal using the primitive unit cell. The spacegroup number
+# 225 is inferred.
+latvecs = [1 1 0;
+            1 0 1;
+            0 1 1] / 2
+positions = [[0, 0, 0]]
+Crystal(latvecs, positions)
+
+# Build a CsCl crystal (two cubic sublattices). By providing distinct type
+# strings, the spacegroup number 221 is inferred.
+latvecs = lattice_vectors(1, 1, 1, 90, 90, 90)
+positions = [[0,0,0], [0.5,0.5,0.5]]
+types = ["Na", "Cl"]
+cryst = Crystal(latvecs, positions; types)
+
+# Build a diamond cubic crystal from its spacegroup number 227. This
+# spacegroup has two possible settings ("1" or "2"), which determine an
+# overall unit cell translation.
+latvecs = lattice_vectors(1, 1, 1, 90, 90, 90)
+positions = [[1, 1, 1] / 4]
+cryst = Crystal(latvecs, positions, 227; setting="1")

      See also lattice_vectors.

      source
      Sunny.FormFactor — Method
      FormFactor(ion::String; g_lande=2)

      The magnetic form factor for a given magnetic ion and charge state. When passed to an intensity_formula, determines a $|𝐪|$-dependent scaling of the structure factor.

      The parameter ion must be one of the following strings:

      Am2, Am3, Am4, Am5, Am6, Am7, Au1, Au2, Au3, Au4, Au5, Ce2, Co0, Co1, Co2, Co3,
+Co4, Cr0, Cr1, Cr2, Cr3, Cr4, Cu0, Cu1, Cu2, Cu3, Cu4, Dy2, Dy3, Er2, Er3, Eu2,
+Eu3, Fe0, Fe1, Fe2, Fe3, Fe4, Gd2, Gd3, Hf2, Hf3, Ho2, Ho3, Ir0a, Ir0b, Ir0c,
+Ir1a, Ir1b, Ir2, Ir3, Ir4, Ir5, Ir6, Mn0, Mn1, Mn2, Mn3, Mn4, Mo0, Mo1, Nb0,
+Nb1, Nd2, Nd3, Ni0, Ni1, Ni2, Ni3, Ni4, Np3, Np4, Np5, Np6, Os0a, Os0b, Os0c,
+Os1a, Os1b, Os2, Os3, Os4, Os5, Os6, Os7, Pd0, Pd1, Pr3, Pt1, Pt2, Pt3, Pt4,
+Pt5, Pt6, Pu3, Pu4, Pu5, Pu6, Re0a, Re0b, Re0c, Re1a, Re1b, Re2, Re3, Re4, Re5,
+Re6, Rh0, Rh1, Ru0, Ru1, Sc0, Sc1, Sc2, Sm2, Sm3, Ta2, Ta3, Ta4, Tb2, Tb3, Tc0,
+Tc1, Ti0, Ti1, Ti2, Ti3, Tm2, Tm3, U3, U4, U5, V0, V1, V2, V3, V4, W0a, W0b,
+W0c, W1a, W1b, W2c, W3, W4, W5, Y0, Yb2, Yb3, Zr0, Zr1

      The trailing number denotes ionization state. For example, "Fe0" denotes a neutral iron atom, while "Fe2" denotes Fe²⁺. If multiple electronic configurations are possible, they will be distinguished by a trailing letter (a, b, ...). Omitting this letter will print an informative error,

      FormFactor("Ir0")
+
+ERROR: Disambiguate form factor according to electronic configuration:
+    "Ir0a" -- 6s⁰5d⁹
+    "Ir0b" -- 6s¹5d⁸
+    "Ir0c" -- 6s²5d⁷

      The form factor is approximated as

      $F(s) = ⟨j_0(s)⟩ + \frac{2-g}{g} ⟨j_2(s)⟩ s^2$,

      involving the Landé $g$-factor. The $⟨j_l(s)⟩$ are radial integrals associated with the $l$th Bessel function of the magnetic dipole, where $s = |k|/4π$, and $|k|$ is the magnitude of momentum transfer.

      The radial integrals have been calculated using Hartree-Fock for transition metals, or Dirac-Fock for the rare earths and actinide series [1–3]. Sunny uses approximate fits as a sum of Gaussians,

      \[⟨j_0(s)⟩ = A e^{-as^2} + B e^{-bs^2} + C e^{-cs^2} + D e^{-ds^2} + E \ +⟨j_l(s)⟩ = (A e^{-as^2} + B e^{-bs^2} + C e^{-cs^2} + D e^{-ds^2} + E) s^2\]

      References:

      1. https://www.ill.eu/sites/ccsl/ffacts/ffachtml.html
      2. J. Brown, The Neutron Data Booklet, 2nd ed., Sec. 2.5 Magnetic Form Factors (2003)
      3. K. Kobayashi, T. Nagao, M. Ito, Acta Cryst. A, 67 pp 473–480 (2011)
      source
      Sunny.ImplicitMidpoint — Type
      ImplicitMidpoint(Δt::Float64; atol=1e-12) where N

      Energy-conserving spin dynamics. One call to the step! function will advance a System by Δt units of time.

      Uses the spherical midpoint integration scheme for dipole systems and the Schrödinger midpoint integration scheme for SU(N) spin systems. Both integration schemes are symplectic, and therefore avoid energy drift over long periods of simulation time.

      source
      Sunny.Langevin — Type
      Langevin(Δt::Float64; λ::Float64, kT::Float64)

      Spin dynamics with coupling to a Langevin thermostat, which includes damping and noise terms. One call to the step! function will advance a System by Δt units of time.

      Assuming ergodicity, the Langevin dynamics will sample from thermal equilibrium for the target temperature kT. The empirical parameter λ determines the strength of the coupling to the thermal bath. In other words, 1/λ is the decorrelation time-scale. If $λ = 0$, then the spin dynamics coincides with ImplicitMidpoint.

      An alternative approach to sampling is LocalSampler, which may be preferred when the allowed spin values become effective discrete (e.g. Ising spins).

      source
      Sunny.LocalSampler — Type
      LocalSampler(; kT, nsweeps=1.0, propose=propose_uniform)

      Monte Carlo simulation involving Metropolis updates to individual spins. One call to the step! function will perform nsweeps of MCMC sampling for a provided System. The default value of 1.0 means that step! performs, on average, one trial update per spin.

      Assuming ergodicity, the LocalSampler will sample from thermal equilibrium for the target temperature kT.

      The trial spin updates are sampled using the propose function. Built-in options include propose_uniform, propose_flip, and propose_delta. Multiple proposals can be mixed with the macro @mix_proposals.

      The returned object stores fields ΔE and Δs, which represent the cumulative change to the net energy and dipole, respectively.

      An alternative approach to sampling is Langevin, which may be preferred for simulating continuous spins, especially in the presence of long-range dipole-dipole interactions (cf. enable_dipole_dipole!).

      source
      Sunny.SpinInfo — Type
      SpinInfo(atom::Int; S, g=2)

      Characterizes the spin at a given atom index within the crystal unit cell. S is an integer multiple of 1/2 and gives the spin angular momentum in units of ħ. g is the g-factor or tensor, such that an angular momentum dipole $s$ produces a magnetic moment $g s$ in units of the Bohr magneton.

      source
      Sunny.SpinWaveTheory — Type
      SpinWaveTheory(sys, energy_ϵ::Float64=1e-8, energy_tol=1e-6)

      Constructs an object to perform linear spin wave theory. Use it with dispersion and dssf functions.

      The optional parameter energy_ϵ adds a small positive shift to the diagonal of the dynamical matrix $D$ to avoid numerical issues with zero-energy quasi-particle modes. The optional parameter energy_tol relaxes the check on the imaginary part of the eigenvalues.

      source
      Sunny.System — Method
      System(crystal::Crystal, latsize, infos, mode; units=Units.meV, seed::Int)

      Construct a System of spins for a given Crystal symmetry. The latsize parameter determines the number of unit cells in each lattice vector direction. The infos parameter is a list of SpinInfo objects, which determine the magnitude $S$ and $g$-tensor of each spin.

      The two primary options for mode are :SUN and :dipole. In the former, each spin-$S$ degree of freedom is described as an SU(N) coherent state, i.e. a quantum superposition of $N = 2S + 1$ levels. This formalism can be useful to capture multipolar spin fluctuations or local entanglement effects.

      Mode :dipole projects the SU(N) dynamics onto the restricted space of pure dipoles. In practice this means that Sunny will simulate Landau-Lifshitz dynamics, but single-ion anisotropy and biquadratic exchange interactions will be renormalized to improve accuracy. To disable this renormalization, use the mode :dipole_large_S which applies the $S → ∞$ classical limit. For details, see the documentation page: Interaction Strength Renormalization.

      The default units system of (meV, Å, tesla) can be overridden by with the units parameter; see Units.

      An optional seed may be provided to achieve reproducible random number generation.

      All spins are initially polarized in the $z$-direction.

      source
      Sunny.add_sample! — Method
      add_sample!(sc::SampledCorrelations, sys::System)

      add_trajectory uses the spin configuration contained in the System to generate a correlation data and accumulate it into sc. For static structure factors, this involves analyzing the spin-spin correlations of the spin configuration provided. For a dynamic structure factor, a trajectory is calculated using the given spin configuration as an initial condition. The spin-spin correlations are then calculating in time and accumulated into sc.

      This function will change the state of sys when calculating dynamical structure factor data. To preserve the initial state of sys, it must be saved separately prior to calling add_sample!. Alternatively, the initial spin configuration may be copied into a new System and this new System can be passed to add_sample!.

      source
      Sunny.available_energies — Method
      available_energies(sc::SampledCorrelations; negative_energies=false)

      Return the ω values for the energy index of a SampledCorrelations. By default, only returns values for non-negative energies, which corresponds to the default output of intensities. Set negative_energies to true to retrieve all ω values.

      source
      Sunny.available_wave_vectors — Method
      available_wave_vectors(sc::SampledCorrelations; bzsize=(1,1,1))

      Returns all wave vectors for which sc contains exact values. bsize specifies the number of Brillouin zones to be included.

      source
      Sunny.axes_bincenters — Method
      axes_bincenters(params::BinningParameters)

      Returns tick marks which label the bins of the histogram described by BinningParameters by their bin centers.

      The following alternative syntax can be used to compute bin centers for a single axis:

      axes_bincenters(binstart,binend,binwidth)
      source
      Sunny.broaden_energy — Method
      broaden_energy(sc::SampledCorrelations, vals, kernel::Function; negative_energies=false)

      Performs a real-space convolution along the energy axis of an array of intensities. Assumes the format of the intensities array corresponds to what would be returned by intensities_interpolated. kernel must be a function that takes two numbers: kernel(ω, ω₀), where ω is a frequency, and ω₀ is the center frequency of the kernel. Sunny provides lorentzian for the most common use case:

      newvals = broaden_energy(sc, vals, (ω, ω₀) -> lorentzian(ω-ω₀, 0.2))
      source
      Sunny.count_bins — Method
      count_bins(binstart,binend,binwidth)

      Returns the number of bins in the binning scheme implied by binstart, binend, and binwidth. To count the bins in a BinningParameters, use params.numbins.

      This function defines how partial bins are handled, so it should be used preferentially over computing the number of bins manually.

      source
      Sunny.dispersion — Method
      dispersion(swt::SpinWaveTheory, qs)

      Computes the spin excitation energy dispersion relations given a SpinWaveTheory and an array of wave vectors qs. Each element $q$ of qs must be a 3-vector in units of reciprocal lattice units. I.e., $qᵢ$ is given in $2π/|aᵢ|$ with $|aᵢ|$ the lattice constant of the original chemical lattice.

      The first indices of the returned array correspond to those of qs. A final index, corresponding to mode, is added to these. Each entry of the array is an energy.

      source
      Sunny.dmvec — Method
      dmvec(D)

      Antisymmetric matrix representation of the Dzyaloshinskii-Moriya pseudo-vector,

        [  0    D[3] -D[2]
+   -D[3]   0    D[1]
+    D[2] -D[1]   0  ]

      Useful in the context of set_exchange!.

      source
      Sunny.dssf — Method
      dssf(swt::SpinWaveTheory, qs)

      Given a SpinWaveTheory object, computes the dynamical spin structure factor,

      \[ 𝒮^{αβ}(𝐤, ω) = 1/(2πN)∫dt ∑_𝐫 \exp[i(ωt - 𝐤⋅𝐫)] ⟨S^α(𝐫, t)S^β(0, 0)⟩,\]

      using the result from linear spin-wave theory,

      \[ 𝒮^{αβ}(𝐤, ω) = ∑_n |A_n^{αβ}(𝐤)|^2 δ[ω-ω_n(𝐤)].\]

      qs is an array of wave vectors of arbitrary dimension. Each element $q$ of qs must be a 3-vector in reciprocal lattice units (RLU), i.e., in the basis of reciprocal lattice vectors.

      The first indices of the returned array correspond to those of qs. A final index, corresponding to mode, is added to these. Each entry of this array is a tensor (3×3 matrix) corresponding to the indices $α$ and $β$.

      source
      Sunny.dynamical_correlations — Method
      dynamical_correlations(sys::System; Δt, nω, ωmax, 
+    process_trajectory=:none, observables=nothing, correlations=nothing)

      Creates a SampledCorrelations for calculating and storing $𝒮(𝐪,ω)$ data. This information will be obtained by running dynamical spin simulations on equilibrium snapshots and measuring pair-correlations. The $𝒮(𝐪,ω)$ data can be retrieved by calling intensities_interpolated. Alternatively, instant_intensities_interpolated will integrate out $ω$ to obtain $𝒮(𝐪)$, optionally applying classical-to-quantum correction factors.

      The SampleCorrelations that is returned will contain no correlation data. Samples are generated and accumulated by calling add_sample!(sc, sys) where sc is a SampleCorrelations and sys is an appropriately equilibrated System. Note that the sys should be thermalized before each call of add_sample! such that the spin configuration in the system represents a new (fully decorrelated) sample.

      Three keywords are required to specify the dynamics used for the trajectory calculation.

      • Δt: The time step used for calculating the trajectory from which dynamic spin-spin correlations are calculated. The trajectories are calculated with an ImplicitMidpoint integrator.
      • ωmax: The maximum energy, $ω$, that will be resolved.
      • nω: The number of energy bins to calculated between 0 and ωmax.

      Additional keyword options are the following:

      • process_trajectory: Specifies a function that will be applied to the sample trajectory before correlation analysis. Current options are :none and :symmetrize. The latter will symmetrize the trajectory in time, which can be useful for removing Fourier artifacts that arise when calculating the correlations.
      • observables: Allows the user to specify custom observables. The observables must be given as a list of complex N×N matrices or LinearMaps. It's recommended to name each observable, for example: observables = [:A => a_observable_matrix, :B => b_map, ...]. By default, Sunny uses the 3 components of the dipole, :Sx, :Sy and :Sz.
      • correlations: Specify which correlation functions are calculated, i.e. which matrix elements $αβ$ of $𝒮^{αβ}(q,ω)$ are calculated and stored. Specified with a vector of tuples. By default Sunny records all auto- and cross-correlations generated by all observables. To retain only the xx and xy correlations, one would set correlations=[(:Sx,:Sx), (:Sx,:Sy)] or correlations=[(1,1),(1,2)].
      source
      Sunny.enable_dipole_dipole! — Method
      enable_dipole_dipole!(sys::System)

      Enables long-range dipole-dipole interactions,

      \[ -(μ_0/4π) ∑_{⟨ij⟩} (3 (𝐌_j⋅𝐫̂_{ij})(𝐌_i⋅𝐫̂_{ij}) - 𝐌_i⋅𝐌_j) / |𝐫_{ij}|^3\]

      where the sum is over all pairs of spins (singly counted), including periodic images, regularized using the Ewald summation convention. The magnetic moments are $𝐌_i = μ_B g 𝐒_i$ where $g$ is the g-factor or g-tensor, and $𝐒_i$ is the spin angular momentum dipole in units of ħ. The Bohr magneton $μ_B$ and vacuum permeability $μ_0$ are physical constants, with numerical values determined by the unit system.

      source
      Sunny.generate_mantid_script_from_binning_parameters — Method
      generate_mantid_script_from_binning_parameters(params::BinningParameters)

      Generate a Mantid script which bins data according to the given BinningParameters.

      Units

      Take care to ensure the units are correct (R.L.U. or absolute). You may want to call Sunny.bin_rlu_as_absolute_units! or Sunny.bin_absolute_units_as_rlu! first.

      source
      Sunny.global_position — Method
      global_position(sys::System, site::Site)

      Position of a Site in global coordinates.

      To precompute a full list of positions, one can use eachsite as below:

      pos = [global_position(sys, site) for site in eachsite(sys)]
      source
      Sunny.instant_correlations — Method
      instant_correlations(sys::System; process_trajectory=:none, observables=nothing, correlations=nothing)

      Creates a SampledCorrelations object for calculating and storing instantaneous structure factor intensities $𝒮(𝐪)$. This data will be calculated from the spin-spin correlations of equilibrium snapshots, absent any dynamical information. $𝒮(𝐪)$ data can be retrieved by calling instant_intensities_interpolated.

      Important note: When dealing with continuous (non-Ising) spins, consider creating using dynamical_correlations instead of instant_correlations. The former will provide full $𝒮(𝐪,ω)$ data, from which $𝒮(𝐪)$ can be obtained by integrating out $ω$. During this integration step, Sunny can incorporate temperature- and $ω$-dependent classical-to-quantum correction factors to produce more accurate $𝒮(𝐪)$ estimates. See instant_intensities_interpolated for more information.

      Prior to calling instant_correlations, ensure that sys represents a good equilibrium sample. Additional sample data may be accumulated by calling add_sample!(sc, sys) with newly equilibrated sys configurations.

      The following optional keywords are available:

      • process_trajectory: Specifies a function that will be applied to the sample trajectory before correlation analysis. Current options are :none and :symmetrize. The latter will symmetrize the trajectory in time, which can be useful for removing Fourier artifacts that arise when calculating the correlations.
      • observables: Allows the user to specify custom observables. The observables must be given as a list of complex N×N matrices or LinearMaps. It's recommended to name each observable, for example: observables = [:A => a_observable_matrix, :B => b_map, ...]. By default, Sunny uses the 3 components of the dipole, :Sx, :Sy and :Sz.
      • correlations: Specify which correlation functions are calculated, i.e. which matrix elements $αβ$ of $𝒮^{αβ}(q,ω)$ are calculated and stored. Specified with a vector of tuples. By default Sunny records all auto- and cross-correlations generated by all observables. To retain only the xx and xy correlations, one would set correlations=[(:Sx,:Sx), (:Sx,:Sy)] or correlations=[(1,1),(1,2)].
      source
      Sunny.instant_intensities_interpolated — Method
      instant_intensities_interpolated(sc::SampledCorrelations, qs; kwargs...)

      Return $𝒮(𝐪)$ intensities at wave vectors qs. The functionality is very similar to intensities_interpolated, except the returned array has dimensions identical to qs. If called on a SampledCorrelations with dynamical information, i.e., $𝒮(𝐪,ω)$, the $ω$ information is integrated out.

      source
      Sunny.integrate_axes! — Method
      integrate_axes!(params::BinningParameters; axes)

      Integrate over one or more axes of the histogram by setting the number of bins in that axis to 1. Examples:

      integrate_axes!(params; axes = [2,3])
+integrate_axes!(params; axes = 2)
      source
      Sunny.intensities_bands — Method
      dispersion, intensities = intensities_bands(swt::SpinWaveTheory, ks, formula::SpinWaveIntensityFormula)

      Computes the scattering intensities at each energy band for each momentum transfer k in ks, according to Linear Spin Wave Theory and the given intensity formula. The formula must have a delta-function kernel, e.g.:

      formula = intensity_formula(swt, :perp, formula; kernel = delta_function_kernel)

      or else the bands will be broadened, and their intensity can not be computed.

      The outputs will be arrays with indices identical to ks, with the last index giving the band index. dispersions reports the energy of each band, while intensities reports the scattering intensity.

      source
      Sunny.intensities_binned — Method
      intensity, counts = intensities_binned(sc::SampledCorrelations, params::BinningParameters, formula; integrated_kernel)

      Given correlation data contained in a SampledCorrelations and BinningParameters describing the shape of a histogram, compute the intensity and normalization for each histogram bin using a given intensity_formula.

      The BinningParameters are expected to accept (q,ω) in R.L.U. for the (possibly reshaped) crystal associated with sc.

      This is an alternative to intensities_interpolated which bins the scattering intensities into a histogram instead of interpolating between them at specified qs values. See unit_resolution_binning_parameters for a reasonable default choice of BinningParameters which roughly emulates intensities_interpolated with interpolation = :round.

      If a function integrated_kernel(Δω) is passed, it will be used as the CDF of a kernel function for energy broadening. For example, integrated_kernel = Δω -> atan(Δω/η)/pi (c.f. integrated_lorentzian implements Lorentzian broadening with parameter η. Energy-dependent energy broadening can be achieved by providing an integrated_kernel(ω,Δω) whose first argument is the energy transfer ω.

      Currently, energy broadening is only supported if the BinningParameters are such that the first three axes are purely spatial and the last (energy) axis is [0,0,0,1].

      source
      Sunny.intensities_broadened — Method
      intensities_broadened(swt::SpinWaveTheory, ks, ωvals, formula)

      Computes the scattering intensities at each (Q,ω) according to Linear Spin Wave Theory and the given intensity formula. The required formula must have a non-delta-function kernel, e.g.:

      formula = intensity_formula(swt, :perp; kernel = lorentzian(0.05))

      or else the intensity at ωvals which are not exactly on the dispersion curve can not be calculated.

      The intensity is computed at each wave vector in ks and each energy in ωvals. The output will be an array with indices identical to ks, with the last index matching ωvals.

      Note that ks is an array of wave vectors of arbitrary dimension. Each element $k$ of ks must be a 3-wavevector in absolute units.

      source
      Sunny.intensities_interpolated — Method
      intensities_interpolated(sc::SampledCorrelations, qs, formula:ClassicalIntensityFormula; interpolation=nothing, negative_energies=false)

      The basic function for retrieving $𝒮(𝐪,ω)$ information from a SampledCorrelations. Maps an array of wave vectors qs to an array of structure factor intensities, including an additional energy index. The values of $ω$ associated with the energy index can be retrieved by calling available_energies. The three coordinates of each wave vector are measured in reciprocal lattice units, i.e., multiples of the reciprocal lattice vectors.

      • interpolation: Since $𝒮(𝐪, ω)$ is calculated on a finite lattice, data is only available at discrete wave vectors. By default, Sunny will round a requested q to the nearest available wave vector. Linear interpolation can be applied by setting interpolation=:linear.
      • negative_energies: If set to true, Sunny will return the periodic extension of the energy axis. Most users will not want this.
      source
      Sunny.intensity_formula — Method
      formula = intensity_formula(sc::SampledCorrelations)

      Establish a formula for computing the intensity of the discrete scattering modes (q,ω) using the correlation data $𝒮^{αβ}(q,ω)$ stored in the SampledCorrelations. The formula returned from intensity_formula can be passed to intensities_interpolated or intensities_binned.

      intensity_formula(sc,...; kT = Inf, formfactors = ...)

      There are keyword arguments providing temperature and form factor corrections:

      • kT: If a temperature is provided, the intensities will be rescaled by a temperature- and ω-dependent classical-to-quantum factor. kT should be specified when making comparisons with spin wave calculations or experimental data. If kT is not specified, infinite temperature (no correction) is assumed.
      • formfactors: To apply form factor corrections, provide this keyword with a list of FormFactors, one for each symmetry-distinct site in the crystal. The order of FormFactors must correspond to the order of site symmetry classes, e.g., as they appear when printed in display(crystal).
      source
      Sunny.intensity_formula — Method

      A custom intensity formula can be specifed by providing a function intensity = f(q,ω,correlations) and specifying which correlations it requires:

      intensity_formula(f,sc::SampledCorrelations, required_correlations; kwargs...)

      The function is intended to be specified using do notation. For example, this custom formula sums the off-diagonal correlations:

      required = [(:Sx,:Sy),(:Sy,:Sz),(:Sx,:Sz)]
+intensity_formula(sc,required,return_type = ComplexF64) do k, ω, off_diagonal_correlations
+    sum(off_diagonal_correlations)
+end

      If your custom formula returns a type other than Float64, use the return_type keyword argument to flag this.

      source
      Sunny.intensity_formula — Method
      formula = intensity_formula(swt::SpinWaveTheory; kernel = ...)

      Establish a formula for computing the scattering intensity by diagonalizing the hamiltonian $H(q)$ using Linear Spin Wave Theory.

      If kernel = delta_function_kernel, then the resulting formula can be used with intensities_bands.

      If kernel is an energy broadening kernel function, then the resulting formula can be used with intensities_broadened. Energy broadening kernel functions can either be a function of Δω only, e.g.:

      kernel = Δω -> ...

      or a function of both the energy transfer ω and of Δω, e.g.:

      kernel = (ω,Δω) -> ...

      The integral of a properly normalized kernel function over all Δω is one.

      source
      Sunny.intensity_formula — Method
      intensity_formula([swt or sc], contraction_mode::Symbol)

      Sunny has several built-in formulas that can be selected by setting contraction_mode to one of these values:

      • :trace (default), which yields $\operatorname{tr} 𝒮(q,ω) = ∑_α 𝒮^{αα}(q,ω)$
      • :perp, which contracts $𝒮^{αβ}(q,ω)$ with the dipole factor $δ_{αβ} - q_{α}q_{β}$, returning the unpolarized intensity.
      • :full, which will return all elements $𝒮^{αβ}(𝐪,ω)$ without contraction.
      source
      Sunny.lattice_params — Method
      lattice_params(latvecs::Mat3)

      Compute the lattice parameters $(a, b, c, α, β, γ)$ for the three lattice vectors provided as columns of latvecs. The inverse mapping is lattice_vectors.

      source
      Sunny.lattice_vectors — Method
      lattice_vectors(a, b, c, α, β, γ)

      Return the lattice vectors, as columns of the $3×3$ output matrix, that correspond to the conventional unit cell defined by the lattice constants $(a, b, c)$ and the angles $(α, β, γ)$ in degrees. The inverse mapping is lattice_params.

      source
      Sunny.magnetic_moment — Method
      magnetic_moment(sys::System, site::Site)

      Get the magnetic moment for a Site. This is the spin dipole multiplied by the Bohr magneton and the local g-tensor.

      source
      Sunny.merge_correlations — Method
      merge_correlations(scs::Vector{SampledCorrelations)

      Accumulate a list of SampledCorrelations into a single, summary SampledCorrelations. Useful for reducing the results of parallel computations.

      source
      Sunny.minimize_energy! — Method
      minimize_energy!(sys::System{N}; maxiters=100, subiters=20,
+                 method=Optim.ConjugateGradient(), kwargs...) where N

      Optimizes the spin configuration in sys to minimize energy. A total of maxiters iterations will be attempted, with restarts after every subiters iterations. The remaining kwargs will be forwarded to the optimize method of the Optim.jl package.

      source
      Sunny.position_to_site — Method
      position_to_site(sys::System, r)

      Converts a position r to four indices of a Site. The coordinates of r are given in units of the lattice vectors for the original crystal. This function can be useful for working with systems that have been reshaped using reshape_supercell.

      Example

      # Find the `site` at the center of a unit cell which is displaced by four
+# multiples of the first lattice vector
+site = position_to_site(sys, [4.5, 0.5, 0.5])
+
+# Print the dipole at this site
+println(sys.dipoles[site])
      source
      Sunny.powder_average_binned — Method
      powder_average_binned(sc::SampledCorrelations, radial_binning_parameters; formula
+                     ω_binning_parameters, integrated_kernel = nothing, bzsize = nothing)

      This function emulates the experimental situation of "powder averaging," where only the magnitude (and not the direction) of the momentum transfer is resolvable. The intensities are binned similarly to intensities_binned, but the histogram x-axis is |k| in absolute units, which is a nonlinear function of kx,ky,kz. The y-axis is energy.

      Radial binning parameters are specified as tuples (start,end,bin_width), e.g. radial_binning_parameters = (0,6π,6π/55).

      Energy broadening is supported in the same way as intensities_binned, and this function accepts the same kind of intensity_formula.

      source
      Sunny.primitive_cell_shape — Method
      primitive_cell_shape(cryst::Crystal)

      Returns the shape of the primitive cell as a 3×3 matrix, in fractional coordinates of the conventional lattice vectors. May be useful for constructing inputs to reshape_supercell.

      Examples

      # Valid if `cryst` has not been reshaped
+@assert cryst.prim_latvecs ≈ cryst.latvecs * primitive_cell_shape(cryst)
      source
      Sunny.print_bond — Method
      print_bond(cryst::Crystal, bond::Bond; b_ref::Bond)

      Prints symmetry information for bond bond. A symmetry-equivalent reference bond b_ref can optionally be provided to fix the meaning of the coefficients A, B, ...

      source
      Sunny.print_site — Method
      print_site(cryst, i; R=I)

      Print symmetry information for the site i, including allowed g-tensor and allowed anisotropy operator. An optional rotation matrix R can be provided to define the reference frame for expression of the anisotropy.

      source
      Sunny.print_stevens_expansion — Method
      function print_stevens_expansion(op)

      Prints a local Hermitian operator as a linear combination of Stevens operators. The operator op may be a finite-dimensional matrix or an abstract spin polynomial in the large-$S$ limit.

      Examples

      S = spin_matrices(2)
+print_stevens_expansion(S[1]^4 + S[2]^4 + S[3]^4)
+# Prints: (1/20)𝒪₄₀ + (1/4)𝒪₄₄ + 102/5
+
+S = spin_matrices(Inf)
+print_stevens_expansion(S[1]^4 + S[2]^4 + S[3]^4)
+# Prints: (1/20)𝒪₄₀ + (1/4)𝒪₄₄ + (3/5)𝒮⁴
      source
      Sunny.print_suggested_frame — Method
      print_suggested_frame(cryst, i; digits=4)

      Print a suggested reference frame, as a rotation matrix R, that can be used as input to print_site(). The purpose is to simplify the description of allowed anisotropies.

      source
      Sunny.print_symmetry_table — Method
      print_symmetry_table(cryst::Crystal, max_dist)

      Print symmetry information for all equivalence classes of sites and bonds, up to a maximum bond distance of max_dist. Equivalent to calling print_bond(cryst, b) for every bond b in reference_bonds(cryst, max_dist), where Bond(i, i, [0,0,0]) refers to a single site i.

      source
      Sunny.print_wrapped_intensities — Method
      print_wrapped_intensities(sys::System; nmax=10)

      For Bravais lattices: Prints up to nmax wavevectors according to their instantaneous (static) structure factor intensities, listed in descending order. For non-Bravais lattices: Performs the same analysis for each spin sublattice independently; the output weights are naïvely averaged over sublattices, without incorporating phase shift information. This procedure therefore wraps all wavevectors into the first Brillouin zone. Each wavevector coordinate is given between $-1/2$ and $1/2$ in reciprocal lattice units (RLU). The output from this function will typically be used as input to suggest_magnetic_supercell.

      Because this function does not incorporate phase information in its averaging over sublattices, the printed weights are not directly comparable with experiment. For that purpose, use instant_correlations instead.

      source
      Sunny.propose_delta — Method
      propose_delta(magnitude)

      Generate a proposal function that adds a Gaussian perturbation to the existing spin state. In :dipole mode, the procedure is to first introduce a random three-vector perturbation $𝐬′ = 𝐬 + |𝐬| ξ$ and then return the properly normalized spin $|𝐬| (𝐬′/|𝐬′|)$. Each component of the random vector $ξ$ is Gaussian distributed with a standard deviation of magnitude; the latter is dimensionless and typically smaller than one.

      In :SUN mode, the procedure is analogous, but now involving Gaussian perturbations to each of the $N$ complex components of an SU(N) coherent state.

      In the limit of very large magnitude, this function coincides with propose_uniform.

      For use with LocalSampler.

      source
      Sunny.propose_flip — Method
      propose_flip

      Function to propose pure spin flip updates in the context of a LocalSampler. Dipoles are flipped as $𝐬 → -𝐬$. SU(N) coherent states are flipped using the time-reversal operator.

      source
      Sunny.propose_uniform — Function
      propose_uniform

      Function to propose a uniformly random spin update in the context of a LocalSampler. In :dipole mode, the result is a random three-vector with appropriate normalization. In :SUN mode, the result is a random SU(N) coherent state with appropriate normalization.

      source
      Sunny.reciprocal_space_path — Method
      reciprocal_space_path(cryst::Crystal, qs, density)

      Returns a pair (path, xticks). The path return value is a list of wavevectors that samples linearly between the provided wavevectors qs. The xticks return value can be used to label the special $𝐪$ values on the x-axis of a plot.

      Special note about units: the wavevectors qs must be provided in reciprocal lattice units (RLU) for the given crystal, but the sampling density must be specified in the global frame. Specifically, the density is given as number of sample points per unit of radian inverse length, where the unit of length is the same as that used to specify the lattice vectors of the Crystal. The path will therefore include more samples between q-points that are further apart in absolute Fourier distance.

      source
      Sunny.reciprocal_space_path_bins — Method
      reciprocal_space_path_bins(sc,qs,density,args...;kwargs...)

      Takes a list of wave vectors, qs in R.L.U., and builds a series of histogram BinningParameters whose first axis traces a path through the provided points. The second and third axes are integrated over according to the args and kwargs, which are passed through to slice_2D_binning_parameters.

      Also returned is a list of marker indices corresponding to the input points, and a list of ranges giving the indices of each histogram x-axis within a concatenated histogram. The density parameter is given in samples per reciprocal lattice unit (R.L.U.).

      source
      Sunny.reciprocal_space_shell — Method
      reciprocal_space_shell(cryst::Crystal, radius, n)

      Sample n points on the reciprocal space sphere with a given radius (units of inverse length).

      Examples

      # Sample wavevectors on the sphere at fixed density
+reciprocal_space_shell(cryst, r, 4π*r^2*density)
      source
      Sunny.reference_bonds — Method
      reference_bonds(cryst::Crystal, max_dist)

      Returns a full list of bonds, one for each symmetry equivalence class, up to distance max_dist. The reference bond b for each equivalence class is selected according to a scoring system that prioritizes simplification of the elements in basis_for_symmetry_allowed_couplings(cryst, b).

      source
      Sunny.remove_periodicity! — Method
      remove_periodicity!(sys::System, dims)

      Remove periodic interactions along the dimensions where dims is true. The system must support inhomogeneous interactions via to_inhomogeneous.

      Example

      # Remove periodic boundaries along the 1st and 3rd dimensions
+remove_periodicity!(sys::System, (true, false, true))
      source
      Sunny.reshape_supercell — Method
      reshape_supercell(sys::System, shape)

      Maps an existing System to a new one that has the shape and periodicity of a requested supercell. The columns of the $3×3$ integer matrix shape represent the supercell lattice vectors measured in units of the original crystal lattice vectors.

      source
      Sunny.resize_supercell — Method
      resize_supercell(sys::System{N}, latsize::NTuple{3,Int}) where N

      Creates a System with a given number of conventional unit cells in each lattice vector direction. Interactions and other settings will be inherited from sys.

      Convenience function for:

      reshape_supercell(sys, [latsize[1] 0 0; 0 latsize[2] 0; 0 0 latsize[3]])

      See also reshape_supercell.

      source
      Sunny.rotate_operator — Method
      rotate_operator(A, R)

      Rotates the local quantum operator A according to the $3×3$ rotation matrix R.

      source
      Sunny.rotation_in_rlu — Method
      rotation_in_rlu(cryst::Crystal, axis, angle)

      Returns a $3×3$ matrix that rotates wavevectors in reciprocal lattice units (RLU). The axis vector is a real-space direction in absolute units (but arbitrary magnitude), and the angle is in radians.

      source
      Sunny.set_coherent! — Method
      set_coherent!(sys::System, Z, site::Site)

      Set a coherent spin state at a Site using the $N$ complex amplitudes in Z.

      For a standard SpinInfo, these amplitudes will be interpreted in the eigenbasis of $𝒮̂ᶻ$. That is, Z[1] represents the amplitude for the basis state fully polarized along the $ẑ$-direction, and subsequent components represent states with decreasing angular momentum along this axis ($m = S, S-1, …, -S$).

      source
      Sunny.set_exchange! — Method
      set_exchange!(sys::System, J, bond::Bond; biquad=0, large_S=false)

      Sets a 3×3 spin-exchange matrix J along bond, yielding a pairwise interaction energy $𝐒_i⋅J 𝐒_j$. This interaction will be propagated to equivalent bonds in consistency with crystal symmetry. Any previous interactions on these bonds will be overwritten. The parameter bond has the form Bond(i, j, offset), where i and j are atom indices within the unit cell, and offset is a displacement in unit cells.

      The parameter J may be scalar or matrix-valued. As a convenience, dmvec(D) can be used to construct the antisymmetric part of the exchange, where D is the Dzyaloshinskii-Moriya pseudo-vector. The resulting interaction will be $𝐃⋅(𝐒_i×𝐒_j)$.

      The optional parameter biquad defines the strength $b$ for scalar biquadratic interactions of the form $b (𝐒_i⋅𝐒_j)²$. For systems restricted to dipoles, $b$ will be automatically renormalized for maximum consistency with the more variationally accurate SU(N) mode. Set large_S=true to work in the large-$S$ limit and disable this renormalization.

      Examples

      # An explicit exchange matrix
+J1 = [2 3 0;
+     -3 2 0;
+      0 0 2]
+set_exchange!(sys, J1, bond)
+
+# An equivalent Heisenberg + DM exchange 
+J2 = 2*I + dmvec([0,0,3])
+set_exchange!(sys, J2, bond)
      source
      Sunny.set_exchange_at! — Method
      set_exchange_at!(sys::System, J, site1::Site, site2::Site; biquad=0, large_S=false, offset=nothing)

      Sets the exchange interaction along the single bond connecting two Sites, ignoring crystal symmetry. The system must support inhomogeneous interactions via to_inhomogeneous.

      If the system is relatively small, the direction of the bond can be ambiguous due to possible periodic wrapping. Resolve this ambiguity by passing an explicit offset vector, in multiples of unit cells.

      See also set_exchange!.

      source
      Sunny.set_external_field_at! — Method
      set_external_field_at!(sys::System, B::Vec3, site::Site)

      Sets a Zeeman coupling between a field B and a single spin. Site includes a unit cell and a sublattice index.

      source
      Sunny.set_onsite_coupling! — Method
      set_onsite_coupling!(sys::System, op, i::Int)

      Set the single-ion anisotropy for the ith atom of every unit cell, as well as all symmetry-equivalent atoms. The local operator op will may be constructed as an abstract function of the local spin operators, as a polynomial of spin_matrices, or as a linear combination of stevens_matrices.

      Examples

      # An easy axis anisotropy in the z-direction
+set_onsite_coupling!(sys, S -> -D*S[3]^3, i)
+
+# The unique quartic single-ion anisotropy for a site with cubic point group
+# symmetry
+set_onsite_coupling!(sys, S -> 20*(S[1]^4 + S[2]^4 + S[3]^4), i)
+
+# An equivalent expression of this quartic anisotropy, up to a constant shift
+O = stevens_matrices(spin_irrep_label(sys, i))
+set_onsite_coupling!(sys, O[4,0] + 5*O[4,4], i)
      source
      Sunny.set_pair_coupling! — Method
      set_pair_coupling!(sys::System, coupling, bond)

      Sets an arbitrary coupling along bond. This coupling will be propagated to equivalent bonds in consistency with crystal symmetry. Any previous interactions on these bonds will be overwritten. The parameter bond has the form Bond(i, j, offset), where i and j are atom indices within the unit cell, and offset is a displacement in unit cells. The coupling is a represented as a matrix acting in the tensor product space of the two sites, and typically originates from to_product_space.

      Examples

      # Add a bilinear and biquadratic exchange
+S = spin_matrices(1/2)
+Si, Sj = to_product_space(S, S)
+set_pair_coupling!(sys, Si'*J1*Sj + (Si'*J2*Sj)^2, bond)
      source
      Sunny.set_spiral_order! — Method
      set_spiral_order!(sys; q, axis, S0)

      Initializes the system with a spiral order described by the wavevector q, an axis of rotation axis, and an initial dipole direction S0 at the real-space origin. The wavevector is expected in repicrocal lattice units (RLU), while the direction vectors axis and S0 are expected in global Cartesian coordinates.

      Example

      # Spiral order for a wavevector propagating in the direction of the first
+# reciprocal lattice vector (i.e., orthogonal to the lattice vectors ``𝐚_2``
+# and ``𝐚_3``), repeating with a period of 10 lattice constants, and spiraling
+# about the ``ẑ``-axis. The spin at the origin will point in the direction
+# ``𝐒_0 = ŷ + ẑ``.  Here, ``(x̂, ŷ, ẑ)`` are the axes of Cartesian coordinate
+# system in the global frame.
+set_spiral_order!(sys; q=[1/10, 0, 0], axis=[0, 0, 1], S0=[0, 1, 1])

      See also set_spiral_order_on_sublattice!.

      source
      Sunny.set_spiral_order_on_sublattice! — Method
      set_spiral_order_on_sublattice!(sys, i; q, axis, S0)

      Initializes sublattice i with a spiral order described by the wavevector q, an axis of rotation axis, and an initial dipole direction S0. The phase is selected such that the spin at sys.dipole[1,1,1,i] will point in the direction of S0. The wavevector is expected in repicrocal lattice units (RLU), while the direction vectors axis and S0 are expected in global Cartesian coordinates.

      This function is not available for systems with reshaped unit cells.

      See also set_spiral_order!.

      source
      Sunny.set_vacancy_at! — Method
      set_vacancy_at!(sys::System, site::Site)

      Make a single site nonmagnetic. Site includes a unit cell and a sublattice index.

      source
      Sunny.slice_2D_binning_parameters — Method
      slice_2D_binning_parameter(sc::SampledCorrelations, cut_from_q, cut_to_q, cut_bins::Int64, cut_width::Float64; plane_normal = [0,0,1],cut_height = cutwidth)

      Creates BinningParameters which make a cut along one dimension of Q-space.

      The x-axis of the resulting histogram consists of cut_bins-many bins ranging from cut_from_q to cut_to_q. The width of the bins in the transverse direciton is controlled by cut_width and cut_height.

      The binning in the transverse directions is defined in the following way, which sets their normalization and orthogonality properties:

      cut_covector = normalize(cut_to_q - cut_from_q)
+transverse_covector = normalize(plane_normal × cut_covector)
+cotransverse_covector = normalize(transverse_covector × cut_covector)

      In other words, the axes are orthonormal with respect to the Euclidean metric.

      If the cut is too narrow, there will be very few scattering vectors per bin, or the number per bin will vary substantially along the cut. If the output appears under-resolved, try increasing cut_width.

      The four axes of the resulting histogram are:

      1. Along the cut
      2. Fist transverse Q direction
      3. Second transverse Q direction
      4. Energy

      This function can be used without reference to a SampledCorrelations using this alternate syntax to manually specify the bin centers for the energy axis:

      slice_2D_binning_parameter(ω_bincenters, cut_from, cut_to,...)

      where ω_bincenters specifies the energy axis, and both cut_from and cut_to are arbitrary covectors, in any units.

      source
      Sunny.spin_irrep_label — Method
      spin_irrep_label(sys::System, i::Int)

      If atom i or site site carries a single spin-$S$ moment, then returns $S$. Otherwise, throws an error.

      source
      Sunny.spin_matrices — Method
      spin_matrices(S)

      Returns a triple of $N×N$spin matrices, whereN = 2S+1. These are the generators of SU(2) in the spin-S` representation.

      If S == Inf, then the return values are abstract symbols denoting infinite-dimensional matrices that commute. These can be useful for repeating historical studies, or modeling micromagnetic systems. A technical discussion appears in the Sunny documentation page: Interaction Strength Renormalization.

      Example

      S = spin_matrices(3/2)
+@assert S'*S ≈ (3/2)*(3/2+1)*I
+@assert S[1]*S[2] - S[2]*S[1] ≈ im*S[3]
+
+S = spin_matrices(Inf)
+@assert S[1]*S[2] - S[2]*S[1] == 0

      See also print_stevens_expansion.

      source
      Sunny.step! — Function
      step!(sys::System, dynamics)

      Advance the spin configuration one dynamical time-step. The dynamics object may be a continuous spin dynamics, such as Langevin or ImplicitMidpoint, or it may be a discrete Monte Carlo sampling scheme such as LocalSampler.

      source
      Sunny.stevens_matrices — Method
      stevens_matrices(S)

      Returns a generator of Stevens operators in the spin-S representation. The return value O can be indexed as O[k,q], where $0 ≤ k ≤ 6$ labels an irrep of SO(3) and $q = -k, …, k$. This will produce an $N×N$ matrix where $N = 2S + 1$. Linear combinations of Stevens operators can be used as a "physical basis" for decomposing local observables. To see this decomposition, use print_stevens_expansion.

      If S == Inf, then symbolic operators will be returned. In this infinite dimensional limit, the Stevens operators become homogeneous polynomials of commuting spin operators.

      Example

      O = stevens_matrices(2)
+S = spin_matrices(2)
+
+A = (1/20)O[4,0] + (1/4)O[4,4] + (102/5)I
+B = S[1]^4 + S[2]^4 + S[3]^4
+@assert A ≈ B

      See also spin_matrices and Interaction Strength Renormalization.

      source
      Sunny.subcrystal — Method
      subcrystal(cryst, types) :: Crystal

      Filters sublattices of a Crystal by atom types, keeping the space group unchanged.

      subcrystal(cryst, classes) :: Crystal

      Filters sublattices of Crystal by equivalence classes, keeping the space group unchanged.

      source
      Sunny.suggest_magnetic_supercell — Method
      suggest_magnetic_supercell(qs; tol=1e-12, maxsize=100)

      Suggests a magnetic supercell, in units of the crystal lattice vectors, that is consistent with periodicity of the wavevectors qs in RLU. If the wavevectors are incommensurate (with respect to the maximum supercell size maxsize), one can select a larger error tolerance tol to find a supercell that is almost commensurate.

      Prints a $3×3$ matrix of integers that is suitable for use in reshape_supercell.

      Examples

      # A magnetic supercell for a single-Q structure. Will print
+q1 = [0, -1/4, 1/4]
+suggest_magnetic_supercell([q1])       # [1 0 0; 0 2 1; 0 -2 1]
+
+# A larger magnetic supercell for a double-Q structure
+q2 = [1/4, 0, 1/4]
+suggest_magnetic_supercell([q1, q2])   # [1 2 2; -1 2 -2; -1 2 2]
+
+# If given incommensurate wavevectors, find an approximate supercell that
+# is exactly commensurate for nearby wavevectors.
+suggest_magnetic_supercell([[0, 0, 1/√5], [0, 0, 1/√7]]; tol=1e-2)
+
+# This prints [1 0 0; 0 1 0; 0 0 16], which becomes commensurate under the
+# approximations `1/√5 ≈ 7/16` and `1/√7 ≈ 3/8`.
      source
      Sunny.symmetry_equivalent_bonds — Method
      symmetry_equivalent_bonds(sys::System, bond::Bond)

      Given a Bond for the original (unreshaped) crystal, return all symmetry equivalent bonds in the System. Each returned bond is represented as a pair of Sites, which may be used as input to set_exchange_at!. Reverse bonds are not included (no double counting).

      Example

      for (site1, site2, offset) in symmetry_equivalent_bonds(sys, bond)
+    @assert site1 < site2
+    set_exchange_at!(sys, J, site1, site2; offset)
+end
      source
      Sunny.to_product_space — Method
      to_product_space(A, B, Cs...)

      Given lists of operators acting on local Hilbert spaces individually, return the corresponding operators that act on the tensor product space. In typical usage, the inputs will represent local physical observables and the outputs will be used to define quantum couplings.

      source
      Sunny.unit_resolution_binning_parameters — Method
      unit_resolution_binning_parameters(sc::SampledCorrelations)

      Create BinningParameters which place one histogram bin centered at each possible (q,ω) scattering vector of the crystal. This is the finest possible binning without creating bins with zero scattering vectors in them.

      This function can be used without reference to a SampledCorrelations using an alternate syntax to manually specify the bin centers for the energy axis and the lattice size:

      unit_resolution_binning_parameters(ω_bincenters,latsize,[reciprocal lattice vectors])

      The last argument may be a 3x3 matrix specifying the reciprocal lattice vectors, or a Crystal.

      Lastly, binning parameters for a single axis may be specifed by their bin centers:

      (binstart,binend,binwidth) = unit_resolution_binning_parameters(bincenters::Vector{Float64})
      source
      Sunny.@mix_proposals — Macro
      @mix_proposals weight1 propose1 weight2 propose2 ...

      Macro to generate a proposal function that randomly selects among the provided functions according to the provided probability weights. For use with LocalSampler.

      Example

      # A proposal function that proposes a spin flip 40% of the time, and a
+# Gaussian perturbation 60% of the time.
+@mix_proposals 0.4 propose_flip 0.6 propose_delta(0.2)
      source

      Optional Makie extensions

      The following will be enabled through a package extension if either GLMakie or WGLMakie is loaded.

      Sunny.plot_spins — Function
      plot_spins(sys::System; arrowscale=1.0, color=:red, show_cell=true,
+           orthographic=false, ghost_radius=0, dims=3)

      Plot the spin configuration defined by sys. Optional parameters include:

      • arrowscale: Scale all arrows by dimensionless factor.
      • color: Arrow color. May be a numeric value per site in system.
      • show_cell: Show original crystallographic unit cell.
      • orthographic: Use camera with orthographic projection.
      • ghost_radius: Show translucent periodic images up to a given distance (length units).
      • dims: Spatial dimensions of system (1, 2, or 3).
      source
      Sunny.view_crystal — Function
      view_crystal(crystal::Crystal, max_dist::Real; show_axis=true, orthographic=false)

      An interactive crystal viewer, with bonds up to max_dist.

      source

      Optional WriteVTK extensions

      The following will be enabled through a package extension if WriteVTK is loaded.

      Sunny.export_vtk — Function
      export_vtk(filename,params::BinningParameters,data)

      Export a VTK-compatible file to filename (do not include file extension when specifying the file name) which contains the data as VTK Cell Data on a grid parameterized by params.

      At least one axis of the BinningParameters must be integrated over, since VTK does not support 4D data. See integrate_axes!.

      source
      diff --git a/previews/PR178/parallelism.html b/previews/PR178/parallelism.html new file mode 100644 index 000000000..785bcb82f --- /dev/null +++ b/previews/PR178/parallelism.html @@ -0,0 +1,66 @@ + +Parallelizing Calculations · Sunny documentation
      GitHub

      Parallelizing Calculations

      Calculating structure factors with classical dynamics is computationally expensive, and Sunny does not currently parallelize these calculations at a fine-grained level. However, Julia provides facilities that allow users to run multiple simulations in parallel with only a little extra effort. We will look at two approaches to doing this: multithreading and Julia's Distributed package. We'll present these approaches in a series of code snippets that can be copied and pasted into your preferred Julia development environment.

      Review of the serial workflow

      The serial approach to calculating a structure factor, covered in FeI₂ at Finite Temperature, involves thermalizing a spin System and then calling add_sample!. add_sample! uses the state of the System as an initial condition for the calculation of a dynamical trajectory. The correlations of the trajectory are calculated and accumulated into a running average of the $𝒮(𝐪,ω)$. This sequence is repeated to generate additional samples.

      To illustrate, we'll set up a a simple model: a spin-1 antiferromagnet on an FCC crystal. 

      using Sunny, GLMakie
+
+function make_system(cryst; J, dims, seed=nothing)
+    sys = System(cryst, dims, [SpinInfo(1, S=1, g=2)], :dipole; seed)
+    set_exchange!(sys, J, Bond(1,1,[1,0,0]))
+    return sys
+end
+
+J = 1.0 # meV 
+cryst = Sunny.fcc_primitive_crystal()
+sys = make_system(cryst; J, dims=(10,10,2))

      To thermalize and generate equilibrium samples, we'll need a Langevin integrator. 

      kT = 0.5    # meV
+Δt = 0.05/J
+integrator = Langevin(Δt; kT, λ=0.1)

      Finally, call dynamical_correlations to configure a SampledCorrelations.

      sc = dynamical_correlations(sys; Δt=0.1, nω=100, ωmax=10.0)

      The serial calculation can now be performed as follows:

      nsamples = 10
+
+# Thermalize the system
+for _ in 1:5000  # Sufficient number of steps to thermalize
+    step!(sys, integrator)
+end
+
+for _ in 1:nsamples
+    # Generate new sample by running Langevin dynamics
+    for _ in 1:1000  # Sufficient number of steps to decorrelate
+        step!(sys, integrator)
+    end
+    # Add the sample to the correlation data
+    add_sample!(sc, sys)
+end

      This will take a second or two on a modern workstation, resulting in a single SampledCorrelations that contains 10 samples.

      Multithreading approach

      To use threads in Julia, you must launch your Julia environment appropriately.

      • From the command line, launch Julia with julia --threads=N, where N is the number of threads. If you don't know how many threads you'd like, you can let Julia decide with --threads=auto.

      • Jupyter notebook users will need to to set up a multithreaded Julia kernel and restart into this kernel. The kernel can be created inside Julia with the following:

          using IJulia
+    IJulia.installkernel("Julia Multithreaded",
+        env=Dict("JULIA_NUM_THREADS" => "auto"))
      • VSCode users should open their settings and search for Julia: Additional Args. There will be link called Edit in settings.json. Click on this, add "--threads=auto" to the list julia.additionalArgs, and start a new REPL.

      Before going further, make sure that Threads.nthreads() returns a number greater than 1.

      We will use multithreading in a very simple way, essentially employing a distributed memory approach to avoid conflicts around memory access. First preallocate a number of systems and correlations.

      npar = Threads.nthreads()
+systems = [make_system(cryst; J, dims=(10,10,2), seed=i) for i in 1:npar]
+scs = [dynamical_correlations(sys; Δt=0.1, nω=100, ωmax=10.0) for _ in 1:npar]
      Dealing with memory constraints

      If you have many threads available and are working with a large system, you may not have enough memory to store all these systems and correlations. In that case, simply reduce npar to a small enough value that you can make the necessary allocations.

      When the Threads.@threads macro is applied before a for loop, the iterations of the loop will execute in parallel using the available threads. We will put the entire thermalization and sampling process inside the loop, with each thread acting on a unique System and SampledCorrelations.

      Threads.@threads for id in 1:npar
+    integrator = Langevin(Δt; kT, λ=0.1)
+    for _ in 1:5000
+        step!(systems[id], integrator)
+    end
+    for _ in 1:nsamples
+        for _ in 1:1000
+            step!(systems[id], integrator)
+        end
+        add_sample!(scs[id], systems[id])
+    end
+end

      You may find this takes a little bit longer than the serial example, but, at the end of it, you will have generated npar correlation objects, each with 10 samples. We can merge these into a summary SampledCorrelations with 10*npar samples with merge_correlations:

      sc = merge_correlations(scs)

      Using Distributed

      Julia also provides a distributed memory approach to parallelism through the standard library package Distributed. This works by launching independent Julia environments on different "processes." An advantage of this approach is that it scales naturally to clusters since the processes are easily distributed across many different compute nodes. A disadvantage, especially when working on a single computer, is the increased memory overhead associated with launching all these environments.

      We begin by importing the package,

      using Distributed

      and launching some new processes. It is often sensible to create as many processes as there are cores.

      ncores = length(Sys.cpu_info())
+addprocs(ncores)

      You can think of each process as a separate computer running a fresh Julia environment, so we'll need to import Sunny and define our function in each of these environments. This is easily achieved with the @everywhere macro.

      @everywhere using Sunny
+
+@everywhere function make_system(cryst; J, dims, seed=nothing)
+    sys = System(cryst, dims, [SpinInfo(1, S=1, g=2)], :dipole; seed)
+    set_exchange!(sys, J, Bond(1,1,[1,0,0]))
+    return sys
+end

      A simple way to perform work on these processes is to use the parallel map function, pmap. This will apply a function to each element of some iterable, such as a list of numbers, and return a list of the results. It is a parallel map because these function calls may occur at the same time on different Julia processes. The pmap function takes care of distributing the work among the different processes and retrieving the results.

      In the example below, we give pmap a list of RNG seeds to iterate over, and we define the function that will be applied to each of these seeds in a do block. The contents of this block are essentially the same as what we put inside our parallel for loop in the multithreading example. The main difference is that the Systems and SampledCorrelations are not created in advance of the parallelization but are instead created inside each Julia process. The do block returns a SampledCorrelations, and the output of all the parallel computations are collected into list of SampledCorrelations called scs.

      scs = pmap(1:ncores) do seed
+    sys = make_system(Sunny.fcc_primitive_crystal(); J=1.0, dims=(10,10,2), seed)
+    sc = dynamical_correlations(sys; Δt=0.1, nω=100, ωmax=10.0)
+    integrator = Langevin(Δt; kT, λ=0.1)
+
+    for _ in 1:5000      # Thermalize
+        step!(sys, integrator)
+    end
+    for _ in 1:nsamples 
+        for _ in 1:1000 
+            step!(sys, integrator)
+        end
+        add_sample!(sc, sys)
+    end
+
+    return sc
+end

      Finally, merge the results into a summary SampledCorrelations.

      sc = merge_correlations(scs)
      diff --git a/previews/PR178/search_index.js b/previews/PR178/search_index.js new file mode 100644 index 000000000..97a13fdd6 --- /dev/null +++ b/previews/PR178/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"parallelism.html#Parallelizing-Calculations","page":"Parallelizing Calculations","title":"Parallelizing Calculations","text":"","category":"section"},{"location":"parallelism.html","page":"Parallelizing Calculations","title":"Parallelizing Calculations","text":"Calculating structure factors with classical dynamics is computationally expensive, and Sunny does not currently parallelize these calculations at a fine-grained level. However, Julia provides facilities that allow users to run multiple simulations in parallel with only a little extra effort. We will look at two approaches to doing this: multithreading and Julia's Distributed package. 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Quadrupolar fluctuations give rise to a single-ion bound state that cannot be described by a dipole-only model. This tutorial illustrates how to use the linear spin wave theory of SU(3) coherent states (i.e. 2-flavor bosons) to model the magnetic behavior in FeI₂. The original study was performed in Bai et al., Nature Physics 17, 467–472 (2021).","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The Fe atoms are arranged in stacked triangular layers. The effective spin Hamiltonian takes the form,","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"mathcalH=sum_(ij) 𝐒_i J_ij 𝐒_j - Dsum_i left(S_i^zright)^2","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"where the set of exchange matrices J_ij between bonded sites (ij) includes competing ferromagnetic and antiferromagnetic interactions. This model also includes a strong easy axis anisotropy, D 0.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"We will formulate this Hamiltonian in Sunny and then calculate its dynamic structure factor.","category":"page"},{"location":"examples/fei2_tutorial.html#Get-Julia-and-Sunny","page":"Case Study: FeI₂","title":"Get Julia and Sunny","text":"","category":"section"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"Sunny is implemented in Julia. This is a relatively new programming language that allows for interactive development (like Python or Matlab) while also providing high numerical efficiency (like C++ or Fortran). New Julia users may wish to take a look at our Getting Started with Julia guide. Sunny requires Julia 1.9 or later.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"From the Julia prompt, load Sunny. For plotting, one can choose either GLMakie (a pop-up window) or WGLMakie (inline plots for a Jupyter notebook or VSCode).","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"using Sunny, GLMakie","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"If these packages are not yet installed, Julia should offer to install them using its built-in package management system. If old versions are installed, you may need to update them to run this tutorial.","category":"page"},{"location":"examples/fei2_tutorial.html#Crystals","page":"Case Study: FeI₂","title":"Crystals","text":"","category":"section"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"A Crystal describes the crystallographic unit cell and will usually be loaded from a .cif file. Here, we instead build a crystal by listing all atoms and their types.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"a = b = 4.05012 # Lattice constants for triangular lattice\nc = 6.75214 # Spacing in the z-direction\n\nlatvecs = lattice_vectors(a, b, c, 90, 90, 120) # A 3x3 matrix of lattice vectors that\n # define the conventional unit cell\npositions = [[0, 0, 0], [1/3, 2/3, 1/4], [2/3, 1/3, 3/4]] # Positions of atoms in fractions\n # of lattice vectors\ntypes = [\"Fe\", \"I\", \"I\"]\nFeI2 = Crystal(latvecs, positions; types)","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"Observe that Sunny inferred the space group, 'P -3 m 1' (164) and labeled the atoms according to their point group symmetries.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"Only the Fe atoms are magnetic, so we discard the I ions using subcrystal.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"cryst = subcrystal(FeI2, \"Fe\")","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"Observe that cryst retains the spacegroup symmetry of the full FeI₂ crystal. This information will be used, for example, to propagate exchange interactions between symmetry-equivalent bonds.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"In a running Julia environment, the crystal can be viewed interactively using view_crystal.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"view_crystal(cryst, 8.0)","category":"page"},{"location":"examples/fei2_tutorial.html#Symmetry-analysis","page":"Case Study: FeI₂","title":"Symmetry analysis","text":"","category":"section"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The command print_symmetry_table provides a list of all the symmetry-allowed interactions up to a cutoff distance.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"print_symmetry_table(cryst, 8.0)","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The allowed g-tensor is expressed as a 3×3 matrix in the free coefficients A, B, ... The allowed single-ion anisotropy is expressed as a linear combination of Stevens operators. The latter correspond to polynomials of the spin operators, as we will describe below.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The allowed exchange interactions are given as a 3×3 matrix for representative bonds. The notation Bond(i, j, n) indicates a bond between atom indices i and j, with cell offset n. In the general case, it will be necessary to associate atom indices with their positions in the unit cell; these can be viewed with display(cryst). Note that the order of the pair (i j) is significant if the exchange tensor contains antisymmetric Dzyaloshinskii–Moriya (DM) interactions.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"In the case of FeI₂, Bond(1, 1, [1,0,0]) is one of the 6 nearest-neighbor Fe-Fe bonds on a triangular lattice layer, and Bond(1, 1, [0,0,1]) is an Fe-Fe bond between layers.","category":"page"},{"location":"examples/fei2_tutorial.html#Building-a-spin-System","page":"Case Study: FeI₂","title":"Building a spin System","text":"","category":"section"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"In constructing a spin System, we must provide several additional details about the spins.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"sys = System(cryst, (4, 4, 4), [SpinInfo(1, S=1, g=2)], :SUN, seed=2)","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"This system includes 444 unit cells, i.e. 64 Fe atoms, each with spin S=1 and a g-factor of 2. Quantum mechanically, spin S=1 involves a superposition of 2S+1=3 distinct angular momentum states. In :SUN mode, this superposition will be modeled explicitly using the formalism of SU(3) coherent states, which captures both dipolar and quadrupolar fluctuations. For the more traditional dipole dynamics, use :dipole mode instead.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"Next we will use set_exchange! to assign interaction to bonds. Sunny will automatically propagate each interaction to all symmetry-equivalent bonds in the unit cell. The FeI₂ interactions below follow Bai et al.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"J1pm = -0.236\nJ1pmpm = -0.161\nJ1zpm = -0.261\nJ2pm = 0.026\nJ3pm = 0.166\nJ′0pm = 0.037\nJ′1pm = 0.013\nJ′2apm = 0.068\n\nJ1zz = -0.236\nJ2zz = 0.113\nJ3zz = 0.211\nJ′0zz = -0.036\nJ′1zz = 0.051\nJ′2azz = 0.073\n\nJ1xx = J1pm + J1pmpm\nJ1yy = J1pm - J1pmpm\nJ1yz = J1zpm\n\nset_exchange!(sys, [J1xx 0.0 0.0;\n 0.0 J1yy J1yz;\n 0.0 J1yz J1zz], Bond(1,1,[1,0,0]))\nset_exchange!(sys, [J2pm 0.0 0.0;\n 0.0 J2pm 0.0;\n 0.0 0.0 J2zz], Bond(1,1,[1,2,0]))\nset_exchange!(sys, [J3pm 0.0 0.0;\n 0.0 J3pm 0.0;\n 0.0 0.0 J3zz], Bond(1,1,[2,0,0]))\nset_exchange!(sys, [J′0pm 0.0 0.0;\n 0.0 J′0pm 0.0;\n 0.0 0.0 J′0zz], Bond(1,1,[0,0,1]))\nset_exchange!(sys, [J′1pm 0.0 0.0;\n 0.0 J′1pm 0.0;\n 0.0 0.0 J′1zz], Bond(1,1,[1,0,1]))\nset_exchange!(sys, [J′2apm 0.0 0.0;\n 0.0 J′2apm 0.0;\n 0.0 0.0 J′2azz], Bond(1,1,[1,2,1]))","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The function set_onsite_coupling! assigns a single-ion anisotropy. The argument can be constructed using spin_matrices or stevens_matrices. Here we use Julia's anonymous function syntax to assign an easy-axis anisotropy along the direction hatz.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"D = 2.165\nset_onsite_coupling!(sys, S -> -D*S[3]^2, 1)","category":"page"},{"location":"examples/fei2_tutorial.html#Calculating-structure-factor-intensities","page":"Case Study: FeI₂","title":"Calculating structure factor intensities","text":"","category":"section"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"In the remainder of this tutorial, we will examine Sunny's tools for calculating the dynamical structure factor using a multi-boson generalization of linear spin wave theory (LSWT). This theory describes non-interacting quasi-particle excitations that hybridize dipolar and quadrupolar modes.","category":"page"},{"location":"examples/fei2_tutorial.html#Finding-the-ground-state","page":"Case Study: FeI₂","title":"Finding the ground state","text":"","category":"section"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"Begin with a random configuration and use minimize_energy! to find a configuration of the SU(3) coherent states (i.e. spin dipoles and quadrupoles) that locally minimizes energy.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"randomize_spins!(sys)\nminimize_energy!(sys)","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"A positive number above indicates that the procedure has converged to a local energy minimum. The configuration, however, may still have defects. This can be checked by visualizing the spins, colored according to their z-components.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"plot_spins(sys; color=[s[3] for s in sys.dipoles])","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"A different understanding of the magnetic ordering can be obtained by moving to Fourier space. The 'instant' structure factor 𝒮(𝐪) is an experimental observable. To investigate 𝒮(𝐪) as true 3D data, Sunny provides instant_correlations and related functions. Here, however, we will use print_wrapped_intensities, which gives average intensities for the individual Bravais sublattices (in effect, all wavevectors are wrapped to the first Brillouin zone).","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"print_wrapped_intensities(sys)","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The result will likely be approximately consistent with the known zero-field energy-minimizing magnetic structure of FeI₂, which is single-Q (two-up, two-down antiferromagnetic order). Mathematically, spontaneous symmetry breaking should select one of Q = 0 -14 14, 14 0 14, or -141414, associated with the three-fold rotational symmetry of the crystal spacegroup. In nature, however, one will frequently encounter competing \"domains\" associated with the three possible orientations of the ground state.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"If the desired ground state is already known, as with FeI₂, it could be entered by hand using set_dipole!. Alternatively, in the case of FeI₂, we could repeatedly employ the above randomization and minimization procedure until a defect-free configuration is found. Some systems will have more complicated ground states, which can be much more challenging to find. For this, Sunny provides experimental support for powerful simulated annealing via parallel tempering, but that is outside the scope of this tutorial.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"Here, let's break the three-fold symmetry of FeI₂ by hand. Given one or more desired Q modes, Sunny can suggest a magnetic supercell with appropriate periodicity. Let's arbitrarily select one of the three possible ordering wavevectors, Q = 0 -14 14. Sunny suggests a corresponding magnetic supercell in units of the crystal lattice vectors.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"suggest_magnetic_supercell([[0, -1/4, 1/4]])","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The system returned by reshape_supercell is smaller, and is sheared relative to the original system. This makes it much easier to find the global energy minimum.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"sys_min = reshape_supercell(sys, [1 0 0; 0 2 1; 0 -2 1])\nrandomize_spins!(sys_min)\nminimize_energy!(sys_min);\nnothing #hide","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"Plot the system again, now including \"ghost\" spins out to 12Å","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"plot_spins(sys_min; color=[s[3] for s in sys_min.dipoles], ghost_radius=12)","category":"page"},{"location":"examples/fei2_tutorial.html#Linear-spin-wave-theory","page":"Case Study: FeI₂","title":"Linear spin wave theory","text":"","category":"section"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"Now that we have found the ground state for a magnetic supercell, we can immediately proceed to perform zero-temperature calculations using linear spin wave theory. We begin by instantiating a SpinWaveTheory type using the supercell.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"swt = SpinWaveTheory(sys_min)","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"Select a sequence of wavevectors that will define a piecewise linear interpolation in reciprocal lattice units (RLU).","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"q_points = [[0,0,0], [1,0,0], [0,1,0], [1/2,0,0], [0,1,0], [0,0,0]];\nnothing #hide","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The function reciprocal_space_path will linearly sample a path between the provided q-points with a given density. The xticks return value provides labels for use in plotting.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"density = 50\npath, xticks = reciprocal_space_path(cryst, q_points, density);\nnothing #hide","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The dispersion function defines the quasiparticle excitation energies ω_i(𝐪) for each point 𝐪 along the reciprocal space path.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"disp = dispersion(swt, path);\nnothing #hide","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"In addition to the band energies ω_i(𝐪), Sunny can calculate the inelastic neutron scattering intensity I_i(𝐪) for each band i according to an intensity_formula. We choose to apply a polarization correction (1 - 𝐪𝐪) by setting the mode argument to :perp. Selecting delta_function_kernel specifies that we want the energy and intensity of each band individually.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"formula = intensity_formula(swt, :perp; kernel=delta_function_kernel)","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The function intensities_bands uses linear spin wave theory to calculate both the dispersion and intensity data for the provided path.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"disp, intensity = intensities_bands(swt, path, formula);\nnothing #hide","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"These can be plotted in GLMakie.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"fig = Figure()\nax = Axis(fig[1,1]; xlabel=\"Momentum (r.l.u.)\", ylabel=\"Energy (meV)\", xticks, xticklabelrotation=π/6)\nylims!(ax, 0.0, 7.5)\nxlims!(ax, 1, size(disp, 1))\ncolorrange = extrema(intensity)\nfor i in axes(disp)[2]\n lines!(ax, 1:length(disp[:,i]), disp[:,i]; color=intensity[:,i], colorrange)\nend\nfig","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"To make comparisons with inelastic neutron scattering (INS) data, it is helpful to employ an empirical broadening kernel, e.g., a lorentzian.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"γ = 0.15 # width in meV\nbroadened_formula = intensity_formula(swt, :perp; kernel=lorentzian(γ))","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The intensities_broadened function requires an energy range in addition to the 𝐪-space path.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"energies = collect(0:0.01:10) # 0 < ω < 10 (meV).\nis1 = intensities_broadened(swt, path, energies, broadened_formula);\nnothing #hide","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"A real FeI₂ sample will exhibit competing magnetic domains associated with spontaneous symmetry breaking of the 6-fold rotational symmetry of the triangular lattice. Note that the wavevectors 𝐪 and -𝐪 are equivalent in the structure factor, which leaves three distinct domain orientations, which are related by 120° rotations about the z-axis. Rather than rotating the spin configuration directly, on can rotate the 𝐪-space path. Below, we use rotation_in_rlu to average the intensities over all three possible orientations.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"R = rotation_in_rlu(cryst, [0, 0, 1], 2π/3)\nis2 = intensities_broadened(swt, [R*q for q in path], energies, broadened_formula)\nis3 = intensities_broadened(swt, [R*R*q for q in path], energies, broadened_formula)\nis_averaged = (is1 + is2 + is3) / 3\n\nfig = Figure()\nax = Axis(fig[1,1]; xlabel=\"Momentum (r.l.u.)\", ylabel=\"Energy (meV)\", xticks, xticklabelrotation=π/6)\nheatmap!(ax, 1:size(is_averaged, 1), energies, is_averaged)\nfig","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"This result can be directly compared to experimental neutron scattering data from Bai et al.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"(The publication figure accidentally used a non-standard coordinate system to label the wave vectors.)","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"To get this agreement, the use of SU(3) coherent states is essential. In other words, we needed a theory of multi-flavored bosons. The lower band has large quadrupolar character, and arises from the strong easy-axis anisotropy of FeI₂. By setting mode = :SUN, the calculation captures this coupled dipole-quadrupole dynamics.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"An interesting exercise is to repeat the same study, but using mode = :dipole instead of :SUN. That alternative choice would constrain the coherent state dynamics to the space of dipoles only.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The full dynamical spin structure factor (DSSF) can be retrieved as a 33 matrix with the dssf function, for a given path of 𝐪-vectors.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"disp, is = dssf(swt, path);\nnothing #hide","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The first output disp is identical to that obtained from dispersion. The second output is contains a list of 33 matrix of intensities. For example, is[q,n][2,3] yields the (yz) component of the structure factor intensity for nth mode at the qth wavevector in the path.","category":"page"},{"location":"examples/fei2_tutorial.html#What's-next?","page":"Case Study: FeI₂","title":"What's next?","text":"","category":"section"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The multi-boson linear spin wave theory, applied above, can be understood as the quantization of a certain generalization of the Landau-Lifshitz spin dynamics. Rather than dipoles, this dynamics takes places on the space of SU(N) coherent states.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The full SU(N) coherent state dynamics, with appropriate quantum correction factors, can be useful to model finite temperature scattering data. In particular, it captures certain anharmonic effects due to thermal fluctuations. This is the subject of our FeI₂ at Finite Temperature tutorial.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The classical dynamics is also a good starting point to study non-equilibrium phenomena. Empirical noise and damping terms can be used to model coupling to a thermal bath. This yields a Langevin dynamics of SU(N) coherent states. Our CP² Skyrmion Quench tutorial shows how this dynamics gives rise to the formation of novel topological defects in a temperature quench.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"Relative to LSWT calculations, it can take much more time to estimate mathcalS(𝐪ω) intensities using classical dynamics simulation. See the SunnyTutorials notebooks for examples of \"production-scale\" simulations.","category":"page"},{"location":"structure-factor.html#Structure-Factor-Calculations","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"","category":"section"},{"location":"structure-factor.html#Overview","page":"Structure Factor Calculations","title":"Overview","text":"","category":"section"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"The dynamical structure factor is of fundamental importance for characterizing a magnetic system, and facilitates quantitative comparison between theory and experimental scattering data.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"Consider, for example, a two-point dynamical spin correlation function, s^α(𝐱+Δ𝐱 t+Δt) s^β(𝐱 t). Here s^α(𝐱 t) represents the time dynamics of a spin dipole component α at position 𝐱, and brackets represent an average over equilibrium initial conditions and over (𝐱 t). The dynamical structure factor is defined as the Fourier transform of this two-point correlation in both space and time, up to an overall scaling factor. Using the convolution theorem, the result is,","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"𝒮^αβ(𝐪 ω) = frac1V s^α(𝐪 ω)^ast s^β(𝐪 ω) ","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"with V the system volume. We will restrict attention to lattice systems with periodic boundaries.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"Consider a crystal unit cell defined by three lattice vectors 𝐚_1 𝐚_2 𝐚_3, and linear system sizes L_1 L_2 L_3 measured in unit cells. The allowed momentum vectors take on discrete values 𝐪 = sum_α=1^3 m_α 𝐛_α L_α, where m_α are an integers and the reciprocal lattice vectors 𝐛_α are defined to satisfy 𝐚_α 𝐛_β = 2π δ_αβ. For a Bravais lattice, 𝐪 will be periodic in the first Brillouin zone, i.e., under any shift 𝐪 𝐪 𝐛_α. More generally, consider a non-Bravais lattice such that each unit cell may contain multiple spins. By partitioning spins s_j(𝐱t) according to their sublattice index j, the relevant momenta 𝐪 remain discretized as above, but now periodicity in the first Brillouin zone is lost. The structure factor may be written as a phase-average over the displacements between sublattices 𝐫_jk,","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"𝒮^αβ(𝐪 ω) = _jk e^i 𝐫_jk 𝐪 𝒮^αβ_jk(𝐪 ω) ","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"From a theoretical perspective, the quantity","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"𝒮^αβ_jk(𝐪 ω) = frac1V s_j^α(𝐪 ω)^ast s_k^β(𝐪 ω)","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"is fundamental. For each sublattice j, the data s_j^α(𝐪 ω) can be efficiently obtained by fast Fourier tranformation of a real space configuration s_j^α(𝐱 t). Internally, Sunny will calculate and store the discrete 𝒮^αβ_jk(𝐪 ω) correlation data, and use this to construct 𝒮^αβ(𝐪ω) intensities that can be compared with experiment.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"Calculating this structure factor involves several steps, with various possible settings. Sunny provides a number of tools to facilitate this calculation and to extract information from the results. These tools are briefly outlined below. Please see the Examples for a \"real life\" use case. Detailed function information is available in the Library API.","category":"page"},{"location":"structure-factor.html#Estimating-stucture-factors-with-classical-dynamics","page":"Structure Factor Calculations","title":"Estimating stucture factors with classical dynamics","text":"","category":"section"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"Classical dynamics may be used to estimate structure factor data by analyzing the spin-spin correlations of dynamical trajectories. This is fundamentally a Monte Carlo approach, as the trajectories must be started from an initial spin configuration that is sampled at thermal equilibrium. (Note that it is not possible to estimate a true T=0 dynamical structure factor using this method, but the temperature may be very low.) Samples are accumulated into a SampledCorrelations, from which intensity information may be extracted. The user does not typically build their own SampledCorrelations but instead initializes one by calling either dynamical_correlations or instant_correlations, as described below.","category":"page"},{"location":"structure-factor.html#Estimating-a-dynamical-structure-factor:-𝒮(𝐪,ω)","page":"Structure Factor Calculations","title":"Estimating a dynamical structure factor: 𝒮(𝐪ω)","text":"","category":"section"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"A SampledCorrelations for estimating the dynamical structure factor, 𝒮^αβ(𝐪ω), may be created by calling dynamical_correlations. This requires three keyword arguments. These will determine the dynamics used to calculate samples and, consequently, the ω information that will be available. ","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"Δt: Determines the step size used for simulating the dynamics. A smaller number will require proportionally more calculation time. While a smaller Δt will enable the resolution of higher energies, Δt is typically selected to ensure numerical stability rather than to maximize the largest ω value. A safe choice is to use the smaller value of Δt = 0.1/(J* S^2) or Δt = 0.1/(D * S), where S is magnetic moment of the largest local spin (as specified in SpinInfo), J is the parameter governing the largest bilinear interaction (e.g. exchange), and D is the parameter governing the largest single-site term of the Hamiltonian (e.g., anisotropy or Zeeman term).\nωmax: Sets the maximum resolved energy. Note that this is not independent of Δt. If ωmax too large, Sunny will throw an error and ask you to choose a smaller Δt. \nnω: Determines the number of energy bins to resolve. A larger number will require more calculation time.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"A sample may be added by calling add_sample!(sc, sys). The input sys must be a spin configuration in good thermal equilibrium, e.g., using the continuous Langevin dynamics or using single spin flip trials with LocalSampler. The statistical quality of the 𝒮^αβ(𝐪ω) can be improved by repeatedly generating decorrelated spin configurations in sys and calling add_sample! on each configuration.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"The outline of typical use case might look like this:","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"# Make a `SampledCorrelations`\nsc = dynamical_correlations(sys; Δt=0.05, ωmax=10.0, nω=100) \n\n# Add samples\nfor _ in 1:nsamples\n decorrelate_system(sys) # Perform some type of Monte Carlo simulation\n add_sample!(sc, sys) # Use spins to calculate trajectory and accumulate new sample of 𝒮(𝐪,ω)\nend","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"The calculation may be configured in a number of ways; see the dynamical_correlations documentation for a list of all keywords.","category":"page"},{"location":"structure-factor.html#Estimating-an-instantaneous-(\"static\")-structure-factor:-𝒮(𝐪)","page":"Structure Factor Calculations","title":"Estimating an instantaneous (\"static\") structure factor: 𝒮(𝐪)","text":"","category":"section"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"Sunny provides two methods for calculating instantaneous, or static, structure factors: 𝒮^αβ(𝐪). The first involves calculating spatial spin-spin correlations at single time slices. The second involves calculating a dynamic structure factor first and integrating out the ω information. The advantage of the latter approach is that it enables application of an ω-dependent classical-to-quantum rescaling of structure factor intensities, a method that should be preferred whenever comparing results to experimental data or spin wave calculations. A disadvantage of this approach is that it is computationally more expensive. There are also many cases when it is not straightforward to calculate a meaningful dynamics, as when working with Ising spins. In this section we will discuss how to calculate instantaneous structure factors from static spin configurations. Information about calculating instantaneous data from a dynamical correlations can be found in the following section.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"The basic usage for the instantaneous case is very similar to the dynamic case, except one calls instant_correlations instead of dynamical_correlations to configure a SampledCorrelations. Note that there are no required keywords as there is no need to specify any dynamics. instant_correlations will return a SampledCorrelations containing no data. Samples may be added by calling add_sample!(sc, sys), where sc is the SampledCorrelations. When performing a finite-temperature calculation, it is important to ensure that the spin configuration in the sys represents a good equilibrium sample, as in the dynamical case. Note, however, that we recommend calculating instantaneous correlations at finite temperature calculations by using full dynamics (i.e., using dynamical_correlations) and then integrating out the energy axis. An approach to doing this is described in the next section.","category":"page"},{"location":"structure-factor.html#Extracting-information-from-sampled-correlation-data","page":"Structure Factor Calculations","title":"Extracting information from sampled correlation data","text":"","category":"section"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"The basic function for extracting information from a SampledCorrelations at a particular wave vector, 𝐪, is intensities_interpolated. It takes a SampledCorrelations, a list of wave vectors, and an intensity_formula. The intensity_formula specifies how to contract and correct correlation data to arrive at a physical intensity. A simple example is formula = intensity_formula(sc, :perp), which will instruct Sunny apply polarization corrections: sum_αβ(I-q_α q_β) 𝒮^αβ(𝐪ω). An intensity at the wave vector 𝐪 = (𝐛_2 + 𝐛_3)2 may then be retrieved with intensities_interpolated(sf, [[0.0, 0.5, 0.5]], formula) . intensities_interpolated returns a list of nω elements at each wavevector. The corresponding ω values can be retrieved by calling available_energies on sf.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"Since Sunny only calculates the structure factor on a finite lattice when performing classical simulations, it is important to realize that exact information is only available at a discrete set of wave vectors. Specifically, for each axis index i, we will get information at q_i = fracnL_i, where n runs from (frac-L_i2+1) to fracL_i2 and L_i is the linear dimension of the lattice used for the calculation. If you request a wave vector that does not fall into this set, Sunny will automatically round to the nearest 𝐪 that is available. If intensities_interpolated is given the keyword argument interpolation=:linear, Sunny will use trilinear interpolation to determine a result at the requested wave vector. ","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"To retrieve the intensities at all wave vectors for which there is exact data, first call the function available_wave_vectors to generate a list of qs. This takes an optional keyword argument bzsize, which must be given a tuple of three integers specifying the number of Brillouin zones to calculate, e.g., bzsize=(2,2,2). The resulting list of wave vectors may then be passed to intensities_interpolated.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"Alternatively, intensities_binned can be used to place the exact data into histogram bins for comparison with experiment.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"The convenience function reciprocal_space_path returns a list of wavevectors sampled along a path that connects specified 𝐪 points. This list can be used as an input to intensities. Another convenience method, reciprocal_space_shell will generate points on a sphere of a given radius. This is useful for powder averaging. ","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"A number of arguments for intensity_formula are available which modify the calculation of structure factor intensity. It is generally recommended to provide a value of kT corresponding to the temperature of sampled configurations. Given kT, Sunny will include an energy- and temperature-dependent classical-to-quantum rescaling of intensities in the formula.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"To retrieve intensity data from a instantaneous structure factor, use instant_intensities_interpolated, which accepts similar arguments to intensities_interpolated. This function may also be used to calculate instantaneous information from a dynamical correlation data, i.e. from a SampledCorrelations created with dynamical_correlations. Note that it is important to supply a value to kT to reap the benefits of this approach over simply calculating a static structure factor at the outset. ","category":"page"},{"location":"writevtk.html#ParaView-Rendering","page":"ParaView Rendering","title":"ParaView Rendering","text":"","category":"section"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"The 4D correlation data produced by Sunny is too high-dimensional to visualize directly. This page describes how to export 3D slices of correlation data from Sunny to the Visual ToolKit (VTK) format, which is compatible with the ParaView visualization software. ParaView supports volumetric rendering:","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"","category":"page"},{"location":"writevtk.html#Simulation-data","page":"ParaView Rendering","title":"Simulation data","text":"","category":"section"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"First, generate some correlation data in Sunny. We will use a 2D lattice, since the correlation data S(Q_xQ_yomega) is 3D and can be exported in its entirety. The following code sets up the system, thermalizes it, and records the correlation data in a SampledCorrelations called dsf.","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"using Sunny\n\n# Single layer 12x12 periodic square lattice\nlatsize = (12,12,1)\n\nlatvecs = lattice_vectors(8.,8.,12.,90,100,90)\npositions = [[0,0,0]]\ntypes = [\"Cu\"]\nformfactors = [FormFactor(\"Cu2\")]\nxtal = Crystal(latvecs, positions; types)\n\nsys = System(xtal, latsize, [SpinInfo(1, S=1/2, g=2)], :SUN; seed=1)\n\nJ = 10.\nset_exchange!(sys, J, Bond(1,1,[1,0,0]))\nset_exchange!(sys, J, Bond(1,1,[0,1,0]))\n\nΔt = 0.01\nkT = 0.5\nlangevin = Langevin(Δt; λ=0.5, kT)\nrandomize_spins!(sys)\nfor i in 1:10_000 # Long enough to reach equilibrium\n step!(sys, langevin)\nend \n\nωmax=10.\n\ndsf = dynamical_correlations(sys\n ;Δt=Δt\n ,nω=48\n ,ωmax=ωmax\n ,process_trajectory=:symmetrize)\n\nnsamples = 10\nfor _ in 1:nsamples\n for _ in 1:1000 \n step!(sys, langevin)\n end\n add_sample!(dsf, sys)\nend","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"The default histogram BinningParameters are already integrated over the z direction because the system is 2D:","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"unit_resolution_binning_parameters(dsf)","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"⊡ 12 bins from -0.042 to +0.958 along [+1.27 dx] (Δ = 0.065)\n⊡ 12 bins from -0.042 to +0.958 along [+1.27 dy] (Δ = 0.065)\n∫ Integrated from +0.000 to +0.000 along [-0.33 dx +1.88 dz] (Δ = 0.524)\n⊡ 48 bins from -0.107 to +10.134 along [+1.00 dE] (Δ = 0.213)","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"The histogram is very oblong; it's approximately 1x1x10. To make it a nicer shape, we will rescale the energy axis to be be fractions of ωmax:","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"params = unit_resolution_binning_parameters(dsf)\nscale_factor = ωmax\nparams.binend[4] /= scale_factor\nparams.binstart[4] /= scale_factor\nparams.binwidth[4] /= scale_factor\nparams.covectors[4,:] ./= scale_factor","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"Doing this changes the last axis of the histogram to fit in [0,1]:","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"⊡ 49 bins from -0.011 to +1.013 along [+0.10 dE] (Δ = 0.213)","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"Now that our histogram is a cube, we compute the intensity in the histogram bins using the usual intensities_binned:","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"formula = intensity_formula(dsf,:trace)\nsignal, counts = intensities_binned(dsf, params; formula)\nintensity = signal ./ counts","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"Now that we have our intensity data and the binning parameters, we can export to VTK format using export_vtk and move to ParaView for the visualization.","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"# Importing WriteVTK enables Sunny's export-to-VTK functions\nimport WriteVTK\n\n# [1,2,4] specifies that the (x,y,z) axes in ParaView are (Qx,Qy,ω)\nexport_vtk(\"square_lattice\", params, intensity; dims_kept = [1,2,4])\n# Writes a file square_lattice.vti in the current directory","category":"page"},{"location":"writevtk.html#Loading-in-ParaView","page":"ParaView Rendering","title":"Loading in ParaView","text":"","category":"section"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"In ParaView, use File > Open to open square_lattice.vti. This will add the file to the Pipeline Browser with a closed eye icon, indicating that the data is ready to be loaded.","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"In the Properties panel, both bin_centers and data will be selected for import by default. Uncheck bin_centers because we don't need that information for the visualization. Click the green Apply button to load the data.","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"By default, only the outline of the data is shown in the 3D viewport. Since we adjusted the energy axis, the outline is a 1x1x1 cube. Optionally enable the axes grid under \"View\", and customize using the adjacent edit button.","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"To enable the volumetric render:","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"Select \"Volume\" from the \"Representation\" drop-down menu under \"Display\".\nThe \"Coloring\" drop-down should automatically select data because it's the only data loaded.\nOpen the Color Map Editor to adjust the opacity of the fog, which may be too faint to see by default.","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"Depending on your computer and your dataset size, the volumetric rendering may be slow, but our dataset is relatively small, so the render should be fast.","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"If nothing shows up at first, don't despair. Often, there are Bragg-like peaks in the correlation data which outshine everything else. To see this, enable Display Data Histogram in the Color Map Editor panel. To zoom in on the lower-intensity data, click and drag the right side handle of the opacity transfer function box to the middle a few times.","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"After suitable color mapping, the dispersion curve should become visible:","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"","category":"page"},{"location":"writevtk.html#Experiment-data","page":"ParaView Rendering","title":"Experiment data","text":"","category":"section"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"Note that since only the data and binning parameters are required for exporting to VTK, experiment data can be exported in the same way. For example, to visualize S(Q_xQ_yQ_z), do this:","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"# Load 4D histogram data from Mantid\nparams, signal = load_nxs(\"experiment_data.nxs\")\n\n# Integrate out the energy axis so we are 3D\nintegrate_axes!(params; axes = 4)\nsignal = sum(signal; dims = 4)\n\n# Export to ParaView\nexport_vtk(\"experiment_data_as_vtk\", params, signal)","category":"page"},{"location":"examples/ising2d.html","page":"Classical Ising model","title":"Classical Ising model","text":"EditURL = \"../../../examples/ising2d.jl\"","category":"page"},{"location":"examples/ising2d.html","page":"Classical Ising model","title":"Classical Ising model","text":"Download this example as Jupyter notebook or Julia script.","category":"page"},{"location":"examples/ising2d.html#Classical-Ising-model","page":"Classical Ising model","title":"Classical Ising model","text":"","category":"section"},{"location":"examples/ising2d.html","page":"Classical Ising model","title":"Classical Ising model","text":"This tutorial illustrates simulation of the classical 2D Ising model.","category":"page"},{"location":"examples/ising2d.html","page":"Classical Ising model","title":"Classical Ising model","text":"using Sunny, GLMakie","category":"page"},{"location":"examples/ising2d.html","page":"Classical Ising model","title":"Classical Ising model","text":"Sunny expects a 3D Crystal unit cell. To model a square lattice, we create an orthogonal unit cell where the z-spacing is distinct from the x and y spacing.","category":"page"},{"location":"examples/ising2d.html","page":"Classical Ising model","title":"Classical Ising model","text":"a = 1\nlatvecs = lattice_vectors(a,a,10a,90,90,90)\ncrystal = Crystal(latvecs, [[0,0,0]])","category":"page"},{"location":"examples/ising2d.html","page":"Classical Ising model","title":"Classical Ising model","text":"Create a System of spins with linear size L in the x and y directions, and only one layer in the z direction. The option :dipole means that the system will store Heisenberg spins, as opposed to SU(N) coherent states. Polarize the initial spin configuration using polarize_spins!. Following the Ising convention, we will restrict these spins to the z-axis and give them magnitude S=1.","category":"page"},{"location":"examples/ising2d.html","page":"Classical Ising model","title":"Classical Ising model","text":"By default, Sunny uses physical units, e.g. magnetic field in tesla. Here we specify an alternative Units system, so that the Zeeman coupling between the spin dipole 𝐬 and an external field 𝐁 has the dimensionless form -𝐁𝐬.","category":"page"},{"location":"examples/ising2d.html","page":"Classical Ising model","title":"Classical Ising model","text":"L = 128\nsys = System(crystal, (L,L,1), [SpinInfo(1, S=1, g=1)], :dipole, units=Units.theory, seed=0)\npolarize_spins!(sys, (0,0,1))","category":"page"},{"location":"examples/ising2d.html","page":"Classical Ising model","title":"Classical Ising model","text":"Use set_exchange! to include a ferromagnetic Heisenberg interaction along nearest-neighbor bonds. The Bond below connects two spins displaced by one lattice constant in the x-direction. This interaction will be propagated to all nearest-neighbors bonds in the system, consistent with the symmetries of the square lattice.","category":"page"},{"location":"examples/ising2d.html","page":"Classical Ising model","title":"Classical Ising model","text":"set_exchange!(sys, -1.0, Bond(1,1,(1,0,0)))","category":"page"},{"location":"examples/ising2d.html","page":"Classical Ising model","title":"Classical Ising model","text":"If an external field is desired, it can be set using set_external_field!.","category":"page"},{"location":"examples/ising2d.html","page":"Classical Ising model","title":"Classical Ising model","text":"B = 0\nset_external_field!(sys, (0,0,B))","category":"page"},{"location":"examples/ising2d.html","page":"Classical Ising model","title":"Classical Ising model","text":"The critical temperature for the Ising model is known analytically.","category":"page"},{"location":"examples/ising2d.html","page":"Classical Ising model","title":"Classical Ising model","text":"Tc = 2/log(1+√2)","category":"page"},{"location":"examples/ising2d.html","page":"Classical Ising model","title":"Classical Ising model","text":"Use a LocalSampler to perform nsweeps Monte Carlo sweeps. A sweep consists of, on average, one trial update per spin in the system. Each proposed update is accepted or rejected according to the Metropolis acceptance probability. As its name suggests, the propose_flip function will only propose pure spin flips, 𝐬 rightarrow -𝐬.","category":"page"},{"location":"examples/ising2d.html","page":"Classical Ising model","title":"Classical Ising model","text":"nsweeps = 4000\nsampler = LocalSampler(kT=Tc, propose=propose_flip)\nfor i in 1:nsweeps\n step!(sys, sampler)\nend","category":"page"},{"location":"examples/ising2d.html","page":"Classical Ising model","title":"Classical Ising model","text":"Plot the Ising spins by extracting the z-component of the dipoles","category":"page"},{"location":"examples/ising2d.html","page":"Classical Ising model","title":"Classical Ising model","text":"heatmap(reshape([s.z for s in sys.dipoles], (L,L)))","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"EditURL = \"../../../examples/powder_averaging.jl\"","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"Download this example as Jupyter notebook or Julia script.","category":"page"},{"location":"examples/powder_averaging.html#Powder-Averaged-CoRhO","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"","category":"section"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"This tutorial illustrates the calculation of the powder-averaged structure factor by performing an orientational average. We consider a simple model of the diamond-cubic crystal CoRh₂O₄, with parameters extracted from Ge et al., Phys. Rev. B 96, 064413.","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"using Sunny, GLMakie","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"Construct a diamond Crystal in the conventional (non-primitive) cubic unit cell. Sunny will populate all eight symmetry-equivalent sites when given the international spacegroup number 227 (\"Fd-3m\") and the appropriate setting. For this spacegroup, there are two conventional translations of the unit cell, and it is necessary to disambiguate through the setting keyword argument. (On your own: what happens if setting is omitted?)","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"a = 8.5031 # (Å)\nlatvecs = lattice_vectors(a, a, a, 90, 90, 90)\ncryst = Crystal(latvecs, [[0,0,0]], 227, setting=\"1\")","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"In a running Julia environment, the crystal can be viewed interactively using view_crystal.","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"view_crystal(cryst, 8.0)","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"Construct a System with an antiferromagnetic nearest neighbor interaction J. Because the diamond crystal is bipartite, the ground state will have unfrustrated Néel order. Selecting latsize=(1,1,1) is sufficient because the ground state is periodic over each cubic unit cell. By passing an explicit seed, the system's random number generator will give repeatable results.","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"latsize = (2, 2, 2)\nseed = 0\nS = 3/2\nJ = 7.5413*meV_per_K # (~ 0.65 meV)\nsys = System(cryst, latsize, [SpinInfo(1; S, g=2)], :dipole; seed=0)\nset_exchange!(sys, J, Bond(1, 3, [0,0,0]))","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"In the ground state, each spin is exactly anti-aligned with its 4 nearest-neighbors. Because every bond contributes an energy of -JS^2, the energy per site is -2JS^2. In this calculation, a factor of 1/2 avoids double-counting the bonds. Due to lack of frustration, direct energy minimization is successful in finding the ground state.","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"randomize_spins!(sys)\nminimize_energy!(sys)\n\n@assert energy_per_site(sys) ≈ -2J*S^2","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"Plotting the spins confirms the expected Néel order. Note that the overall, global rotation of dipoles is arbitrary.","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"s0 = sys.dipoles[1,1,1,1]\nplot_spins(sys; color=[s'*s0 for s in sys.dipoles])","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"For numerical efficiency, it will be helpful to work with the smallest possible magnetic supercell. Here, it is the primitive unit cell, which contains just two sites. The variable shape below defines the primitive lattice vectors cryst.prim_latvecs in units of the conventional lattice vectors. This result is used as input to reshape_supercell. The energy per site remains the same, which verifies that the magnetic supercell is valid.","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"shape = cryst.latvecs \\ cryst.prim_latvecs\nsys_prim = reshape_supercell(sys, shape)\n@assert energy_per_site(sys_prim) ≈ -2J*S^2","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"Now estimate 𝒮(𝐪ω) with SpinWaveTheory and an intensity_formula. The mode :perp contracts with a dipole factor to return the unpolarized intensity. The formula also employs lorentzian broadening. The isotropic FormFactor for Cobalt(2+) dampens intensities at large 𝐪.","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"swt = SpinWaveTheory(sys_prim)\nη = 0.4 # (meV)\nkernel = lorentzian(η)\nformfactors = [FormFactor(\"Co2\")]\nformula = intensity_formula(swt, :perp; kernel, formfactors)","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"For the \"single crystal\" result, we may use reciprocal_space_path to construct a path that connects high-symmetry points in reciprocal space. The intensities_broadened function collects intensities along this path for the given set of energy values.","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"qpoints = [[0.0, 0.0, 0.0], [0.5, 0.0, 0.0], [0.5, 0.5, 0.0], [0.0, 0.0, 0.0]]\npath, xticks = reciprocal_space_path(cryst, qpoints, 50)\nenergies = collect(0:0.01:6)\nis = intensities_broadened(swt, path, energies, formula)\n\nfig = Figure()\nax = Axis(fig[1,1]; aspect=1.4, ylabel=\"ω (meV)\", xlabel=\"𝐪 (r.l.u.)\",\n xticks, xticklabelrotation=π/10)\nheatmap!(ax, 1:size(is, 1), energies, is, colormap=:gnuplot2)\nfig","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"A powder measurement effectively involves an average over all possible crystal orientations. We use the function reciprocal_space_shell to sample n wavevectors on a sphere of a given radius (inverse angstroms), and then calculate the spherically-averaged intensity.","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"radii = 0.01:0.02:3 # (1/Å)\noutput = zeros(Float64, length(radii), length(energies))\nfor (i, radius) in enumerate(radii)\n n = 300\n qs = reciprocal_space_shell(cryst, radius, n)\n is = intensities_broadened(swt, qs, energies, formula)\n output[i, :] = sum(is, dims=1) / size(is, 1)\nend\n\nfig = Figure()\nax = Axis(fig[1,1]; xlabel=\"Q (Å⁻¹)\", ylabel=\"ω (meV)\")\nheatmap!(ax, radii, energies, output, colormap=:gnuplot2)\nfig","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"This result can be compared to experimental neutron scattering data from Fig. 5 of Ge et al.","category":"page"},{"location":"examples/powder_averaging.html","page":"Powder Averaged CoRh₂O₄","title":"Powder Averaged CoRh₂O₄","text":"","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"EditURL = \"../../../examples/out_of_equilibrium.jl\"","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"Download this example as Jupyter notebook or Julia script.","category":"page"},{"location":"examples/out_of_equilibrium.html#CP-Skyrmion-Quench","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"","category":"section"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"This example demonstrates Sunny's ability to simulate the out-of-equilibrium dynamics of generalized spin systems. We will implement the model Hamiltonian of Zhang et al., Nature Communications 14, 3626 (2023), which supports a novel type of topological defect, a CP² skyrmion, that involves both the dipolar and quadrupolar parts of a quantum spin.","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"Beginning from an initial high-temperature state, a disordered gas of CP² skyrmions can be formed by rapidly quenching to low temperature. To model the coupled dynamics of dipoles and quadrupoles, Sunny uses a recently developed generalization of the Landau-Lifshitz spin dynamics, Dahlbom et al., Phys. Rev. B 106, 235154 (2022).","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"using Sunny, GLMakie","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"The Hamiltonian we will implement,","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"mathcalH = sum_langle ij rangle J_ij( hatS_i^x hatS_j^x + hatS_i^y hatS_j^y + DeltahatS_i^z hatS_j^z) - hsum_ihatS_i^z + Dsum_i(hatS_i^z)^2","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"contains competing ferromagnetic nearest-neightbor and antiferromagnetic next-nearest-neighbor exchange terms on a triangular lattice. Both exchanges exhibit anisotropy on the z-term. Additionally, there is an external magnetic field, h, and easy-plane single-ion anisotropy, D 0. We begin by implementing the Crystal.","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"lat_vecs = lattice_vectors(1.0, 1.0, 2.0, 90, 90, 120)\nbasis_vecs = [[0,0,0]]\ncryst = Crystal(lat_vecs, basis_vecs)","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"The crystal is then used to create a spin System. All parameters in this model system are dimensionless, so we select \"theory\" units and set the g-factor to one.","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"L = 40\ndims = (L, L, 1)\nsys = System(cryst, dims, [SpinInfo(1, S=1, g=1)], :SUN; seed=101, units=Units.theory)","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"We proceed to implement each term of the Hamiltonian, selecting our parameters so that the system occupies a region of the phase diagram that supports skyrmions. The exchange interactions are set as follows.","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"J1 = -1 # Nearest-neighbor ferromagnetic\nJ2 = (2.0/(1+√5)) # Tune competing exchange to set skyrmion scale length\nΔ = 2.6 # Exchange anisotropy\n\nex1 = J1 * [1.0 0.0 0.0;\n 0.0 1.0 0.0;\n 0.0 0.0 Δ]\nex2 = J2 * [1.0 0.0 0.0;\n 0.0 1.0 0.0;\n 0.0 0.0 Δ]\nset_exchange!(sys, ex1, Bond(1, 1, [1, 0, 0]))\nset_exchange!(sys, ex2, Bond(1, 1, [1, 2, 0]))","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"Next we add the external field,","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"h = 15.5\nfield = set_external_field!(sys, [0.0 0.0 h])","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"and finally an easy-plane single-ion anisotropy,","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"D = 19.0\nset_onsite_coupling!(sys, S -> D*S[3]^2, 1)","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"Initialize system to an infinite temperature (fully randomized) initial condition.","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"randomize_spins!(sys)","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"We are now ready to simulate the quenching process using a generalized Langevin spin dynamics. If we were working with spin dipoles only, then Langevin dynamics would be the usual Landau-Lifshitz spin dynamics, augmented with damping and noise terms. In the present study, we are instead working with quantum spin-1 (an (N=3)-level system that includes both dipoles and quadrupoles). Here, Langevin captures the coupled dipole-quadrupole dynamics using the formalism of SU(N) coherent states.","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"Selecting kT = 0 in the Langevin dynamics will effective disable the noise term. Then the parameter λ effectively determines the damping time-scale.","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"Δt = 0.2/D # Integration time step (inverse meV). Typically this will be\n # inversely proportional to the largest energy scale in the\n # system. We can use a fairly large time-step here because\n # accuracy isn't critical.\nkT = 0 # Target equilibrium temperature (meV)\nλ = 0.1 # Magnitude of coupling to thermal bath (dimensionless)\nintegrator = Langevin(Δt; kT, λ)","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"Finally we run the dynamics. We will record the state of the system at three different times during the quenching process by copying the coherents field of the System.","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"τs = [4., 16, 256] # Times to record snapshots\nframes = [] # Empty array to store snapshots\nfor i in eachindex(τs)\n dur = i == 1 ? τs[1] : τs[i] - τs[i-1] # Determine the length of time to simulate\n numsteps = round(Int, dur/Δt)\n for _ in 1:numsteps # Perform the integration\n step!(sys, integrator)\n end\n push!(frames, copy(sys.coherents)) # Save a snapshot spin configuration\nend","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"To visualize the state of the system contained in each snapshot, we will calculate and plot the skyrmion density on each plaquette of our lattice. The function plot_triangular_plaquettes is not part of the core Sunny package, but rather something you could define yourself. We are using the definition in plotting2d.jl from the Sunny examples/extra directory.","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"include(pkgdir(Sunny, \"examples\", \"extra\", \"plotting2d.jl\"))\n\nfunction sun_berry_curvature(z₁, z₂, z₃)\n z₁, z₂, z₃ = normalize.((z₁, z₂, z₃))\n n₁ = z₁ ⋅ z₂\n n₂ = z₂ ⋅ z₃\n n₃ = z₃ ⋅ z₁\n return angle(n₁ * n₂ * n₃)\nend\n\nplot_triangular_plaquettes(sun_berry_curvature, frames; resolution=(1800,600),\n offset_spacing=10, texts = [\"\\tt = \"*string(τ) for τ in τs], text_offset = (0.0, 6.0)\n)","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"The times are given in hbarJ_1. The white background corresponds to a quantum paramagnetic state, where the local spin exhibits a strong quadrupole moment and little or no dipole moment. Observe that the process has generated a number of well-formed skyrmions of both positive (red) and negative (blue) charge in addition to a number of other metastable spin configurations. A full-sized version of this figure is available in Dahlbom et al..","category":"page"},{"location":"library.html#Library-API","page":"Library API","title":"Library API","text":"","category":"section"},{"location":"library.html","page":"Library API","title":"Library API","text":"This page describes the public types and functions exported by Sunny. This documentation can be also be accessed using the Julia help system (enter ? at the Julia command prompt).","category":"page"},{"location":"library.html","page":"Library API","title":"Library API","text":"","category":"page"},{"location":"library.html","page":"Library API","title":"Library API","text":"Sunny.plot_spins\nSunny.view_crystal\nSunny.export_vtk","category":"page"},{"location":"library.html","page":"Library API","title":"Library API","text":"Modules = [Sunny]\nPrivate = false","category":"page"},{"location":"library.html#Sunny.Site","page":"Library API","title":"Sunny.Site","text":"(cell1, cell2, cell3, i) :: Site\n\nFour indices identifying a single site in a System. The first three indices select the lattice cell and the last selects the sublattice (i.e., the atom within the unit cell).\n\nThis object can be used to index dipoles and coherents fields of a System. A Site is also required to specify inhomogeneous interactions via functions such as set_external_field_at! or set_exchange_at!.\n\nNote that the definition of a cell may change when a system is reshaped. In this case, it is convenient to construct the Site using position_to_site, which always takes a position in fractional coordinates of the original lattice vectors.\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.Units","page":"Library API","title":"Sunny.Units","text":"Units.meV\nUnits.theory\n\nThe unit system is implicitly determined by the definition of two physical constants: the vacuum permeability μ₀ and the Bohr magneton μ_B. Temperatures are effectively measured in units of energy (k_B = 1) and time is effectively measured in units of inverse energy (ħ = 1). The default unit system, Units.meV, employs (meV, Å, tesla). Select alternatively Units.theory for a units system defined so that μ₀ = μ_B = 1.\n\nSee also meV_per_K\n\n\n\n\n\n","category":"constant"},{"location":"library.html#Sunny.meV_per_K","page":"Library API","title":"Sunny.meV_per_K","text":"meV_per_K = 0.086173332621451774\n\nA physical constant. Useful for converting kelvin into the default energy units, meV.\n\n\n\n\n\n","category":"constant"},{"location":"library.html#Sunny.BinningParameters","page":"Library API","title":"Sunny.BinningParameters","text":"BinningParameters(binstart,binend,binwidth;covectors = I(4))\nBinningParameters(binstart,binend;numbins,covectors = I(4))\n\nDescribes a 4D parallelepided histogram in a format compatible with experimental Inelasitic Neutron Scattering data. See generate_mantid_script_from_binning_parameters to convert BinningParameters to a format understandable by the Mantid software, or load_nxs to load BinningParameters from a Mantid .nxs file.\n\nThe coordinates of the histogram axes are specified by multiplication of (q,ω) with each row of the covectors matrix, with q given in [R.L.U.]. Since the default covectors matrix is the identity matrix, the default axes are (qx,qy,qz,ω) in absolute units.\n\nThe convention for the binning scheme is that:\n\nThe left edge of the first bin starts at binstart\nThe bin width is binwidth\nThe last bin contains binend\nThere are no \"partial bins;\" the last bin may contain values greater than binend. C.f. count_bins.\n\nA value can be binned by computing its bin index:\n\ncoords = covectors * value\nbin_ix = 1 .+ floor.(Int64,(coords .- binstart) ./ binwidth)\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.Bond","page":"Library API","title":"Sunny.Bond","text":"Bond(i, j, n)\n\nRepresents a bond between atom indices i and j. n is a vector of three integers specifying unit cell displacement in terms of lattice vectors.\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.Crystal","page":"Library API","title":"Sunny.Crystal","text":"An object describing a crystallographic unit cell and its space group symmetry. Constructors are as follows:\n\nCrystal(filename; symprec=1e-5)\n\nReads the crystal from a .cif file located at the path filename. The optional parameter symprec controls the precision tolerance for spacegroup symmetries.\n\nCrystal(latvecs, positions; types=nothing, symprec=1e-5)\n\nConstructs a crystal from the complete list of atom positions positions, with coordinates (between 0 and 1) in units of lattice vectors latvecs. Spacegroup symmetry information is automatically inferred. The optional parameter types is a list of strings, one for each atom, and can be used to break symmetry-equivalence between atoms.\n\nCrystal(latvecs, positions, spacegroup_number; types=nothing, setting=nothing, symprec=1e-5)\n\nBuilds a crystal by applying symmetry operators for a given international spacegroup number. For certain spacegroups, there are multiple possible unit cell settings; in this case, a warning message will be printed, and a list of crystals will be returned, one for every possible setting. Alternatively, the optional setting string will disambiguate between unit cell conventions.\n\nCurrently, crystals built using only the spacegroup number will be missing some symmetry information. It is generally preferred to build a crystal from a .cif file or from the full specification of the unit cell.\n\nExamples\n\n# Read a Crystal from a .cif file\nCrystal(\"filename.cif\")\n\n# Build an FCC crystal using the primitive unit cell. The spacegroup number\n# 225 is inferred.\nlatvecs = [1 1 0;\n 1 0 1;\n 0 1 1] / 2\npositions = [[0, 0, 0]]\nCrystal(latvecs, positions)\n\n# Build a CsCl crystal (two cubic sublattices). By providing distinct type\n# strings, the spacegroup number 221 is inferred.\nlatvecs = lattice_vectors(1, 1, 1, 90, 90, 90)\npositions = [[0,0,0], [0.5,0.5,0.5]]\ntypes = [\"Na\", \"Cl\"]\ncryst = Crystal(latvecs, positions; types)\n\n# Build a diamond cubic crystal from its spacegroup number 227. This\n# spacegroup has two possible settings (\"1\" or \"2\"), which determine an\n# overall unit cell translation.\nlatvecs = lattice_vectors(1, 1, 1, 90, 90, 90)\npositions = [[1, 1, 1] / 4]\ncryst = Crystal(latvecs, positions, 227; setting=\"1\")\n\nSee also lattice_vectors.\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.FormFactor-Tuple{String}","page":"Library API","title":"Sunny.FormFactor","text":"FormFactor(ion::String; g_lande=2)\n\nThe magnetic form factor for a given magnetic ion and charge state. When passed to an intensity_formula, determines a 𝐪-dependent scaling of the structure factor.\n\nThe parameter ion must be one of the following strings:\n\nAm2, Am3, Am4, Am5, Am6, Am7, Au1, Au2, Au3, Au4, Au5, Ce2, Co0, Co1, Co2, Co3,\nCo4, Cr0, Cr1, Cr2, Cr3, Cr4, Cu0, Cu1, Cu2, Cu3, Cu4, Dy2, Dy3, Er2, Er3, Eu2,\nEu3, Fe0, Fe1, Fe2, Fe3, Fe4, Gd2, Gd3, Hf2, Hf3, Ho2, Ho3, Ir0a, Ir0b, Ir0c,\nIr1a, Ir1b, Ir2, Ir3, Ir4, Ir5, Ir6, Mn0, Mn1, Mn2, Mn3, Mn4, Mo0, Mo1, Nb0,\nNb1, Nd2, Nd3, Ni0, Ni1, Ni2, Ni3, Ni4, Np3, Np4, Np5, Np6, Os0a, Os0b, Os0c,\nOs1a, Os1b, Os2, Os3, Os4, Os5, Os6, Os7, Pd0, Pd1, Pr3, Pt1, Pt2, Pt3, Pt4,\nPt5, Pt6, Pu3, Pu4, Pu5, Pu6, Re0a, Re0b, Re0c, Re1a, Re1b, Re2, Re3, Re4, Re5,\nRe6, Rh0, Rh1, Ru0, Ru1, Sc0, Sc1, Sc2, Sm2, Sm3, Ta2, Ta3, Ta4, Tb2, Tb3, Tc0,\nTc1, Ti0, Ti1, Ti2, Ti3, Tm2, Tm3, U3, U4, U5, V0, V1, V2, V3, V4, W0a, W0b,\nW0c, W1a, W1b, W2c, W3, W4, W5, Y0, Yb2, Yb3, Zr0, Zr1\n\nThe trailing number denotes ionization state. For example, \"Fe0\" denotes a neutral iron atom, while \"Fe2\" denotes Fe²⁺. If multiple electronic configurations are possible, they will be distinguished by a trailing letter (a, b, ...). Omitting this letter will print an informative error,\n\nFormFactor(\"Ir0\")\n\nERROR: Disambiguate form factor according to electronic configuration:\n \"Ir0a\" -- 6s⁰5d⁹\n \"Ir0b\" -- 6s¹5d⁸\n \"Ir0c\" -- 6s²5d⁷\n\nThe form factor is approximated as\n\nF(s) = j_0(s) + frac2-gg j_2(s) s^2,\n\ninvolving the Landé g-factor. The j_l(s) are radial integrals associated with the lth Bessel function of the magnetic dipole, where s = k4π, and k is the magnitude of momentum transfer. \n\nThe radial integrals have been calculated using Hartree-Fock for transition metals, or Dirac-Fock for the rare earths and actinide series [1–3]. Sunny uses approximate fits as a sum of Gaussians,\n\nj_0(s) = A e^-as^2 + B e^-bs^2 + C e^-cs^2 + D e^-ds^2 + E \nj_l(s) = (A e^-as^2 + B e^-bs^2 + C e^-cs^2 + D e^-ds^2 + E) s^2\n\nReferences:\n\nhttps://www.ill.eu/sites/ccsl/ffacts/ffachtml.html\nJ. Brown, The Neutron Data Booklet, 2nd ed., Sec. 2.5 Magnetic Form Factors (2003)\nK. Kobayashi, T. Nagao, M. Ito, Acta Cryst. A, 67 pp 473–480 (2011)\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.ImplicitMidpoint","page":"Library API","title":"Sunny.ImplicitMidpoint","text":"ImplicitMidpoint(Δt::Float64; atol=1e-12) where N\n\nEnergy-conserving spin dynamics. One call to the step! function will advance a System by Δt units of time.\n\nUses the spherical midpoint integration scheme for dipole systems and the Schrödinger midpoint integration scheme for SU(N) spin systems. Both integration schemes are symplectic, and therefore avoid energy drift over long periods of simulation time.\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.Langevin","page":"Library API","title":"Sunny.Langevin","text":"Langevin(Δt::Float64; λ::Float64, kT::Float64)\n\nSpin dynamics with coupling to a Langevin thermostat, which includes damping and noise terms. One call to the step! function will advance a System by Δt units of time.\n\nAssuming ergodicity, the Langevin dynamics will sample from thermal equilibrium for the target temperature kT. The empirical parameter λ determines the strength of the coupling to the thermal bath. In other words, 1/λ is the decorrelation time-scale. If λ = 0, then the spin dynamics coincides with ImplicitMidpoint.\n\nAn alternative approach to sampling is LocalSampler, which may be preferred when the allowed spin values become effective discrete (e.g. Ising spins).\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.LocalSampler","page":"Library API","title":"Sunny.LocalSampler","text":"LocalSampler(; kT, nsweeps=1.0, propose=propose_uniform)\n\nMonte Carlo simulation involving Metropolis updates to individual spins. One call to the step! function will perform nsweeps of MCMC sampling for a provided System. The default value of 1.0 means that step! performs, on average, one trial update per spin.\n\nAssuming ergodicity, the LocalSampler will sample from thermal equilibrium for the target temperature kT. \n\nThe trial spin updates are sampled using the propose function. Built-in options include propose_uniform, propose_flip, and propose_delta. Multiple proposals can be mixed with the macro @mix_proposals.\n\nThe returned object stores fields ΔE and Δs, which represent the cumulative change to the net energy and dipole, respectively.\n\nAn alternative approach to sampling is Langevin, which may be preferred for simulating continuous spins, especially in the presence of long-range dipole-dipole interactions (cf. enable_dipole_dipole!).\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.SampledCorrelations","page":"Library API","title":"Sunny.SampledCorrelations","text":"SampledCorrelations\n\nBasic data type for storing sampled correlation data. A SampleCorrelations is initialized by calling either dynamical_correlations or instant_correlations.\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.SpinInfo","page":"Library API","title":"Sunny.SpinInfo","text":"SpinInfo(atom::Int; S, g=2)\n\nCharacterizes the spin at a given atom index within the crystal unit cell. S is an integer multiple of 1/2 and gives the spin angular momentum in units of ħ. g is the g-factor or tensor, such that an angular momentum dipole s produces a magnetic moment g s in units of the Bohr magneton.\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.SpinWaveTheory","page":"Library API","title":"Sunny.SpinWaveTheory","text":"SpinWaveTheory(sys, energy_ϵ::Float64=1e-8, energy_tol=1e-6)\n\nConstructs an object to perform linear spin wave theory. Use it with dispersion and dssf functions.\n\nThe optional parameter energy_ϵ adds a small positive shift to the diagonal of the dynamical matrix D to avoid numerical issues with zero-energy quasi-particle modes. The optional parameter energy_tol relaxes the check on the imaginary part of the eigenvalues.\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.System-Tuple{Crystal, Tuple{Int64, Int64, Int64}, Vector{SpinInfo}, Symbol}","page":"Library API","title":"Sunny.System","text":"System(crystal::Crystal, latsize, infos, mode; units=Units.meV, seed::Int)\n\nConstruct a System of spins for a given Crystal symmetry. The latsize parameter determines the number of unit cells in each lattice vector direction. The infos parameter is a list of SpinInfo objects, which determine the magnitude S and g-tensor of each spin.\n\nThe two primary options for mode are :SUN and :dipole. In the former, each spin-S degree of freedom is described as an SU(N) coherent state, i.e. a quantum superposition of N = 2S + 1 levels. This formalism can be useful to capture multipolar spin fluctuations or local entanglement effects. \n\nMode :dipole projects the SU(N) dynamics onto the restricted space of pure dipoles. In practice this means that Sunny will simulate Landau-Lifshitz dynamics, but single-ion anisotropy and biquadratic exchange interactions will be renormalized to improve accuracy. To disable this renormalization, use the mode :dipole_large_S which applies the S classical limit. For details, see the documentation page: Interaction Strength Renormalization.\n\nThe default units system of (meV, Å, tesla) can be overridden by with the units parameter; see Units. \n\nAn optional seed may be provided to achieve reproducible random number generation.\n\nAll spins are initially polarized in the z-direction.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.add_sample!-Tuple{SampledCorrelations, System}","page":"Library API","title":"Sunny.add_sample!","text":"add_sample!(sc::SampledCorrelations, sys::System)\n\nadd_trajectory uses the spin configuration contained in the System to generate a correlation data and accumulate it into sc. For static structure factors, this involves analyzing the spin-spin correlations of the spin configuration provided. For a dynamic structure factor, a trajectory is calculated using the given spin configuration as an initial condition. The spin-spin correlations are then calculating in time and accumulated into sc. \n\nThis function will change the state of sys when calculating dynamical structure factor data. To preserve the initial state of sys, it must be saved separately prior to calling add_sample!. Alternatively, the initial spin configuration may be copied into a new System and this new System can be passed to add_sample!.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.available_energies-Tuple{SampledCorrelations}","page":"Library API","title":"Sunny.available_energies","text":"available_energies(sc::SampledCorrelations; negative_energies=false)\n\nReturn the ω values for the energy index of a SampledCorrelations. By default, only returns values for non-negative energies, which corresponds to the default output of intensities. Set negative_energies to true to retrieve all ω values.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.available_wave_vectors-Tuple{SampledCorrelations}","page":"Library API","title":"Sunny.available_wave_vectors","text":"available_wave_vectors(sc::SampledCorrelations; bzsize=(1,1,1))\n\nReturns all wave vectors for which sc contains exact values. bsize specifies the number of Brillouin zones to be included.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.axes_bincenters-Tuple{Any, Any, Any}","page":"Library API","title":"Sunny.axes_bincenters","text":"axes_bincenters(params::BinningParameters)\n\nReturns tick marks which label the bins of the histogram described by BinningParameters by their bin centers.\n\nThe following alternative syntax can be used to compute bin centers for a single axis:\n\naxes_bincenters(binstart,binend,binwidth)\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.broaden_energy-Tuple{SampledCorrelations, Any, Function}","page":"Library API","title":"Sunny.broaden_energy","text":"broaden_energy(sc::SampledCorrelations, vals, kernel::Function; negative_energies=false)\n\nPerforms a real-space convolution along the energy axis of an array of intensities. Assumes the format of the intensities array corresponds to what would be returned by intensities_interpolated. kernel must be a function that takes two numbers: kernel(ω, ω₀), where ω is a frequency, and ω₀ is the center frequency of the kernel. Sunny provides lorentzian for the most common use case:\n\nnewvals = broaden_energy(sc, vals, (ω, ω₀) -> lorentzian(ω-ω₀, 0.2))\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.count_bins-Tuple{Any, Any, Any}","page":"Library API","title":"Sunny.count_bins","text":"count_bins(binstart,binend,binwidth)\n\nReturns the number of bins in the binning scheme implied by binstart, binend, and binwidth. To count the bins in a BinningParameters, use params.numbins.\n\nThis function defines how partial bins are handled, so it should be used preferentially over computing the number of bins manually.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.dispersion-Tuple{SpinWaveTheory, Any}","page":"Library API","title":"Sunny.dispersion","text":"dispersion(swt::SpinWaveTheory, qs)\n\nComputes the spin excitation energy dispersion relations given a SpinWaveTheory and an array of wave vectors qs. Each element q of qs must be a 3-vector in units of reciprocal lattice units. I.e., qᵢ is given in 2πaᵢ with aᵢ the lattice constant of the original chemical lattice.\n\nThe first indices of the returned array correspond to those of qs. A final index, corresponding to mode, is added to these. Each entry of the array is an energy.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.dmvec-Tuple{Any}","page":"Library API","title":"Sunny.dmvec","text":"dmvec(D)\n\nAntisymmetric matrix representation of the Dzyaloshinskii-Moriya pseudo-vector,\n\n [ 0 D[3] -D[2]\n -D[3] 0 D[1]\n D[2] -D[1] 0 ]\n\nUseful in the context of set_exchange!.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.dssf-Tuple{SpinWaveTheory, Any}","page":"Library API","title":"Sunny.dssf","text":"dssf(swt::SpinWaveTheory, qs)\n\nGiven a SpinWaveTheory object, computes the dynamical spin structure factor,\n\n 𝒮^αβ(𝐤 ω) = 1(2πN)dt _𝐫 expi(ωt - 𝐤𝐫) S^α(𝐫 t)S^β(0 0)\n\nusing the result from linear spin-wave theory,\n\n 𝒮^αβ(𝐤 ω) = _n A_n^αβ(𝐤)^2 δω-ω_n(𝐤)\n\nqs is an array of wave vectors of arbitrary dimension. Each element q of qs must be a 3-vector in reciprocal lattice units (RLU), i.e., in the basis of reciprocal lattice vectors.\n\nThe first indices of the returned array correspond to those of qs. A final index, corresponding to mode, is added to these. Each entry of this array is a tensor (3×3 matrix) corresponding to the indices α and β.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.dynamical_correlations-Union{Tuple{System{N}}, Tuple{N}} where N","page":"Library API","title":"Sunny.dynamical_correlations","text":"dynamical_correlations(sys::System; Δt, nω, ωmax, \n process_trajectory=:none, observables=nothing, correlations=nothing)\n\nCreates a SampledCorrelations for calculating and storing 𝒮(𝐪ω) data. This information will be obtained by running dynamical spin simulations on equilibrium snapshots and measuring pair-correlations. The 𝒮(𝐪ω) data can be retrieved by calling intensities_interpolated. Alternatively, instant_intensities_interpolated will integrate out ω to obtain 𝒮(𝐪), optionally applying classical-to-quantum correction factors.\n\nThe SampleCorrelations that is returned will contain no correlation data. Samples are generated and accumulated by calling add_sample!(sc, sys) where sc is a SampleCorrelations and sys is an appropriately equilibrated System. Note that the sys should be thermalized before each call of add_sample! such that the spin configuration in the system represents a new (fully decorrelated) sample.\n\nThree keywords are required to specify the dynamics used for the trajectory calculation.\n\nΔt: The time step used for calculating the trajectory from which dynamic spin-spin correlations are calculated. The trajectories are calculated with an ImplicitMidpoint integrator.\nωmax: The maximum energy, ω, that will be resolved.\nnω: The number of energy bins to calculated between 0 and ωmax.\n\nAdditional keyword options are the following:\n\nprocess_trajectory: Specifies a function that will be applied to the sample trajectory before correlation analysis. Current options are :none and :symmetrize. The latter will symmetrize the trajectory in time, which can be useful for removing Fourier artifacts that arise when calculating the correlations.\nobservables: Allows the user to specify custom observables. The observables must be given as a list of complex N×N matrices or LinearMaps. It's recommended to name each observable, for example: observables = [:A => a_observable_matrix, :B => b_map, ...]. By default, Sunny uses the 3 components of the dipole, :Sx, :Sy and :Sz.\ncorrelations: Specify which correlation functions are calculated, i.e. which matrix elements αβ of 𝒮^αβ(qω) are calculated and stored. Specified with a vector of tuples. By default Sunny records all auto- and cross-correlations generated by all observables. To retain only the xx and xy correlations, one would set correlations=[(:Sx,:Sx), (:Sx,:Sy)] or correlations=[(1,1),(1,2)].\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.eachsite-Tuple{System}","page":"Library API","title":"Sunny.eachsite","text":"eachsite(sys::System)\n\nAn iterator over all Sites in the system. \n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.enable_dipole_dipole!-Union{Tuple{System{N}}, Tuple{N}} where N","page":"Library API","title":"Sunny.enable_dipole_dipole!","text":"enable_dipole_dipole!(sys::System)\n\nEnables long-range dipole-dipole interactions,\n\n -(μ_04π) _ij (3 (𝐌_j𝐫_ij)(𝐌_i𝐫_ij) - 𝐌_i𝐌_j) 𝐫_ij^3\n\nwhere the sum is over all pairs of spins (singly counted), including periodic images, regularized using the Ewald summation convention. The magnetic moments are 𝐌_i = μ_B g 𝐒_i where g is the g-factor or g-tensor, and 𝐒_i is the spin angular momentum dipole in units of ħ. The Bohr magneton μ_B and vacuum permeability μ_0 are physical constants, with numerical values determined by the unit system.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.energy-Union{Tuple{System{N}}, Tuple{N}} where N","page":"Library API","title":"Sunny.energy","text":"energy(sys::System)\n\nThe total system energy. See also energy_per_site.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.energy_per_site-Union{Tuple{System{N}}, Tuple{N}} where N","page":"Library API","title":"Sunny.energy_per_site","text":"energy_per_site(sys::System)\n\nThe total system energy divided by the number of sites.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.generate_mantid_script_from_binning_parameters-Tuple{Any}","page":"Library API","title":"Sunny.generate_mantid_script_from_binning_parameters","text":"generate_mantid_script_from_binning_parameters(params::BinningParameters)\n\nGenerate a Mantid script which bins data according to the given BinningParameters.\n\nwarning: Units\nTake care to ensure the units are correct (R.L.U. or absolute). You may want to call Sunny.bin_rlu_as_absolute_units! or Sunny.bin_absolute_units_as_rlu! first.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.global_position-Tuple{System, Any}","page":"Library API","title":"Sunny.global_position","text":"global_position(sys::System, site::Site)\n\nPosition of a Site in global coordinates.\n\nTo precompute a full list of positions, one can use eachsite as below:\n\npos = [global_position(sys, site) for site in eachsite(sys)]\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.instant_correlations-Tuple{System}","page":"Library API","title":"Sunny.instant_correlations","text":"instant_correlations(sys::System; process_trajectory=:none, observables=nothing, correlations=nothing)\n\nCreates a SampledCorrelations object for calculating and storing instantaneous structure factor intensities 𝒮(𝐪). This data will be calculated from the spin-spin correlations of equilibrium snapshots, absent any dynamical information. 𝒮(𝐪) data can be retrieved by calling instant_intensities_interpolated.\n\nImportant note: When dealing with continuous (non-Ising) spins, consider creating using dynamical_correlations instead of instant_correlations. The former will provide full 𝒮(𝐪ω) data, from which 𝒮(𝐪) can be obtained by integrating out ω. During this integration step, Sunny can incorporate temperature- and ω-dependent classical-to-quantum correction factors to produce more accurate 𝒮(𝐪) estimates. See instant_intensities_interpolated for more information.\n\nPrior to calling instant_correlations, ensure that sys represents a good equilibrium sample. Additional sample data may be accumulated by calling add_sample!(sc, sys) with newly equilibrated sys configurations.\n\nThe following optional keywords are available:\n\nprocess_trajectory: Specifies a function that will be applied to the sample trajectory before correlation analysis. Current options are :none and :symmetrize. The latter will symmetrize the trajectory in time, which can be useful for removing Fourier artifacts that arise when calculating the correlations.\nobservables: Allows the user to specify custom observables. The observables must be given as a list of complex N×N matrices or LinearMaps. It's recommended to name each observable, for example: observables = [:A => a_observable_matrix, :B => b_map, ...]. By default, Sunny uses the 3 components of the dipole, :Sx, :Sy and :Sz.\ncorrelations: Specify which correlation functions are calculated, i.e. which matrix elements αβ of 𝒮^αβ(qω) are calculated and stored. Specified with a vector of tuples. By default Sunny records all auto- and cross-correlations generated by all observables. To retain only the xx and xy correlations, one would set correlations=[(:Sx,:Sx), (:Sx,:Sy)] or correlations=[(1,1),(1,2)].\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.instant_intensities_interpolated-Tuple{SampledCorrelations, Any, Any}","page":"Library API","title":"Sunny.instant_intensities_interpolated","text":"instant_intensities_interpolated(sc::SampledCorrelations, qs; kwargs...)\n\nReturn 𝒮(𝐪) intensities at wave vectors qs. The functionality is very similar to intensities_interpolated, except the returned array has dimensions identical to qs. If called on a SampledCorrelations with dynamical information, i.e., 𝒮(𝐪ω), the ω information is integrated out.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.integrate_axes!-Tuple{BinningParameters}","page":"Library API","title":"Sunny.integrate_axes!","text":"integrate_axes!(params::BinningParameters; axes)\n\nIntegrate over one or more axes of the histogram by setting the number of bins in that axis to 1. Examples:\n\nintegrate_axes!(params; axes = [2,3])\nintegrate_axes!(params; axes = 2)\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.integrated_lorentzian-Tuple{Float64}","page":"Library API","title":"Sunny.integrated_lorentzian","text":"integrated_lorentzian(η)\n\nReturns x mapsto atan(xη)π for use with intensities_binned.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.intensities_bands-Tuple{SpinWaveTheory, Any, Sunny.SpinWaveIntensityFormula}","page":"Library API","title":"Sunny.intensities_bands","text":"dispersion, intensities = intensities_bands(swt::SpinWaveTheory, ks, formula::SpinWaveIntensityFormula)\n\nComputes the scattering intensities at each energy band for each momentum transfer k in ks, according to Linear Spin Wave Theory and the given intensity formula. The formula must have a delta-function kernel, e.g.:\n\nformula = intensity_formula(swt, :perp, formula; kernel = delta_function_kernel)\n\nor else the bands will be broadened, and their intensity can not be computed.\n\nThe outputs will be arrays with indices identical to ks, with the last index giving the band index. dispersions reports the energy of each band, while intensities reports the scattering intensity.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.intensities_binned-Tuple{SampledCorrelations, BinningParameters, Sunny.ClassicalIntensityFormula}","page":"Library API","title":"Sunny.intensities_binned","text":"intensity, counts = intensities_binned(sc::SampledCorrelations, params::BinningParameters, formula; integrated_kernel)\n\nGiven correlation data contained in a SampledCorrelations and BinningParameters describing the shape of a histogram, compute the intensity and normalization for each histogram bin using a given intensity_formula.\n\nThe BinningParameters are expected to accept (q,ω) in R.L.U. for the (possibly reshaped) crystal associated with sc.\n\nThis is an alternative to intensities_interpolated which bins the scattering intensities into a histogram instead of interpolating between them at specified qs values. See unit_resolution_binning_parameters for a reasonable default choice of BinningParameters which roughly emulates intensities_interpolated with interpolation = :round.\n\nIf a function integrated_kernel(Δω) is passed, it will be used as the CDF of a kernel function for energy broadening. For example, integrated_kernel = Δω -> atan(Δω/η)/pi (c.f. integrated_lorentzian implements Lorentzian broadening with parameter η. Energy-dependent energy broadening can be achieved by providing an integrated_kernel(ω,Δω) whose first argument is the energy transfer ω.\n\nCurrently, energy broadening is only supported if the BinningParameters are such that the first three axes are purely spatial and the last (energy) axis is [0,0,0,1].\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.intensities_broadened-Tuple{SpinWaveTheory, Any, Any, Any}","page":"Library API","title":"Sunny.intensities_broadened","text":"intensities_broadened(swt::SpinWaveTheory, ks, ωvals, formula)\n\nComputes the scattering intensities at each (Q,ω) according to Linear Spin Wave Theory and the given intensity formula. The required formula must have a non-delta-function kernel, e.g.:\n\nformula = intensity_formula(swt, :perp; kernel = lorentzian(0.05))\n\nor else the intensity at ωvals which are not exactly on the dispersion curve can not be calculated.\n\nThe intensity is computed at each wave vector in ks and each energy in ωvals. The output will be an array with indices identical to ks, with the last index matching ωvals.\n\nNote that ks is an array of wave vectors of arbitrary dimension. Each element k of ks must be a 3-wavevector in absolute units.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.intensities_interpolated-Tuple{SampledCorrelations, Any, Sunny.ClassicalIntensityFormula}","page":"Library API","title":"Sunny.intensities_interpolated","text":"intensities_interpolated(sc::SampledCorrelations, qs, formula:ClassicalIntensityFormula; interpolation=nothing, negative_energies=false)\n\nThe basic function for retrieving 𝒮(𝐪ω) information from a SampledCorrelations. Maps an array of wave vectors qs to an array of structure factor intensities, including an additional energy index. The values of ω associated with the energy index can be retrieved by calling available_energies. The three coordinates of each wave vector are measured in reciprocal lattice units, i.e., multiples of the reciprocal lattice vectors.\n\ninterpolation: Since 𝒮(𝐪 ω) is calculated on a finite lattice, data is only available at discrete wave vectors. By default, Sunny will round a requested q to the nearest available wave vector. Linear interpolation can be applied by setting interpolation=:linear.\nnegative_energies: If set to true, Sunny will return the periodic extension of the energy axis. Most users will not want this.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.intensity_formula-Tuple{Function, SampledCorrelations, AbstractVector{Int64}}","page":"Library API","title":"Sunny.intensity_formula","text":"formula = intensity_formula(sc::SampledCorrelations)\n\nEstablish a formula for computing the intensity of the discrete scattering modes (q,ω) using the correlation data 𝒮^αβ(qω) stored in the SampledCorrelations. The formula returned from intensity_formula can be passed to intensities_interpolated or intensities_binned.\n\nintensity_formula(sc,...; kT = Inf, formfactors = ...)\n\nThere are keyword arguments providing temperature and form factor corrections:\n\nkT: If a temperature is provided, the intensities will be rescaled by a temperature- and ω-dependent classical-to-quantum factor. kT should be specified when making comparisons with spin wave calculations or experimental data. If kT is not specified, infinite temperature (no correction) is assumed.\nformfactors: To apply form factor corrections, provide this keyword with a list of FormFactors, one for each symmetry-distinct site in the crystal. The order of FormFactors must correspond to the order of site symmetry classes, e.g., as they appear when printed in display(crystal).\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.intensity_formula-Tuple{Function, SampledCorrelations, Any}","page":"Library API","title":"Sunny.intensity_formula","text":"A custom intensity formula can be specifed by providing a function intensity = f(q,ω,correlations) and specifying which correlations it requires:\n\nintensity_formula(f,sc::SampledCorrelations, required_correlations; kwargs...)\n\nThe function is intended to be specified using do notation. For example, this custom formula sums the off-diagonal correlations:\n\nrequired = [(:Sx,:Sy),(:Sy,:Sz),(:Sx,:Sz)]\nintensity_formula(sc,required,return_type = ComplexF64) do k, ω, off_diagonal_correlations\n sum(off_diagonal_correlations)\nend\n\nIf your custom formula returns a type other than Float64, use the return_type keyword argument to flag this.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.intensity_formula-Tuple{Function, SpinWaveTheory, AbstractVector{Int64}}","page":"Library API","title":"Sunny.intensity_formula","text":"formula = intensity_formula(swt::SpinWaveTheory; kernel = ...)\n\nEstablish a formula for computing the scattering intensity by diagonalizing the hamiltonian H(q) using Linear Spin Wave Theory.\n\nIf kernel = delta_function_kernel, then the resulting formula can be used with intensities_bands.\n\nIf kernel is an energy broadening kernel function, then the resulting formula can be used with intensities_broadened. Energy broadening kernel functions can either be a function of Δω only, e.g.:\n\nkernel = Δω -> ...\n\nor a function of both the energy transfer ω and of Δω, e.g.:\n\nkernel = (ω,Δω) -> ...\n\nThe integral of a properly normalized kernel function over all Δω is one.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.intensity_formula-Tuple{SpinWaveTheory, Symbol}","page":"Library API","title":"Sunny.intensity_formula","text":"intensity_formula([swt or sc], contraction_mode::Symbol)\n\nSunny has several built-in formulas that can be selected by setting contraction_mode to one of these values:\n\n:trace (default), which yields operatornametr 𝒮(qω) = _α 𝒮^αα(qω)\n:perp, which contracts 𝒮^αβ(qω) with the dipole factor δ_αβ - q_αq_β, returning the unpolarized intensity.\n:full, which will return all elements 𝒮^αβ(𝐪ω) without contraction.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.lattice_params-Tuple{StaticArraysCore.SMatrix{3, 3, Float64, 9}}","page":"Library API","title":"Sunny.lattice_params","text":"lattice_params(latvecs::Mat3)\n\nCompute the lattice parameters (a b c α β γ) for the three lattice vectors provided as columns of latvecs. The inverse mapping is lattice_vectors.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.lattice_vectors-NTuple{6, Any}","page":"Library API","title":"Sunny.lattice_vectors","text":"lattice_vectors(a, b, c, α, β, γ)\n\nReturn the lattice vectors, as columns of the 33 output matrix, that correspond to the conventional unit cell defined by the lattice constants (a b c) and the angles (α β γ) in degrees. The inverse mapping is lattice_params.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.load_nxs-Tuple{Any}","page":"Library API","title":"Sunny.load_nxs","text":"params, signal = load_nxs(filename)\n\nGiven the name of a Mantid-exported MDHistoWorkspace file, load the BinningParameters and the signal from that file.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.lorentzian-Tuple{Any, Any}","page":"Library API","title":"Sunny.lorentzian","text":"lorentzian(x, η)\n\nReturns η(π(x^2 + η^2)).\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.magnetic_moment-Tuple{System, Any}","page":"Library API","title":"Sunny.magnetic_moment","text":"magnetic_moment(sys::System, site::Site)\n\nGet the magnetic moment for a Site. This is the spin dipole multiplied by the Bohr magneton and the local g-tensor.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.merge_correlations-Union{Tuple{Array{SampledCorrelations{N}, 1}}, Tuple{N}} where N","page":"Library API","title":"Sunny.merge_correlations","text":"merge_correlations(scs::Vector{SampledCorrelations)\n\nAccumulate a list of SampledCorrelations into a single, summary SampledCorrelations. Useful for reducing the results of parallel computations.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.minimize_energy!-Union{Tuple{System{N}}, Tuple{N}} where N","page":"Library API","title":"Sunny.minimize_energy!","text":"minimize_energy!(sys::System{N}; maxiters=100, subiters=20,\n method=Optim.ConjugateGradient(), kwargs...) where N\n\nOptimizes the spin configuration in sys to minimize energy. A total of maxiters iterations will be attempted, with restarts after every subiters iterations. The remaining kwargs will be forwarded to the optimize method of the Optim.jl package.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.polarize_spins!-Union{Tuple{N}, Tuple{System{N}, Any}} where N","page":"Library API","title":"Sunny.polarize_spins!","text":"polarize_spins!(sys::System, dir)\n\nPolarize all spins in the system along the direction dir.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.position_to_site-Tuple{System, Any}","page":"Library API","title":"Sunny.position_to_site","text":"position_to_site(sys::System, r)\n\nConverts a position r to four indices of a Site. The coordinates of r are given in units of the lattice vectors for the original crystal. This function can be useful for working with systems that have been reshaped using reshape_supercell.\n\nExample\n\n# Find the `site` at the center of a unit cell which is displaced by four\n# multiples of the first lattice vector\nsite = position_to_site(sys, [4.5, 0.5, 0.5])\n\n# Print the dipole at this site\nprintln(sys.dipoles[site])\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.powder_average_binned-Tuple{SampledCorrelations, Any, Sunny.ClassicalIntensityFormula}","page":"Library API","title":"Sunny.powder_average_binned","text":"powder_average_binned(sc::SampledCorrelations, radial_binning_parameters; formula\n ω_binning_parameters, integrated_kernel = nothing, bzsize = nothing)\n\nThis function emulates the experimental situation of \"powder averaging,\" where only the magnitude (and not the direction) of the momentum transfer is resolvable. The intensities are binned similarly to intensities_binned, but the histogram x-axis is |k| in absolute units, which is a nonlinear function of kx,ky,kz. The y-axis is energy.\n\nRadial binning parameters are specified as tuples (start,end,bin_width), e.g. radial_binning_parameters = (0,6π,6π/55).\n\nEnergy broadening is supported in the same way as intensities_binned, and this function accepts the same kind of intensity_formula.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.primitive_cell_shape-Tuple{Crystal}","page":"Library API","title":"Sunny.primitive_cell_shape","text":"primitive_cell_shape(cryst::Crystal)\n\nReturns the shape of the primitive cell as a 3×3 matrix, in fractional coordinates of the conventional lattice vectors. May be useful for constructing inputs to reshape_supercell.\n\nExamples\n\n# Valid if `cryst` has not been reshaped\n@assert cryst.prim_latvecs ≈ cryst.latvecs * primitive_cell_shape(cryst)\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.print_bond-Tuple{Crystal, Bond}","page":"Library API","title":"Sunny.print_bond","text":"print_bond(cryst::Crystal, bond::Bond; b_ref::Bond)\n\nPrints symmetry information for bond bond. A symmetry-equivalent reference bond b_ref can optionally be provided to fix the meaning of the coefficients A, B, ...\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.print_site-Tuple{Any, Any}","page":"Library API","title":"Sunny.print_site","text":"print_site(cryst, i; R=I)\n\nPrint symmetry information for the site i, including allowed g-tensor and allowed anisotropy operator. An optional rotation matrix R can be provided to define the reference frame for expression of the anisotropy.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.print_stevens_expansion-Tuple{AbstractMatrix}","page":"Library API","title":"Sunny.print_stevens_expansion","text":"function print_stevens_expansion(op)\n\nPrints a local Hermitian operator as a linear combination of Stevens operators. The operator op may be a finite-dimensional matrix or an abstract spin polynomial in the large-S limit.\n\nExamples\n\nS = spin_matrices(2)\nprint_stevens_expansion(S[1]^4 + S[2]^4 + S[3]^4)\n# Prints: (1/20)𝒪₄₀ + (1/4)𝒪₄₄ + 102/5\n\nS = spin_matrices(Inf)\nprint_stevens_expansion(S[1]^4 + S[2]^4 + S[3]^4)\n# Prints: (1/20)𝒪₄₀ + (1/4)𝒪₄₄ + (3/5)𝒮⁴\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.print_suggested_frame-Tuple{Crystal, Int64}","page":"Library API","title":"Sunny.print_suggested_frame","text":"print_suggested_frame(cryst, i; digits=4)\n\nPrint a suggested reference frame, as a rotation matrix R, that can be used as input to print_site(). The purpose is to simplify the description of allowed anisotropies.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.print_symmetry_table-Tuple{Crystal, Any}","page":"Library API","title":"Sunny.print_symmetry_table","text":"print_symmetry_table(cryst::Crystal, max_dist)\n\nPrint symmetry information for all equivalence classes of sites and bonds, up to a maximum bond distance of max_dist. Equivalent to calling print_bond(cryst, b) for every bond b in reference_bonds(cryst, max_dist), where Bond(i, i, [0,0,0]) refers to a single site i.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.print_wrapped_intensities-Union{Tuple{System{N}}, Tuple{N}} where N","page":"Library API","title":"Sunny.print_wrapped_intensities","text":"print_wrapped_intensities(sys::System; nmax=10)\n\nFor Bravais lattices: Prints up to nmax wavevectors according to their instantaneous (static) structure factor intensities, listed in descending order. For non-Bravais lattices: Performs the same analysis for each spin sublattice independently; the output weights are naïvely averaged over sublattices, without incorporating phase shift information. This procedure therefore wraps all wavevectors into the first Brillouin zone. Each wavevector coordinate is given between -12 and 12 in reciprocal lattice units (RLU). The output from this function will typically be used as input to suggest_magnetic_supercell.\n\nBecause this function does not incorporate phase information in its averaging over sublattices, the printed weights are not directly comparable with experiment. For that purpose, use instant_correlations instead.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.propose_delta-Tuple{Any}","page":"Library API","title":"Sunny.propose_delta","text":"propose_delta(magnitude)\n\nGenerate a proposal function that adds a Gaussian perturbation to the existing spin state. In :dipole mode, the procedure is to first introduce a random three-vector perturbation 𝐬 = 𝐬 + 𝐬 ξ and then return the properly normalized spin 𝐬 (𝐬𝐬). Each component of the random vector ξ is Gaussian distributed with a standard deviation of magnitude; the latter is dimensionless and typically smaller than one. \n\nIn :SUN mode, the procedure is analogous, but now involving Gaussian perturbations to each of the N complex components of an SU(N) coherent state.\n\nIn the limit of very large magnitude, this function coincides with propose_uniform.\n\nFor use with LocalSampler.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.propose_flip-Union{Tuple{N}, Tuple{System{N}, Any}} where N","page":"Library API","title":"Sunny.propose_flip","text":"propose_flip\n\nFunction to propose pure spin flip updates in the context of a LocalSampler. Dipoles are flipped as 𝐬 -𝐬. SU(N) coherent states are flipped using the time-reversal operator.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.propose_uniform","page":"Library API","title":"Sunny.propose_uniform","text":"propose_uniform\n\nFunction to propose a uniformly random spin update in the context of a LocalSampler. In :dipole mode, the result is a random three-vector with appropriate normalization. In :SUN mode, the result is a random SU(N) coherent state with appropriate normalization.\n\n\n\n\n\n","category":"function"},{"location":"library.html#Sunny.randomize_spins!-Union{Tuple{System{N}}, Tuple{N}} where N","page":"Library API","title":"Sunny.randomize_spins!","text":"randomize_spins!(sys::System)\n\nRandomizes all spins under appropriate the uniform distribution.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.reciprocal_space_path-Tuple{Crystal, Any, Any}","page":"Library API","title":"Sunny.reciprocal_space_path","text":"reciprocal_space_path(cryst::Crystal, qs, density)\n\nReturns a pair (path, xticks). The path return value is a list of wavevectors that samples linearly between the provided wavevectors qs. The xticks return value can be used to label the special 𝐪 values on the x-axis of a plot.\n\nSpecial note about units: the wavevectors qs must be provided in reciprocal lattice units (RLU) for the given crystal, but the sampling density must be specified in the global frame. Specifically, the density is given as number of sample points per unit of radian inverse length, where the unit of length is the same as that used to specify the lattice vectors of the Crystal. The path will therefore include more samples between q-points that are further apart in absolute Fourier distance.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.reciprocal_space_path_bins-Tuple{Any, Any, Any, Vararg{Any}}","page":"Library API","title":"Sunny.reciprocal_space_path_bins","text":"reciprocal_space_path_bins(sc,qs,density,args...;kwargs...)\n\nTakes a list of wave vectors, qs in R.L.U., and builds a series of histogram BinningParameters whose first axis traces a path through the provided points. The second and third axes are integrated over according to the args and kwargs, which are passed through to slice_2D_binning_parameters.\n\nAlso returned is a list of marker indices corresponding to the input points, and a list of ranges giving the indices of each histogram x-axis within a concatenated histogram. The density parameter is given in samples per reciprocal lattice unit (R.L.U.).\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.reciprocal_space_shell-Tuple{Crystal, Any, Any}","page":"Library API","title":"Sunny.reciprocal_space_shell","text":"reciprocal_space_shell(cryst::Crystal, radius, n)\n\nSample n points on the reciprocal space sphere with a given radius (units of inverse length).\n\nExamples\n\n# Sample wavevectors on the sphere at fixed density\nreciprocal_space_shell(cryst, r, 4π*r^2*density)\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.reference_bonds-Tuple{Crystal, Float64}","page":"Library API","title":"Sunny.reference_bonds","text":"reference_bonds(cryst::Crystal, max_dist)\n\nReturns a full list of bonds, one for each symmetry equivalence class, up to distance max_dist. The reference bond b for each equivalence class is selected according to a scoring system that prioritizes simplification of the elements in basis_for_symmetry_allowed_couplings(cryst, b).\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.remove_periodicity!-Union{Tuple{N}, Tuple{System{N}, Any}} where N","page":"Library API","title":"Sunny.remove_periodicity!","text":"remove_periodicity!(sys::System, dims)\n\nRemove periodic interactions along the dimensions where dims is true. The system must support inhomogeneous interactions via to_inhomogeneous.\n\nExample\n\n# Remove periodic boundaries along the 1st and 3rd dimensions\nremove_periodicity!(sys::System, (true, false, true))\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.repeat_periodically-Union{Tuple{N}, Tuple{System{N}, Tuple{Int64, Int64, Int64}}} where N","page":"Library API","title":"Sunny.repeat_periodically","text":"repeat_periodically(sys::System{N}, counts::NTuple{3,Int}) where N\n\nCreates a System identical to sys but repeated a given number of times in each dimension, specified by the tuple counts.\n\nSee also reshape_supercell.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.reshape_supercell-Union{Tuple{N}, Tuple{System{N}, Any}} where N","page":"Library API","title":"Sunny.reshape_supercell","text":"reshape_supercell(sys::System, shape)\n\nMaps an existing System to a new one that has the shape and periodicity of a requested supercell. The columns of the 33 integer matrix shape represent the supercell lattice vectors measured in units of the original crystal lattice vectors.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.resize_supercell-Union{Tuple{N}, Tuple{System{N}, Tuple{Int64, Int64, Int64}}} where N","page":"Library API","title":"Sunny.resize_supercell","text":"resize_supercell(sys::System{N}, latsize::NTuple{3,Int}) where N\n\nCreates a System with a given number of conventional unit cells in each lattice vector direction. Interactions and other settings will be inherited from sys.\n\nConvenience function for:\n\nreshape_supercell(sys, [latsize[1] 0 0; 0 latsize[2] 0; 0 0 latsize[3]])\n\nSee also reshape_supercell.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.rotate_operator-Tuple{LinearAlgebra.Hermitian{ComplexF64, Matrix{ComplexF64}}, Any}","page":"Library API","title":"Sunny.rotate_operator","text":"rotate_operator(A, R)\n\nRotates the local quantum operator A according to the 33 rotation matrix R.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.rotation_in_rlu-Tuple{Crystal, Any, Any}","page":"Library API","title":"Sunny.rotation_in_rlu","text":"rotation_in_rlu(cryst::Crystal, axis, angle)\n\nReturns a 33 matrix that rotates wavevectors in reciprocal lattice units (RLU). The axis vector is a real-space direction in absolute units (but arbitrary magnitude), and the angle is in radians.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_coherent!-Union{Tuple{N}, Tuple{System{N}, Any, Any}} where N","page":"Library API","title":"Sunny.set_coherent!","text":"set_coherent!(sys::System, Z, site::Site)\n\nSet a coherent spin state at a Site using the N complex amplitudes in Z.\n\nFor a standard SpinInfo, these amplitudes will be interpreted in the eigenbasis of 𝒮ᶻ. That is, Z[1] represents the amplitude for the basis state fully polarized along the z-direction, and subsequent components represent states with decreasing angular momentum along this axis (m = S S-1 -S).\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_dipole!-Union{Tuple{N}, Tuple{System{N}, Any, Any}} where N","page":"Library API","title":"Sunny.set_dipole!","text":"set_dipole!(sys::System, dir, site::Site)\n\nPolarize the spin at a Site along the direction dir.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_exchange!-Union{Tuple{N}, Tuple{System{N}, Any, Bond}} where N","page":"Library API","title":"Sunny.set_exchange!","text":"set_exchange!(sys::System, J, bond::Bond; biquad=0, large_S=false)\n\nSets a 3×3 spin-exchange matrix J along bond, yielding a pairwise interaction energy 𝐒_iJ 𝐒_j. This interaction will be propagated to equivalent bonds in consistency with crystal symmetry. Any previous interactions on these bonds will be overwritten. The parameter bond has the form Bond(i, j, offset), where i and j are atom indices within the unit cell, and offset is a displacement in unit cells.\n\nThe parameter J may be scalar or matrix-valued. As a convenience, dmvec(D) can be used to construct the antisymmetric part of the exchange, where D is the Dzyaloshinskii-Moriya pseudo-vector. The resulting interaction will be 𝐃(𝐒_i𝐒_j).\n\nThe optional parameter biquad defines the strength b for scalar biquadratic interactions of the form b (𝐒_i𝐒_j)². For systems restricted to dipoles, b will be automatically renormalized for maximum consistency with the more variationally accurate SU(N) mode. Set large_S=true to work in the large-S limit and disable this renormalization.\n\nExamples\n\n# An explicit exchange matrix\nJ1 = [2 3 0;\n -3 2 0;\n 0 0 2]\nset_exchange!(sys, J1, bond)\n\n# An equivalent Heisenberg + DM exchange \nJ2 = 2*I + dmvec([0,0,3])\nset_exchange!(sys, J2, bond)\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_exchange_at!-Union{Tuple{N}, Tuple{System{N}, Any, Union{NTuple{4, Int64}, CartesianIndex{4}}, Union{NTuple{4, Int64}, CartesianIndex{4}}}} where N","page":"Library API","title":"Sunny.set_exchange_at!","text":"set_exchange_at!(sys::System, J, site1::Site, site2::Site; biquad=0, large_S=false, offset=nothing)\n\nSets the exchange interaction along the single bond connecting two Sites, ignoring crystal symmetry. The system must support inhomogeneous interactions via to_inhomogeneous.\n\nIf the system is relatively small, the direction of the bond can be ambiguous due to possible periodic wrapping. Resolve this ambiguity by passing an explicit offset vector, in multiples of unit cells.\n\nSee also set_exchange!.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_external_field!-Tuple{System, Any}","page":"Library API","title":"Sunny.set_external_field!","text":"set_external_field!(sys::System, B::Vec3)\n\nSets the external field B that couples to all spins.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_external_field_at!-Tuple{System, Any, Any}","page":"Library API","title":"Sunny.set_external_field_at!","text":"set_external_field_at!(sys::System, B::Vec3, site::Site)\n\nSets a Zeeman coupling between a field B and a single spin. Site includes a unit cell and a sublattice index.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_onsite_coupling!-Tuple{System, Any, Int64}","page":"Library API","title":"Sunny.set_onsite_coupling!","text":"set_onsite_coupling!(sys::System, op, i::Int)\n\nSet the single-ion anisotropy for the ith atom of every unit cell, as well as all symmetry-equivalent atoms. The local operator op will may be constructed as an abstract function of the local spin operators, as a polynomial of spin_matrices, or as a linear combination of stevens_matrices.\n\nExamples\n\n# An easy axis anisotropy in the z-direction\nset_onsite_coupling!(sys, S -> -D*S[3]^3, i)\n\n# The unique quartic single-ion anisotropy for a site with cubic point group\n# symmetry\nset_onsite_coupling!(sys, S -> 20*(S[1]^4 + S[2]^4 + S[3]^4), i)\n\n# An equivalent expression of this quartic anisotropy, up to a constant shift\nO = stevens_matrices(spin_irrep_label(sys, i))\nset_onsite_coupling!(sys, O[4,0] + 5*O[4,4], i)\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_onsite_coupling_at!-Tuple{System, Any, Union{NTuple{4, Int64}, CartesianIndex{4}}}","page":"Library API","title":"Sunny.set_onsite_coupling_at!","text":"set_onsite_coupling_at!(sys::System, op, site::Site)\n\nSets the single-ion anisotropy operator op for a single Site, ignoring crystal symmetry. The system must support inhomogeneous interactions via to_inhomogeneous.\n\nSee also set_onsite_coupling!.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_pair_coupling!-Union{Tuple{N}, Tuple{System{N}, Any, Any}} where N","page":"Library API","title":"Sunny.set_pair_coupling!","text":"set_pair_coupling!(sys::System, coupling, bond)\n\nSets an arbitrary coupling along bond. This coupling will be propagated to equivalent bonds in consistency with crystal symmetry. Any previous interactions on these bonds will be overwritten. The parameter bond has the form Bond(i, j, offset), where i and j are atom indices within the unit cell, and offset is a displacement in unit cells. The coupling is a represented as a matrix acting in the tensor product space of the two sites, and typically originates from to_product_space.\n\nExamples\n\n# Add a bilinear and biquadratic exchange\nS = spin_matrices(1/2)\nSi, Sj = to_product_space(S, S)\nset_pair_coupling!(sys, Si'*J1*Sj + (Si'*J2*Sj)^2, bond)\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_spiral_order!-Tuple{Any}","page":"Library API","title":"Sunny.set_spiral_order!","text":"set_spiral_order!(sys; q, axis, S0)\n\nInitializes the system with a spiral order described by the wavevector q, an axis of rotation axis, and an initial dipole direction S0 at the real-space origin. The wavevector is expected in repicrocal lattice units (RLU), while the direction vectors axis and S0 are expected in global Cartesian coordinates.\n\nExample\n\n# Spiral order for a wavevector propagating in the direction of the first\n# reciprocal lattice vector (i.e., orthogonal to the lattice vectors ``𝐚_2``\n# and ``𝐚_3``), repeating with a period of 10 lattice constants, and spiraling\n# about the ``ẑ``-axis. The spin at the origin will point in the direction\n# ``𝐒_0 = ŷ + ẑ``. Here, ``(x̂, ŷ, ẑ)`` are the axes of Cartesian coordinate\n# system in the global frame.\nset_spiral_order!(sys; q=[1/10, 0, 0], axis=[0, 0, 1], S0=[0, 1, 1])\n\nSee also set_spiral_order_on_sublattice!.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_spiral_order_on_sublattice!-Tuple{Any, Any}","page":"Library API","title":"Sunny.set_spiral_order_on_sublattice!","text":"set_spiral_order_on_sublattice!(sys, i; q, axis, S0)\n\nInitializes sublattice i with a spiral order described by the wavevector q, an axis of rotation axis, and an initial dipole direction S0. The phase is selected such that the spin at sys.dipole[1,1,1,i] will point in the direction of S0. The wavevector is expected in repicrocal lattice units (RLU), while the direction vectors axis and S0 are expected in global Cartesian coordinates.\n\nThis function is not available for systems with reshaped unit cells.\n\nSee also set_spiral_order!.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_vacancy_at!-Union{Tuple{N}, Tuple{System{N}, Any}} where N","page":"Library API","title":"Sunny.set_vacancy_at!","text":"set_vacancy_at!(sys::System, site::Site)\n\nMake a single site nonmagnetic. Site includes a unit cell and a sublattice index.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.slice_2D_binning_parameters-Tuple{Vector{Float64}, Any, Any, Int64, Any}","page":"Library API","title":"Sunny.slice_2D_binning_parameters","text":"slice_2D_binning_parameter(sc::SampledCorrelations, cut_from_q, cut_to_q, cut_bins::Int64, cut_width::Float64; plane_normal = [0,0,1],cut_height = cutwidth)\n\nCreates BinningParameters which make a cut along one dimension of Q-space.\n\nThe x-axis of the resulting histogram consists of cut_bins-many bins ranging from cut_from_q to cut_to_q. The width of the bins in the transverse direciton is controlled by cut_width and cut_height.\n\nThe binning in the transverse directions is defined in the following way, which sets their normalization and orthogonality properties:\n\ncut_covector = normalize(cut_to_q - cut_from_q)\ntransverse_covector = normalize(plane_normal × cut_covector)\ncotransverse_covector = normalize(transverse_covector × cut_covector)\n\nIn other words, the axes are orthonormal with respect to the Euclidean metric.\n\nIf the cut is too narrow, there will be very few scattering vectors per bin, or the number per bin will vary substantially along the cut. If the output appears under-resolved, try increasing cut_width.\n\nThe four axes of the resulting histogram are:\n\nAlong the cut\nFist transverse Q direction\nSecond transverse Q direction\nEnergy\n\nThis function can be used without reference to a SampledCorrelations using this alternate syntax to manually specify the bin centers for the energy axis:\n\nslice_2D_binning_parameter(ω_bincenters, cut_from, cut_to,...)\n\nwhere ω_bincenters specifies the energy axis, and both cut_from and cut_to are arbitrary covectors, in any units.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.spin_irrep_label-Tuple{System, Int64}","page":"Library API","title":"Sunny.spin_irrep_label","text":"spin_irrep_label(sys::System, i::Int)\n\nIf atom i or site site carries a single spin-S moment, then returns S. Otherwise, throws an error.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.spin_matrices-Tuple{Any}","page":"Library API","title":"Sunny.spin_matrices","text":"spin_matrices(S)\n\nReturns a triple of NNspin matrices, whereN = 2S+1. These are the generators of SU(2) in the spin-S` representation.\n\nIf S == Inf, then the return values are abstract symbols denoting infinite-dimensional matrices that commute. These can be useful for repeating historical studies, or modeling micromagnetic systems. A technical discussion appears in the Sunny documentation page: Interaction Strength Renormalization.\n\nExample\n\nS = spin_matrices(3/2)\n@assert S'*S ≈ (3/2)*(3/2+1)*I\n@assert S[1]*S[2] - S[2]*S[1] ≈ im*S[3]\n\nS = spin_matrices(Inf)\n@assert S[1]*S[2] - S[2]*S[1] == 0\n\nSee also print_stevens_expansion.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.step!","page":"Library API","title":"Sunny.step!","text":"step!(sys::System, dynamics)\n\nAdvance the spin configuration one dynamical time-step. The dynamics object may be a continuous spin dynamics, such as Langevin or ImplicitMidpoint, or it may be a discrete Monte Carlo sampling scheme such as LocalSampler.\n\n\n\n\n\n","category":"function"},{"location":"library.html#Sunny.stevens_matrices-Tuple{Any}","page":"Library API","title":"Sunny.stevens_matrices","text":"stevens_matrices(S)\n\nReturns a generator of Stevens operators in the spin-S representation. The return value O can be indexed as O[k,q], where 0 k 6 labels an irrep of SO(3) and q = -k k. This will produce an NN matrix where N = 2S + 1. Linear combinations of Stevens operators can be used as a \"physical basis\" for decomposing local observables. To see this decomposition, use print_stevens_expansion.\n\nIf S == Inf, then symbolic operators will be returned. In this infinite dimensional limit, the Stevens operators become homogeneous polynomials of commuting spin operators.\n\nExample\n\nO = stevens_matrices(2)\nS = spin_matrices(2)\n\nA = (1/20)O[4,0] + (1/4)O[4,4] + (102/5)I\nB = S[1]^4 + S[2]^4 + S[3]^4\n@assert A ≈ B\n\nSee also spin_matrices and Interaction Strength Renormalization.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.subcrystal-Union{Tuple{N}, Tuple{Crystal, Vararg{String, N}}} where N","page":"Library API","title":"Sunny.subcrystal","text":"subcrystal(cryst, types) :: Crystal\n\nFilters sublattices of a Crystal by atom types, keeping the space group unchanged.\n\nsubcrystal(cryst, classes) :: Crystal\n\nFilters sublattices of Crystal by equivalence classes, keeping the space group unchanged.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.suggest_magnetic_supercell-Tuple{Any}","page":"Library API","title":"Sunny.suggest_magnetic_supercell","text":"suggest_magnetic_supercell(qs; tol=1e-12, maxsize=100)\n\nSuggests a magnetic supercell, in units of the crystal lattice vectors, that is consistent with periodicity of the wavevectors qs in RLU. If the wavevectors are incommensurate (with respect to the maximum supercell size maxsize), one can select a larger error tolerance tol to find a supercell that is almost commensurate.\n\nPrints a 33 matrix of integers that is suitable for use in reshape_supercell.\n\nExamples\n\n# A magnetic supercell for a single-Q structure. Will print\nq1 = [0, -1/4, 1/4]\nsuggest_magnetic_supercell([q1]) # [1 0 0; 0 2 1; 0 -2 1]\n\n# A larger magnetic supercell for a double-Q structure\nq2 = [1/4, 0, 1/4]\nsuggest_magnetic_supercell([q1, q2]) # [1 2 2; -1 2 -2; -1 2 2]\n\n# If given incommensurate wavevectors, find an approximate supercell that\n# is exactly commensurate for nearby wavevectors.\nsuggest_magnetic_supercell([[0, 0, 1/√5], [0, 0, 1/√7]]; tol=1e-2)\n\n# This prints [1 0 0; 0 1 0; 0 0 16], which becomes commensurate under the\n# approximations `1/√5 ≈ 7/16` and `1/√7 ≈ 3/8`.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.symmetry_equivalent_bonds-Tuple{System, Bond}","page":"Library API","title":"Sunny.symmetry_equivalent_bonds","text":"symmetry_equivalent_bonds(sys::System, bond::Bond)\n\nGiven a Bond for the original (unreshaped) crystal, return all symmetry equivalent bonds in the System. Each returned bond is represented as a pair of Sites, which may be used as input to set_exchange_at!. Reverse bonds are not included (no double counting).\n\nExample\n\nfor (site1, site2, offset) in symmetry_equivalent_bonds(sys, bond)\n @assert site1 < site2\n set_exchange_at!(sys, J, site1, site2; offset)\nend\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.to_inhomogeneous-Union{Tuple{System{N}}, Tuple{N}} where N","page":"Library API","title":"Sunny.to_inhomogeneous","text":"to_inhomogeneous(sys::System)\n\nReturns a copy of the system that allows for inhomogeneous interactions, which can be set using set_onsite_coupling_at!, set_exchange_at!, and set_vacancy_at!.\n\nInhomogeneous systems do not support symmetry-propagation of interactions or system reshaping.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.to_product_space-Tuple{Any, Any, Vararg{Any}}","page":"Library API","title":"Sunny.to_product_space","text":"to_product_space(A, B, Cs...)\n\nGiven lists of operators acting on local Hilbert spaces individually, return the corresponding operators that act on the tensor product space. In typical usage, the inputs will represent local physical observables and the outputs will be used to define quantum couplings.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.unit_resolution_binning_parameters-Tuple{Any, Any, Vararg{Any}}","page":"Library API","title":"Sunny.unit_resolution_binning_parameters","text":"unit_resolution_binning_parameters(sc::SampledCorrelations)\n\nCreate BinningParameters which place one histogram bin centered at each possible (q,ω) scattering vector of the crystal. This is the finest possible binning without creating bins with zero scattering vectors in them.\n\nThis function can be used without reference to a SampledCorrelations using an alternate syntax to manually specify the bin centers for the energy axis and the lattice size:\n\nunit_resolution_binning_parameters(ω_bincenters,latsize,[reciprocal lattice vectors])\n\nThe last argument may be a 3x3 matrix specifying the reciprocal lattice vectors, or a Crystal.\n\nLastly, binning parameters for a single axis may be specifed by their bin centers:\n\n(binstart,binend,binwidth) = unit_resolution_binning_parameters(bincenters::Vector{Float64})\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.@mix_proposals-Tuple","page":"Library API","title":"Sunny.@mix_proposals","text":"@mix_proposals weight1 propose1 weight2 propose2 ...\n\nMacro to generate a proposal function that randomly selects among the provided functions according to the provided probability weights. For use with LocalSampler.\n\nExample\n\n# A proposal function that proposes a spin flip 40% of the time, and a\n# Gaussian perturbation 60% of the time.\n@mix_proposals 0.4 propose_flip 0.6 propose_delta(0.2)\n\n\n\n\n\n","category":"macro"},{"location":"library.html#Optional-Makie-extensions","page":"Library API","title":"Optional Makie extensions","text":"","category":"section"},{"location":"library.html","page":"Library API","title":"Library API","text":"The following will be enabled through a package extension if either GLMakie or WGLMakie is loaded.","category":"page"},{"location":"library.html","page":"Library API","title":"Library API","text":"plot_spins\nview_crystal","category":"page"},{"location":"library.html#Sunny.plot_spins","page":"Library API","title":"Sunny.plot_spins","text":"plot_spins(sys::System; arrowscale=1.0, color=:red, show_cell=true,\n orthographic=false, ghost_radius=0, dims=3)\n\nPlot the spin configuration defined by sys. Optional parameters include:\n\narrowscale: Scale all arrows by dimensionless factor.\ncolor: Arrow color. May be a numeric value per site in system.\nshow_cell: Show original crystallographic unit cell.\northographic: Use camera with orthographic projection.\nghost_radius: Show translucent periodic images up to a given distance (length units).\ndims: Spatial dimensions of system (1, 2, or 3).\n\n\n\n\n\n","category":"function"},{"location":"library.html#Sunny.view_crystal","page":"Library API","title":"Sunny.view_crystal","text":"view_crystal(crystal::Crystal, max_dist::Real; show_axis=true, orthographic=false)\n\nAn interactive crystal viewer, with bonds up to max_dist.\n\n\n\n\n\n","category":"function"},{"location":"library.html#Optional-WriteVTK-extensions","page":"Library API","title":"Optional WriteVTK extensions","text":"","category":"section"},{"location":"library.html","page":"Library API","title":"Library API","text":"The following will be enabled through a package extension if WriteVTK is loaded.","category":"page"},{"location":"library.html","page":"Library API","title":"Library API","text":"export_vtk","category":"page"},{"location":"library.html#Sunny.export_vtk","page":"Library API","title":"Sunny.export_vtk","text":"export_vtk(filename,params::BinningParameters,data)\n\nExport a VTK-compatible file to filename (do not include file extension when specifying the file name) which contains the data as VTK Cell Data on a grid parameterized by params.\n\nAt least one axis of the BinningParameters must be integrated over, since VTK does not support 4D data. See integrate_axes!.\n\n\n\n\n\n","category":"function"},{"location":"examples/spinw/08_Kagome_AFM.html","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"EditURL = \"../../../../examples/spinw_ports/08_Kagome_AFM.jl\"","category":"page"},{"location":"examples/spinw/08_Kagome_AFM.html","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"Download this example as Jupyter notebook or Julia script.","category":"page"},{"location":"examples/spinw/08_Kagome_AFM.html#Kagome-Antiferromagnet","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"","category":"section"},{"location":"examples/spinw/08_Kagome_AFM.html","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"This is a Sunny port of SpinW Tutorial 8, authored by Bjorn Fak and Sandor Toth. The goal is to calculate the linear spin wave theory spectrum for the sqrt3 times sqrt3 order of a Kagome antiferromagnet.","category":"page"},{"location":"examples/spinw/08_Kagome_AFM.html","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"Load Packages","category":"page"},{"location":"examples/spinw/08_Kagome_AFM.html","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"using Sunny, GLMakie","category":"page"},{"location":"examples/spinw/08_Kagome_AFM.html","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"Build a Crystal with Poverline3 space group and Cr⁺ ions on each site.","category":"page"},{"location":"examples/spinw/08_Kagome_AFM.html","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"a = 1\nlatvecs = lattice_vectors(a, a, 10a, 90, 90, 120)\ncrystal = Crystal(latvecs, [[1/2,0,0]], 147)","category":"page"},{"location":"examples/spinw/08_Kagome_AFM.html","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"Build a System with antiferrogmanetic nearest neighbor exchange J=1.","category":"page"},{"location":"examples/spinw/08_Kagome_AFM.html","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"S = 1\nsys = System(crystal, (3,3,1), [SpinInfo(1; S, g=2)], :dipole)\nJ = 1.0\nset_exchange!(sys, J, Bond(2,3,[0,0,0]))","category":"page"},{"location":"examples/spinw/08_Kagome_AFM.html","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"Initialize to an energy minimizing magnetic structure, for which nearest-neighbor spins are at 120° angles.","category":"page"},{"location":"examples/spinw/08_Kagome_AFM.html","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"q = -[1/3, 1/3, 0]\naxis = [0,0,1]\nset_spiral_order_on_sublattice!(sys, 1; q, axis, S0=[cos(0),sin(0),0])\nset_spiral_order_on_sublattice!(sys, 2; q, axis, S0=[cos(0),sin(0),0])\nset_spiral_order_on_sublattice!(sys, 3; q, axis, S0=[cos(2π/3),sin(2π/3),0])\nplot_spins(sys; dims=2)","category":"page"},{"location":"examples/spinw/08_Kagome_AFM.html","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"Check energy. Each site participates in 4 bonds with energy JS^2cos(2π3). Factor of 1/2 avoids double counting.","category":"page"},{"location":"examples/spinw/08_Kagome_AFM.html","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"@assert energy_per_site(sys) ≈ (4/2)*J*S^2*cos(2π/3)","category":"page"},{"location":"examples/spinw/08_Kagome_AFM.html","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"Define a path in reciprocal space.","category":"page"},{"location":"examples/spinw/08_Kagome_AFM.html","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"points_rlu = [[-1/2, 0, 0], [0, 0, 0], [1/2, 1/2, 0]]\ndensity = 100\npath, xticks = reciprocal_space_path(crystal, points_rlu, density);\nnothing #hide","category":"page"},{"location":"examples/spinw/08_Kagome_AFM.html","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"Calculate discrete intensities","category":"page"},{"location":"examples/spinw/08_Kagome_AFM.html","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"swt = SpinWaveTheory(sys)\nformula = intensity_formula(swt, :perp; kernel=delta_function_kernel)\ndisp, intensity = intensities_bands(swt, path, formula);\nnothing #hide","category":"page"},{"location":"examples/spinw/08_Kagome_AFM.html","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"Plot over a restricted color range from [0,1e-2]. Note that the intensities of the flat band at zero-energy are off-scale.","category":"page"},{"location":"examples/spinw/08_Kagome_AFM.html","page":"Kagome Antiferromagnet","title":"Kagome Antiferromagnet","text":"fig = Figure()\nax = Axis(fig[1,1]; xlabel=\"Momentum (r.l.u.)\", ylabel=\"Energy (meV)\", xticks, xticklabelrotation=π/6)\nylims!(ax, -1e-1, 2.3)\nfor i in axes(disp)[2]\n lines!(ax, 1:length(disp[:,i]), disp[:,i]; color=intensity[:,i], colorrange=(0,1e-2))\nend\nfig","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"EditURL = \"../../../examples/fei2_classical.jl\"","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"Download this example as Jupyter notebook or Julia script.","category":"page"},{"location":"examples/fei2_classical.html#FeI-at-Finite-Temperature","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"","category":"section"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"using Sunny, LinearAlgebra, GLMakie","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"In our previous Case Study: FeI₂, we used linear spin wave theory (LSWT) to calculate the dynamical structure factor. Here, we perform a similar calculation using classical spin dynamics. Because we are interested in the coupled dynamics of spin dipoles and quadrupoles, we employ a classical dynamics of SU(3) coherent states that generalizes the Landau-Lifshitz equation.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"Compared to LSWT, simulations using classical dynamics are much slower, and are limited in k-space resolution. However, they make it is possible to capture nonlinear effects associated with finite temperature fluctuations. Classical dynamics are also appealing for studying out-of-equilibrium systems (e.g., relaxation of spin glasses), or systems with quenched inhomogeneities that require large simulation volumes.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"In this tutorial, we show how to study the finite temperature dynamics of FeI₂ using the classical approach. It is important to stress that the estimation of S(𝐪ω) with classical dynamics is fundamentally a Monte Carlo calculation: sample spin configurations are drawn from thermal equilibrium and used as initial conditions for generating dissipationless trajectories. The correlations of these trajectories are then averaged and used to calculate scattering intensities. It is therefore important to ensure that the initial spin configurations are sampled appropriately and that sufficient statistics are collected. We will demonstrate one approach here.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"As an overview, we will:","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"Identify the ground state\nMeasure correlation data describing the excitations around that ground state\nUse the correlation data to compute scattering intensities","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"As the implementation of the FeI₂ model is already covered in detail in the LSWT tutorial, we will not repeat it below. Instead, we will assume that you already have defined a sys in the same way with lattice dimensions 444.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"a = b = 4.05012#hide\nc = 6.75214#hide\nlatvecs = lattice_vectors(a, b, c, 90, 90, 120)#hide\npositions = [[0,0,0], [1/3, 2/3, 1/4], [2/3, 1/3, 3/4]]#hide\ntypes = [\"Fe\", \"I\", \"I\"]#hide\nFeI2 = Crystal(latvecs, positions; types)#hide\ncryst = subcrystal(FeI2, \"Fe\")#hide\nsys = System(cryst, (4,4,4), [SpinInfo(1,S=1,g=2)], :SUN, seed=2)#hide\nJ1pm = -0.236#hide\nJ1pmpm = -0.161#hide\nJ1zpm = -0.261#hide\nJ2pm = 0.026#hide\nJ3pm = 0.166#hide\nJ′0pm = 0.037#hide\nJ′1pm = 0.013#hide\nJ′2apm = 0.068#hide\nJ1zz = -0.236#hide\nJ2zz = 0.113#hide\nJ3zz = 0.211#hide\nJ′0zz = -0.036#hide\nJ′1zz = 0.051#hide\nJ′2azz = 0.073#hide\nJ1xx = J1pm + J1pmpm#hide\nJ1yy = J1pm - J1pmpm#hide\nJ1yz = J1zpm#hide\nset_exchange!(sys, [J1xx 0.0 0.0; 0.0 J1yy J1yz; 0.0 J1yz J1zz], Bond(1,1,[1,0,0]))#hide\nset_exchange!(sys, [J2pm 0.0 0.0; 0.0 J2pm 0.0; 0.0 0.0 J2zz], Bond(1,1,[1,2,0]))#hide\nset_exchange!(sys, [J3pm 0.0 0.0; 0.0 J3pm 0.0; 0.0 0.0 J3zz], Bond(1,1,[2,0,0]))#hide\nset_exchange!(sys, [J′0pm 0.0 0.0; 0.0 J′0pm 0.0; 0.0 0.0 J′0zz], Bond(1,1,[0,0,1]))#hide\nset_exchange!(sys, [J′1pm 0.0 0.0; 0.0 J′1pm 0.0; 0.0 0.0 J′1zz], Bond(1,1,[1,0,1]))#hide\nset_exchange!(sys, [J′2apm 0.0 0.0; 0.0 J′2apm 0.0; 0.0 0.0 J′2azz], Bond(1,1,[1,2,1]))#hide\nD = 2.165#hide\nset_onsite_coupling!(sys, S -> -D*S[3]^2, 1)#hide\nsys","category":"page"},{"location":"examples/fei2_classical.html#Finding-a-ground-state","page":"FeI₂ at Finite Temperature","title":"Finding a ground state","text":"","category":"section"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"Sunny uses the Langevin dynamics of SU(N) coherent states to sample spin configurations from the thermal equlibrium. One first constructs a Langevin integrator. This requires a time step, temperature, and a phenomenological damping parameter λ that sets the coupling to the thermal bath.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"Δt = 0.05/D # Should be inversely proportional to the largest energy scale\n # in the system. For FeI2, this is the easy-axis anisotropy,\n # `D = 2.165` (meV). The prefactor 0.05 is relatively small,\n # and achieves high accuracy.\nkT = 0.2 # Temperature of the thermal bath (meV).\nλ = 0.1 # This value is typically good for Monte Carlo sampling,\n # independent of system details.\n\nlangevin = Langevin(Δt; kT, λ);\nnothing #hide","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"Langevin dynamics can be used to search for a magnetically ordered state. For this, the temperature kT must be below the ordering temperature, but large enough that the dynamical sampling procedure can overcome local energy barriers and eliminate defects.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"randomize_spins!(sys)\nfor _ in 1:20_000\n step!(sys, langevin)\nend","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"Although thermal fluctuations are present, the correct antiferromagnetic order (2 up, 2 down) is apparent.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"plot_spins(sys; color=[s[3] for s in sys.dipoles])","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"For other systems, it can be much harder to find the magnetic ordering in an unbiased way, and more complicated sampling procedures may be necessary.","category":"page"},{"location":"examples/fei2_classical.html#Calculating-Thermal-Averaged-Correlations-\\langle-S{\\alpha\\beta}(𝐪,ω)\\rangle","page":"FeI₂ at Finite Temperature","title":"Calculating Thermal-Averaged Correlations langle S^alphabeta(𝐪ω)rangle","text":"","category":"section"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"Our aim is to study the classical spin dynamics for states sampled in thermal equilibrium. To minimize finite size effects, and achieve sufficient momentum space resolution, we should significantly enlarge the system volume. The function resize_supercell takes new dimensions as multiples of the unit cell lattice vectors.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"sys_large = resize_supercell(sys, (16,16,4)) # 16x16x4 copies of the original unit cell\nplot_spins(sys_large; color=[s[3] for s in sys_large.dipoles])","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"Now we will re-thermalize the system to a configuration just above the ordering temperature. Sunny expects energies in meV by default, so we use meV_per_K to convert from kelvin.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"kT = 3.5 * meV_per_K # 3.5K ≈ 0.30 meV\nlangevin.kT = kT\nfor _ in 1:10_000\n step!(sys_large, langevin)\nend","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"The next step is to collect correlation data S^alphabeta. This will involve sampling spin configurations from thermal equilibrium, and then integrating the Hamiltonian dynamics of SU(N) coherent states to collect Fourier-space information about normal modes. Quantization of these modes yields the magnons, and the associated dynamical spin-spin correlations can be compared with neutron scattering intensities S^alphabeta(qomega). Because this a real-space calculation, data is only available for discrete q modes (the resolution scales like inverse system size).","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"To store the correlation data, we initialize a SampledCorrelations object by calling dynamical_correlations. It requires three keyword arguments: an integration step size, a target number of ωs to retain, and a maximum energy ω to resolve. For the time step, twice the value used for the Langevin integrator is usually a good choice.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"sc = dynamical_correlations(sys_large; Δt=2Δt, nω=120, ωmax=7.5)","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"The function add_sample! will collect data by running a dynamical trajectory starting from the current system configuration.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"add_sample!(sc, sys_large) # Accumulate the sample into `sc`","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"To collect additional data, it is required to re-sample the spin configuration from the thermal distribution. For efficiency, the dynamics should be run long enough that consecutive samples are uncorrelated.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"for _ in 1:2\n for _ in 1:1000 # Enough steps to decorrelate spins\n step!(sys_large, langevin)\n end\n add_sample!(sc, sys_large) # Accumulate the sample into `sc`\nend","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"Now, sc has more samples included:","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"sc","category":"page"},{"location":"examples/fei2_classical.html#Computing-Scattering-Intensities","page":"FeI₂ at Finite Temperature","title":"Computing Scattering Intensities","text":"","category":"section"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"With the thermally-averaged correlation data langle S^alphabeta(qomega)rangle in hand, we now need to specify how to extract a scattering intensity from this information. This is done by constructing an intensity_formula. By way of example, we will use a formula which computes the trace of the structure factor and applies a classical-to-quantum temperature-dependent rescaling kT.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"formula = intensity_formula(sc, :trace; kT)","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"Recall that langle S^alphabeta(qomega)rangle is only available at certain discrete q values, due to the finite lattice size. There are two basic approaches to handling this discreteness. The first approach is to interpolate between the available data using intensities_interpolated. For example, we can plot single-q slices at (0,0,0) and (π,π,π) using this method:","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"qs = [[0, 0, 0], [0.5, 0.5, 0.5]]\nis = intensities_interpolated(sc, qs, formula; interpolation = :round)\n\nωs = available_energies(sc)\nfig = lines(ωs, is[1,:]; axis=(xlabel=\"meV\", ylabel=\"Intensity\"), label=\"(0,0,0)\")\nlines!(ωs, is[2,:]; label=\"(π,π,π)\")\naxislegend()\nfig","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"The resolution in energy can be improved by increasing nω (and decreasing Δt), and the general accuracy can be improved by collecting additional samples from the thermal equilibrium.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"For real calculations, one often wants to apply further corrections and more accurate formulas. Here, we apply FormFactor corrections appropriate for Fe2 magnetic ions, and a dipole polarization correction :perp.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"formfactors = [FormFactor(\"Fe2\"; g_lande=3/2)]\nnew_formula = intensity_formula(sc, :perp; kT, formfactors = formfactors)","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"Frequently, one wants to extract energy intensities along lines that connect special wave vectors–a so-called \"spaghetti plot\". The function reciprocal_space_path creates an appropriate horizontal axis for this plot by linearly sampling between provided q-points with a given sample density. The number of sample points between two wavevectors q1 and q2 is given by dist*density where dist = norm(cryst.recipvecs * (q1 - q2)) is measured in the global frame.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"points = [[0, 0, 0], # List of wave vectors that define a path\n [1, 0, 0],\n [0, 1, 0],\n [1/2, 0, 0],\n [0, 1, 0],\n [0, 0, 0]]\ndensity = 40\npath, xticks = reciprocal_space_path(cryst, points, density);\nnothing #hide","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"Again using intensities_interpolated, we can evaluate the (interpolated) intensity at each point on the path. Since scattering intensities are only available at a certain discrete (Qomega) points, the intensity on the path can be calculated by interpolating between these discrete points:","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"is_interpolated = intensities_interpolated(sc, path, new_formula;\n interpolation = :linear, # Interpolate between available wave vectors\n);\n# Add artificial broadening\nis_interpolated_broadened = broaden_energy(sc, is, (ω, ω₀)->lorentzian(ω-ω₀, 0.05));\nnothing #hide","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"The second approach to handle the discreteness of the data is to bin the intensity at the discrete points into the bins of a histogram. First, the five sub-histograms are set up using reciprocal_space_path_bins in analogy to reciprocal_space_path.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"cut_width = 0.3\ndensity = 15\nparamsList, markers, ranges = reciprocal_space_path_bins(sc,points,density,cut_width);\nnothing #hide","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"Then, the intensity data is computed using intensities_binned for each sub-histogram:","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"total_bins = ranges[end][end]\nenergy_bins = paramsList[1].numbins[4]\nis_binned = zeros(Float64,total_bins,energy_bins)\nintegrated_kernel = integrated_lorentzian(0.05) # Lorentzian broadening\nfor k in eachindex(paramsList)\n bin_data, counts = intensities_binned(sc,paramsList[k], new_formula;\n integrated_kernel = integrated_kernel\n )\n is_binned[ranges[k],:] = bin_data[:,1,1,:] ./ counts[:,1,1,:]\nend","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"The graph produced by interpolating (top) is similar to the one produced by binning (bottom):","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"fig = Figure()\nax_top = Axis(fig[1,1],ylabel = \"meV\",xticklabelrotation=π/8,xticklabelsize=12;xticks)\nax_bottom = Axis(fig[2,1],ylabel = \"meV\",xticks = (markers, string.(points)),xticklabelrotation=π/8,xticklabelsize=12)\n\nheatmap!(ax_top,1:size(is_interpolated,1), ωs, is_interpolated;\n colorrange=(0.0,0.07),\n)\n\nheatmap!(ax_bottom,1:size(is_binned,1), ωs, is_binned;\n colorrange=(0.0,0.05),\n)\n\nfig","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"Note that we have clipped the colors in order to make the higher-energy excitations more visible.","category":"page"},{"location":"examples/fei2_classical.html#Unconventional-RLU-Systems-and-Constant-Energy-Cuts","page":"FeI₂ at Finite Temperature","title":"Unconventional RLU Systems and Constant Energy Cuts","text":"","category":"section"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"Often it is useful to plot cuts across multiple wave vectors but at a single energy. We'll pick an energy,","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"ωidx = 60\ntarget_ω = ωs[ωidx]","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"and take a constant-energy cut at that energy. The most straightforward way is to make a plot whose axes are aligned with the conventional reciprocal lattice of the crystal. This is accomplished using unit_resolution_binning_parameters:","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"params = unit_resolution_binning_parameters(sc)\nparams.binstart[1:2] .= -1 # Expand plot range slightly\n\n# Set energy integration range\nomega_width = 0.3\nparams.binstart[4] = target_ω - (omega_width/2)\nparams.binend[4] = target_ω # `binend` should be inside (e.g. at the center) of the range\nparams.binwidth[4] = omega_width\n\nintegrate_axes!(params, axes = 3) # Integrate out z direction entirely\n\nparams","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"In each of the following plots, black dashed lines represent (direct) lattice vectors. Since these plots are in reciprocal space, direct lattice vectors are represented as covectors (i.e. coordinate grids) instead of as arrows.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"is, counts = intensities_binned(sc,params,new_formula)\n\nfig = Figure()\nax = Axis(fig[1,1];\n title=\"Δω=0.3 meV (Binned)\", aspect=true,\n xlabel = \"[H, 0, 0]\",\n ylabel = \"[0, K, 0]\"\n)\nbcs = axes_bincenters(params)\nhm = heatmap!(ax,bcs[1],bcs[2],is[:,:,1,1] ./ counts[:,:,1,1])\nfunction add_lines!(ax,params)#hide\n bes = Sunny.axes_binedges(params)#hide\n hrange = range(-2,2,length=17)#hide\n linesegments!(ax,[(Point2f(params.covectors[1,1:3] ⋅ [h,-10,0],params.covectors[2,1:3] ⋅ [h,-10,0]),Point2f(params.covectors[1,1:3] ⋅ [h,10,0],params.covectors[2,1:3] ⋅ [h,10,0])) for h = hrange],linestyle=:dash,color=:black)#hide\n krange = range(-2,2,length=17)#hide\n linesegments!(ax,[(Point2f(params.covectors[1,1:3] ⋅ [-10,k,0],params.covectors[2,1:3] ⋅ [-10,k,0]),Point2f(params.covectors[1,1:3] ⋅ [10,k,0],params.covectors[2,1:3] ⋅ [10,k,0])) for k = krange],linestyle=:dash,color=:black)#hide\n xlims!(ax,bes[1][1],bes[1][end])#hide\n ylims!(ax,bes[2][1],bes[2][end])#hide\nend#hide\nadd_lines!(ax,params)\nColorbar(fig[1,2], hm);\nfig","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"In the above plot, the dashed-line (direct) lattice vectors are clearly orthogonal. However, we know that in real space, the lattice vectors a and b are not orthogonal, but rather point along the edges of a hexagon (see lower left corner):","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"

      ","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"Thus, plotting the direct lattice vectors as orthogonal (even in reciprocal space) is somewhat misleading. Worse yet, the [H,0,0] by [0,K,0] plot apparently loses the 6-fold symmetry of the crystal! Lastly, if one works out the components of the real-space metric with respect to the axes of the plot, one finds that there are non-zero off-diagonal entries,","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"latvecs = sys.crystal.latvecs\nmetric = latvecs' * I(3) * latvecs","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"so real-space rotations and angles map into reciprocal space rotations angles in a complicated way.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"To resolve these important issues, we want to use axes which are orthogonal (i.e. they diagonalize the metric and solve all of the problems just mentioned). The canonical choice is to use the combination frac12a + b of lattice vectors (equiv. a^* - frac12b^*), which is orthogonal to a:","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"(latvecs * [1/2,1,0]) ⋅ (latvecs * [1,0,0]) == 0","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"This new vector frac12a+b is visibly orthogonal to a in real space:","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"f = Figure()#hide\nax = Axis(f[1,1])#hide\narrows!(ax,[Point2f(0,0),Point2f(latvecs[1:2,1] ./ 2)],[Vec2f(latvecs[1:2,1] ./ 2), Vec2f(latvecs[1:2,2])],arrowcolor = :blue,arrowsize = 30.,linewidth = 5.,linecolor = :blue)#hide\narrows!(ax,[Point2f(0,0)],[Vec2f(latvecs[1:2,:] * [1/2,1,0])],arrowcolor = :red,arrowsize = 30.,linewidth = 5.,linecolor = :red, linestyle = :dash)#hide\nscatter!(ax,[Point2f(latvecs[1:2,:] * [a,b,0]) for a in -1:1, b in -1:1][:],color = :black)#hide\nannotations!(ax,[\"0\",\"0+b\",\"0+a\", \"a/2\", \"b\"],[Point2f(0,-0.3),Point2f(latvecs[1:2,2]) .- Vec2f(0,0.3),Point2f(latvecs[1:2,1]) .- Vec2f(0,0.3),Point2f(latvecs[1:2,1] ./ 4) .- Vec2f(0,0.3),Point2f(latvecs[1:2,1] ./ 2) .+ Vec2f(latvecs[1:2,2] ./ 2) .+ Vec2f(0.3,0.3)],color=[:black,:black,:black,:blue,:blue])#hide\nf#hide","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"To use \"projection onto the new vector\" as a histogram axis, only a single change is needed to the binning parameters. The second covector (previously b) must be swapped out for frac12a + b (recall that reciprocal space covectors, such as those used in BinningParameters correspond to direct space vectors).","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"params.covectors[2,1:3] = [1/2,1,0] # [1/2,1,0] times [a;b;c] is (a/2 + b)\nparams#hide","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"The second axis of the histogram now agrees with what is conventionally labelled as [H,-H/2,0].","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"warning: Length of the new vector\nNote that, although frac12a+b is orthogonal to a, it is not the same length as a. Instead, it is sqrt(3/4) times as long. Note the unsymmetrical axes labels in the plots that follow as a direct result of this!","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"# Zoom out horizontal axis\nparams.binstart[1], params.binend[1] = -2, 2\n\n# Adjust vertical axis bounds to account for\n# length of a/2 + b\nparams.binstart[2], params.binend[2] = -2 * sqrt(3/4), 2 * sqrt(3/4)\n\n# Re-compute in the new coordinate system\nis, counts = intensities_binned(sc,params,new_formula)\n\nfig = Figure(; resolution=(1200,500))#hide\nax_right = Axis(fig[1,3];#hide\n title=\"ω≈$(round(target_ω, digits=2)) meV with Δω=0.3 meV (Binned)\", aspect=true,#hide\n xlabel = \"[H, -1/2H, 0]\"#hide\n)#hide\nbcs = axes_bincenters(params)#hide\nhm_right = heatmap!(ax_right,bcs[1],bcs[2],is[:,:,1,1] ./ counts[:,:,1,1])#hide\nadd_lines!(ax_right,params)\nColorbar(fig[1,4], hm_right);#hide\nnothing #hide","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"For comparison purposes, we will make the same plot using intensities_interpolated to emulate zero-width bins. This time, it's more convenient to think in terms of reciprocal vectors a^* and b^*. Now, our coordinate transformation consists of establishing a new, orthogonal basis to specify our wave vectors: a^* - frac12b^*, b^* and c^*. Writing this in matrix form allows us to sample a rectilinear grid of wave vectors in this frame. Finally, we'll convert these back into the original RLU system for input into Sunny.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"# New basis matrix\nA = [1 0 0\n -1/2 1 0\n 0 0 1]\n\n# Define our grid of wave vectors\nnpoints = 60\nas = range(-2, 2, npoints)\nbs = range(-3/√3, 3/√3, npoints)\nqs_ortho = [[a, b, 0] for a in as, b in bs]\n\n# Convert to original RLU system for input to Sunny\nqs = [A * q for q in qs_ortho]\n\n# Use interpolation to get intensities\nis = intensities_interpolated(sc, qs, new_formula; interpolation=:linear)\n\nax_left = Axis(fig[1,2];#hide\n title=\"ω≈$(round(ωs[ωidx], digits=2)) meV (Interpolated)\", aspect=true,#hide\n xlabel = \"[H, -1/2H, 0]\", ylabel = \"[0, K, 0]\"#hide\n)#hide\nhm_left = heatmap!(ax_left, as, bs, is[:,:,ωidx])#hide\nadd_lines!(ax_left,params)\nColorbar(fig[1,1], hm_left);#hide\nfig","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"Now, not only are the dashed-line lattice vectors no longer misleadingly orthogonal, but the six-fold symmetry has been restored as well! Further, the metric has been diagonalized:","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"metric = (latvecs * inv(A'))' * I(3) * (latvecs * inv(A'))","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"Finally, we note that instantaneous structure factor data, 𝒮(𝐪), can be obtained from a dynamic structure factor with instant_intensities_interpolated. Here we'll reuse the grid of wave vectors we generated above.","category":"page"},{"location":"examples/fei2_classical.html","page":"FeI₂ at Finite Temperature","title":"FeI₂ at Finite Temperature","text":"is_static = instant_intensities_interpolated(sc, qs, new_formula; interpolation = :linear)\n\nhm = heatmap(as, bs, is_static;\n axis=(\n title=\"Instantaneous Structure Factor\",\n xlabel = \"[H, -1/2H, 0]\",\n ylabel = \"[0, K, 0]\",\n aspect=true\n )\n)\nColorbar(hm.figure[1,2], hm.plot)\nhm","category":"page"},{"location":"anisotropy.html#Interaction-Strength-Renormalization","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"","category":"section"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"A unique feature of Sunny is its support for building classical models where each quantum spin is represented as a full N-level system, rather than just an expected dipole. This formalism enables more accurate modeling of quantum spin Hamiltonians that include, e.g., a single-ion anisotropy, or a biquadratic coupling between sites.","category":"page"},{"location":"anisotropy.html#Local-operators","page":"Interaction Strength Renormalization","title":"Local operators","text":"","category":"section"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"A quantum spin-S state has N = 2S + 1 levels. Each local spin operator hatS^xyz is faithfully represented as an NN matrix. Access these matrices using spin_matrices for a given label S. For example, spin_matrices(1/2) returns the Pauli matrices divided by 2.","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"When S 12, it is possible to construct multipole moments beyond the spin-dipole. For example,","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"XXX TODO XXX","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"S = spin_matrices(2)","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"The Stevens operators hatmathcalO_kq are polynomials of the spin operators, and are accessed using stevens_matrices. With these building blocks, a single-ion anisotropy is defined using set_onsite_coupling!. For example:","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"# An easy axis anisotropy in the z-direction\nS = spin_operators(sys, i)\nset_onsite_coupling!(sys, -D*S[3]^3, i)\n\n# The unique quartic single-ion anisotropy for a site with cubic point group\n# symmetry\nO = stevens_operators(sys, i)\nset_onsite_coupling!(sys, O[4,0] + 5*O[4,4], i)\n\n# An equivalent expression of this quartic anisotropy, up to a constant shift\nset_onsite_coupling!(sys, 20*(S[1]^4 + S[2]^4 + S[3]^4), i)","category":"page"},{"location":"anisotropy.html#Renormalization-procedure-for-:dipole-mode","page":"Interaction Strength Renormalization","title":"Renormalization procedure for :dipole mode","text":"","category":"section"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"There are two allowed modes for a System. The mode :SUN models each spin as an SU(N) coherent state (i.e., as a set of N complex amplitudes), and is the most variationally accurate. The mode :dipole constrains the SU(N) coherent-state dynamics to the space of pure dipoles. In either mode, Sunny encourages specifying single-ion anisotropies as NN matrices. In :dipole mode, Sunny will automatically renormalize the anisotropy operator to achieve maximal consistency with :SUN mode. This procedure was derived in D. Dahlbom et al., [arXiv:2304.03874]. Here, we summarize the final results.","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"The starting point is a quantum operator hatmathcalH_mathrmlocal giving the single-ion anisotropy for one site. It can be expanded in Stevens operators,","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"hatmathcal H_mathrmlocal = sum_k q A_kq hatmathcalO_kq","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"See the documentation of print_stevens_expansion for some explicit examples of this expansion.","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"The traditional classical limit of a quantum spin Hamiltonian, which yields the Landau-Lifshitz dynamics, can be derived by taking the formal S toinfty limit, such that each spin operator hatmathbfS is replaced by its dipole expectation value mathbfs. Correspondingly, the Stevens operators hatmathcalO_kq become polynomials mathcalO_kq(mathbfs) in the classical dipole. With this traditional approach, one would arrive at the bare expected energy,","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"H_mathrmbare(mathbfs) = sum_k q A_kq mathcalO_kq(mathbfs)","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"In a real magnetic compound, however, S may not be very large, and one can achieve a better approximation by avoiding the S toinfty limit. The strategy is to begin with the full dynamics of SU(N) coherent states, and then constrain it to the space of dipoles mathbfs. Doing so will again yield the Landau-Lifshitz dynamics, but now involving the renormalized expected energy,","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"H_mathrmrenormalized(mathbfs) = sum_k q c_k A_kq mathcalO_kq(mathbfs)","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"The k-dependent renormalization factors are","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"beginalign*\nc_2 = 1-frac12S^-1 \nc_4 = 1-3S^-1+frac114S^-2-frac34S^-3 \nc_6 = 1-frac152S^-1+frac854S^-2-frac2258S^-3+frac1378S^-4-frac154S^-5\nendalign*","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"Sunny will use H_mathrmrenormalized(mathbfs) in its classical dynamics of dipoles. Because of this renormalization, Sunny is more variationally accurate than traditional codes like SpinW.","category":"page"},{"location":"anisotropy.html#How-and-when-to-disable-renormalization?","page":"Interaction Strength Renormalization","title":"How and when to disable renormalization?","text":"","category":"section"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"Although we generally recommend the above renormalization procedure, there are circumstances where it is not desirable. Examples include reproducing a model-system study, or describing a micromagnetic system for which the Stoinfty limit is quantitatively realized. To get symbolic operators in the large-S limit, use spin_matrices or stevens_matrices with the argument Inf. Sunny will not perform any renormalization on anisotropy operators constructed through these primitives.","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"Note that Sunny will also renormalize scalar biquadratic exchange interactions by default. Disable this renormalization by setting large_S = true in the call to set_exchange!.","category":"page"},{"location":"anisotropy.html#Stevens-operators","page":"Interaction Strength Renormalization","title":"Stevens operators","text":"","category":"section"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"The Stevens operators hatmathcalO_kq are defined as polynomials of angular momentum operators hatS_xyz in some spin-S representation.","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"Using","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"beginalign*\nX = mathbfhatS cdot mathbfhatS = S (S+1) \nhatS_pm = hatS_x pm i hatS_y \nphi_+ = frac14quad phi_- = frac14 i\nendalign*","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"the relevant Stevens operators are defined as,","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"beginalign*\nhatmathcalO_00 = 1 \n\nhatmathcalO_2pm2 =phi_pm(hatS_+^2pm hatS_-^2)+mathrmhc\nhatmathcalO_2pm1 =phi_pm(hatS_+pm hatS_-)hatS_z+mathrmhc\nhatmathcalO_20 =3hatS_z^2-X\n\nhatmathcalO_4pm4 =phi_pm(hatS_+^4pm hatS_-^4)+mathrmhc\nhatmathcalO_4pm3 =phi_pm(hatS_+^3pm hatS_-^3)hatS_z+mathrmhc\nhatmathcalO_4pm2 =phi_pm(hatS_+^2pm hatS_-^2)(7hatS_z^2-(X+5))+mathrmhc\nhatmathcalO_4pm1 =phi_pm(hatS_+pm hatS_-)(7hatS_z^3-(3X+1)hatS_z)+mathrmhc\nhatmathcalO_40 =35hatS_z^4-(30X-25)hatS_z^2+(3X^2-6X)\n\nhatmathcalO_6pm6 =phi_pm(hatS_+^6pm hatS_-^6)+mathrmhc\nhatmathcalO_6pm5 =phi_pm(hatS_+^5pm hatS_-^5)hatS_z+mathrmhc\nhatmathcalO_6pm4 =phi_pm(hatS_+^4pm hatS_-^4)(11hatS_z^2-X-38)+mathrmhc\nhatmathcalO_6pm3 =phi_pm(hatS_+^3pm hatS_-^3)(11hatS_z^3-(3X+59)hatS_z)+mathrmhc\nhatmathcalO_6pm2 =phi_pm(hatS_+^2pm hatS_-^2)(33hatS_z^4-(18X+123)hatS_z^2+X^2+10X+102)+mathrmhc\nhatmathcalO_6pm1 =phi_pm(hatS_+pm hatS_-)(33hatS_z^5-(30X-15)hatS_z^3+(5X^2-10X+12)hatS_z)+mathrmhc\nhatmathcalO_60 =231hatS_z^6-(315X-735)hatS_z^4+(105X^2-525X+294)hatS_z^2-5X^3+40X^2-60X\nendalign*","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"Stevens operators hatmathcalO_kq for odd k are disallowed from the single-ion anisotropy under the assumption of time-reversal symmetry. Computer-generated tables of Stevens operators with larger k are available from C. Rudowicz and C. Y. Chung, J. Phys.: Condens. Matter 16, 5825 (2004).","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"For each k value, the collection of operators hatmathcalO_kq for q = -k dots k is an irreducible representation of the group of rotations O(3). In particular, a physical rotation will transform hatmathcalO_kq into a linear combination of hatmathcalO_kq where q varies but k remains fixed. ","category":"page"},{"location":"anisotropy.html","page":"Interaction Strength Renormalization","title":"Interaction Strength Renormalization","text":"In taking the large-S limit, each dipole operator is replaced by its expectation value mathbfs = langle hatmathbfS rangle, and only leading-order terms are retained. The operator hatmathcalO_kq becomes a homogeneous polynomial O_kq(mathbfs) of order k in the spin components. One can see these polynomials by constructing stevens_matrices with the argument S = Inf. Due to the normalization constraint, each dipole can be expressed in polar angles, (theta phi). Then the Stevens functions O_kq(mathbfs) correspond to the spherical harmonic functions Y_l^m(theta phi) where l=k and m=q, and modulo k and q-dependent rescaling factors.","category":"page"},{"location":"versions.html#Version-History","page":"Version History","title":"Version History","text":"","category":"section"},{"location":"versions.html#v0.5.6","page":"Version History","title":"v0.5.6","text":"","category":"section"},{"location":"versions.html","page":"Version History","title":"Version History","text":"General pair couplings are now supported in set_pair_coupling!. \nDeprecate spin_operators, stevens_operators, large_S_spin_operators and large_S_stevens_operators. Use instead spin_matrices and stevens_matrices, which require a specific spin-S label. To infer this, one can use spin_irrep_label.","category":"page"},{"location":"versions.html#v0.5.5","page":"Version History","title":"v0.5.5","text":"","category":"section"},{"location":"versions.html","page":"Version History","title":"Version History","text":"reshape_supercell now allows reshaping to multiples of the primitive unit cell, which can speed up certain calculations. This is illustrated in the CoRh₂O₄ powder averaging tutorial.\nresize_supercell now allows all resizings.\nAdded energy_per_site.\nset_spiral_order_on_sublattice! cannot work on reshaped systems.\nVarious bug fixes. In particular, an intensity_formula with :full will now uniformly calculate a 3x3 matrix of complex numbers.","category":"page"},{"location":"versions.html#v0.5.4","page":"Version History","title":"v0.5.4","text":"","category":"section"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Various enhancements to view_crystal. Atoms are now labeled by index, and bonds support interactive inspection (GLMakie only). Font sizes work correctly on Makie v0.20-beta. If using Makie v0.19 on a high-resolution display, pass rescale=1.5 to enlarge font sizes.\nThe function suggest_magnetic_supercell now requires only a list of wavevectors, and will return a 33 matrix that can be programmatically passed to reshape_supercell. The new tolerance parameter tol allows suggest_magnetic_supercell to approximate incommensurate wavevectors with nearby commensurate ones.\nNew functions set_spiral_order! and set_spiral_order_on_sublattice! can be used to initialize a spiral, single-Q order.\nSunny now retains all constant energy shifts that have been introduced by anisotropy operators.\nFix export_vtk functionality.","category":"page"},{"location":"versions.html#v0.5.3","page":"Version History","title":"v0.5.3","text":"","category":"section"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Add large_S_spin_operators and large_S_stevens_operators to support single-ion anisotropies in dipole mode without renormalization. Set large_S=true in set_exchange! to avoid renormalization of biquadratics.\nview_crystal has been rewritten in Makie.\nplot_spins now expects ghost_radius in physical length units.\nSpinWaveTheory will (currently) error if provided a system with enable_dipole_dipole!.","category":"page"},{"location":"versions.html#v0.5.2","page":"Version History","title":"v0.5.2","text":"","category":"section"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Form factors for 5d transition ions.\nDownload links for notebooks and scripts on each doc example\nVarious bug fixes.","category":"page"},{"location":"versions.html#v0.5.1","page":"Version History","title":"v0.5.1","text":"","category":"section"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Fix binning edge cases.\nplot_spins accepts resolution argument.","category":"page"},{"location":"versions.html#v0.5.0","page":"Version History","title":"v0.5.0","text":"","category":"section"},{"location":"versions.html","page":"Version History","title":"Version History","text":"New features.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Support for Linear Spin Wave Theory in :dipole and :SUN modes. (Thanks Hao Zhang!)","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"New function minimize_energy! to efficiently find an optimal configuration of spin dipoles or SU(N) coherent states.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Major refactors and enhancements to intensity calculations. This new interface allows unification between LSWT and classical spin dynamics calculations. This interface allows: Custom observables as local quantum operators, better support for linebroadening, and automatic binning to facilitate comparison with experimental data. See intensity_formula for documentation. Use load_nxs to load experimental neutron scattering data.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Breaking changes.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Require Julia 1.9.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Replace set_anisotropy! with a new function set_onsite_coupling! (and similarly set_onsite_coupling_at!). The latter expects an explicit matrix representation for the local Hamiltonian. This can be constructed, e.g., as a linear combination of stevens_operators, or as a polynomial of spin_operators. To understand the mapping between these two, the new function print_stevens_expansion acts on an arbitrary local operator.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Remove set_biquadratic!. Instead, use an optional keyword argument biquad to set_exchange!.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Rename DynamicStructureFactor to dynamical_correlations. Similarly, replace InstantStructureFactor with instant_correlations. The return type has been renamed SampledCorrelations to emphasize that the object may be holding thermodynamic samples, which are collected using add_sample!.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Remove intensities function. Instead, use one of intensities_interpolated or intensities_binned. These will require an intensity_formula, which defines a calculator (e.g., LSWT).","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Rename connected_path to reciprocal_space_path, which now returns an xticks object that can be used in plotting. Replace spherical_shell with reciprocal_space_shell that functions similarly.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Rename polarize_spin! to set_dipole! for consistency with set_coherent!. The behavior of the former function is unchanged: the spin at a given site will still be polarized along the provided direction.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Rename all_sites to eachsite consistent with Julia convention for iterators.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Rename reshape_geometry to reshape_supercell, which is the fundamental reshaping function. Rename resize_periodically to resize_supercell.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"The constructor SpinInfo now requires a g-factor or tensor as a named argument.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"The constructor FormFactor no longer accepts an atom index. Instead, the form factors are associated with site-symmetry classes in order of appearance.","category":"page"},{"location":"versions.html#v0.4.3","page":"Version History","title":"v0.4.3","text":"","category":"section"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Experimental support for linear SpinWaveTheory, implemented in SU(N) mode. This module may evolve rapidly.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Implement renormalization of single-ion anisotropy and biquadratic interactions when in :dipole mode. This makes the model more faithful to the quantum mechanical Hamiltonian, but is also a breaking change.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Various improvements and bugfixes for to_inhomogeneous. Setting inhomogeneous interactions via set_exchange_at! should now infer the correct bond offset direction, or will report an ambiguity error. Ambiguities can be resolved by passing an explicit offset.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"The function remove_periodicity! disables periodicity along specified dimensions.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Rename StaticStructureFactor to InstantStructureFactor.","category":"page"},{"location":"versions.html#v0.4.2","page":"Version History","title":"v0.4.2","text":"","category":"section"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Introduce LocalSampler, a framework for MCMC sampling with local spin updates.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Rename print_dominant_wavevectors to print_wrapped_intensities to reduce confusion with the physical instantaneous intensities.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"The function spherical_shell now takes a radius in physical units of inverse Å.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"New exported functions global_position, magnetic_moment, all_sites.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Remove all uses of Base.deepcopy which resolves crashes.","category":"page"},{"location":"versions.html#v0.4.1","page":"Version History","title":"v0.4.1","text":"","category":"section"},{"location":"versions.html","page":"Version History","title":"Version History","text":"The function to_inhomogeneous creates a system that supports inhomogeneous interactions, which can be set using set_exchange_at!, etc.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"set_biquadratic! replaces set_exchange_with_biquadratic!.","category":"page"},{"location":"versions.html#v0.4.0","page":"Version History","title":"v0.4.0","text":"","category":"section"},{"location":"versions.html","page":"Version History","title":"Version History","text":"This update includes many breaking changes, and is missing some features of 0.3.0.","category":"page"},{"location":"versions.html#Creating-a-spin-System","page":"Version History","title":"Creating a spin System","text":"","category":"section"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Rename SpinSystem to System. Its constructor now has the form,","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"System(crystal, latsize, infos, mode)","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"The parameter infos is now a list of SpinInfo objects. Each defines spin angular momentum S = frac12 1 frac32 , and an optional g-factor or tensor.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"The parameter mode is one of :SUN or :dipole.","category":"page"},{"location":"versions.html#Setting-interactions","page":"Version History","title":"Setting interactions","text":"","category":"section"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Interactions are now added mutably to an existing System using the following functions: set_external_field!, set_exchange!, set_onsite_coupling!, enable_dipole_dipole!.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"As a convenience, one can use dmvec(D) to convert a DM vector to a 33 antisymmetric exchange matrix.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Fully general single-ion anisotropy is now possible. The function set_onsite_coupling! expects the single ion anisotropy to be expressed as a polynomial in symbolic spin operators 𝒮, or as a linear combination of symbolic Stevens operators 𝒪. For example, an easy axis anisotropy in the direction n may be written D*(𝒮⋅n)^2.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Stevens operators 𝒪[k,q] admit polynomial expression in spin operators 𝒮[α]. Conversely, a polynomial of spin operators can be expressed as a linear combination of Stevens operators. To see this expansion use print_anisotropy_as_stevens.","category":"page"},{"location":"versions.html#Inhomogeneous-field","page":"Version History","title":"Inhomogeneous field","text":"","category":"section"},{"location":"versions.html","page":"Version History","title":"Version History","text":"An external field can be applied to a single site with set_external_field_at!. ","category":"page"},{"location":"versions.html#Structure-factor-rewrite","page":"Version History","title":"Structure factor rewrite","text":"","category":"section"},{"location":"versions.html","page":"Version History","title":"Version History","text":"The calculation of structure factors has been completely rewritten. For the new interface, see the FeI₂ at Finite Temperature page.","category":"page"},{"location":"versions.html#Various","page":"Version History","title":"Various","text":"","category":"section"},{"location":"versions.html","page":"Version History","title":"Version History","text":"The \"Sampler\" interface is in flux. Langevin replaces both LangevinHeunP and LangevinSampler. Local spin-flip Monte Carlo sampling methods are temporarily broken.\nrepeat_periodically replaces extend_periodically.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"Additional related functions include resize_periodically and reshape_geometry, the latter being fundamental.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"print_symmetry_table replaces print_bond_table().","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"The new function includes the list of symmetry-allowed single ion anisotropies in addition to exchange interactions.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"When reading CIF files, the field _atom_site_label is now used in place of the field _atom_site_type_symbol.","category":"page"},{"location":"versions.html","page":"Version History","title":"Version History","text":"This is required for correctness. The field _atom_site_label is guaranteed to be present, and is guaranteed to be a distinct label for each symmetry-inequivalent site. Code that explicitly referred to site labels (e.g. in calls to subcrystal) will need to be updated to use the new label.","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"EditURL = \"../../../../examples/spinw_ports/15_Ba3NbFe3Si2O14.jl\"","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"Download this example as Jupyter notebook or Julia script.","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html#BaNbFeSiO","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"","category":"section"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"This is a Sunny port of SpinW Tutorial 15, authored by Toth et al. The goal is to calculate the linear spin wave theory spectrum for Ba₃NbFe₃Si₂O₁₄.","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"Load packages","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"using Sunny, GLMakie","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"Build a Crystal for Ba₃NbFe₃Si₂O₁₄ using the crystal structure from Marty et al., Phys. Rev. Lett. 101, 247201 (2008).","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"a = b = 8.539 # (Å)\nc = 5.2414\nlatvecs = lattice_vectors(a, b, c, 90, 90, 120)\ntypes = [\"Fe\",\"Nb\",\"Ba\",\"Si\",\"O\",\"O\",\"O\"]\npositions = [[0.24964,0,0.5],[0,0,0],[0.56598,0,0],[2/3,1/3,0.5220],[2/3,1/3,0.2162],[0.5259,0.7024,0.3536],[0.7840,0.9002,0.7760]]\nlangasite = Crystal(latvecs, positions, 150; types)\ncrystal = subcrystal(langasite, \"Fe\")\nview_crystal(crystal, 7)","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"Create a System with a lattice size of (117). The magnetic structure of Ba₃NbFe₃Si₂O₁₄ was determined to have the ordering wavevector 𝐐=(0017) and hence the magnetic unit cell has 7 sites.","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"latsize = (1,1,7)\nS = 5/2\nseed = 5\nsys = System(crystal, latsize, [SpinInfo(1; S, g=2)], :dipole)","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"Set exchange interactions as parametrized in Loire et al., Phys. Rev. Lett. 106, 207201 (2011)","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"J₁ = 0.85\nJ₂ = 0.24\nJ₃ = 0.053\nJ₄ = 0.017\nJ₅ = 0.24\nset_exchange!(sys, J₁, Bond(3, 2, [1,1,0]))\nset_exchange!(sys, J₄, Bond(1, 1, [0,0,1]))\nset_exchange!(sys, J₂, Bond(1, 3, [0,0,0]))","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"The final two exchanges define the chirality of the magnetic structure. The crystal chirality, epsilon_T, the chirality of each triangle, ϵ_D and the sense of rotation of the spin helices along c, ϵ_H. The three chiralities are related by ϵ_T=ϵ_D ϵ_H. We now assign J_3 and J_5 according to the crystal chirality.","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"ϵD = -1\nϵH = +1\nϵT = ϵD * ϵH\n\nif ϵT == -1\n set_exchange!(sys, J₃, Bond(2, 3, [-1,-1,1]))\n set_exchange!(sys, J₅, Bond(3, 2, [1,1,1]))\nelseif ϵT == 1\n set_exchange!(sys, J₅, Bond(2, 3, [-1,-1,1]))\n set_exchange!(sys, J₃, Bond(3, 2, [1,1,1]))\nelse\n throw(\"Provide a valid chirality\")\nend","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"Whilst Sunny provides tools to optimize the ground state automatically, in this case we already know the model ground state. Set the spiral magnetic order using set_spiral_order_on_sublattice!. It takes an ordering wavevector q, an axis of rotation for the spins axis, and the initial spin S0 for each sublattice.","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"q = [0, 0, 1/7]\naxis = [0,0,1]\nset_spiral_order_on_sublattice!(sys, 1; q, axis, S0=[1, 0, 0])\nset_spiral_order_on_sublattice!(sys, 2; q, axis, S0=[-1/2, -sqrt(3)/2, 0])\nset_spiral_order_on_sublattice!(sys, 3; q, axis, S0=[-1/2, +sqrt(3)/2, 0])\n\nplot_spins(sys; color=[s[1] for s in sys.dipoles])","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"Define a path in reciprocal space, 01-1+xi for xi = 0 dots 3.","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"points_rlu = [[0,1,-1],[0,1,-1+1],[0,1,-1+2],[0,1,-1+3]];\ndensity = 100\npath, xticks = reciprocal_space_path(crystal, points_rlu, density);\nnothing #hide","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"Calculate broadened intensities","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"swt = SpinWaveTheory(sys)\nγ = 0.15 # width in meV\nbroadened_formula = intensity_formula(swt, :perp; kernel=lorentzian(γ))\nenergies = collect(0:0.01:6) # 0 < ω < 6 (meV).\nis = intensities_broadened(swt, path, energies, broadened_formula);\nnothing #hide","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"Plot","category":"page"},{"location":"examples/spinw/15_Ba3NbFe3Si2O14.html","page":"Ba₃NbFe₃Si₂O₁₄","title":"Ba₃NbFe₃Si₂O₁₄","text":"fig = Figure()\nax = Axis(fig[1,1]; xlabel=\"Momentum (r.l.u.)\", ylabel=\"Energy (meV)\", xticks, xticklabelrotation=π/6)\nheatmap!(ax, 1:size(is,1), energies, is, colorrange=(0,5))\nfig","category":"page"},{"location":"index.html#Overview","page":"Overview","title":"Overview","text":"","category":"section"},{"location":"index.html","page":"Overview","title":"Overview","text":"Sunny is a Julia package for modeling atomic-scale magnetism. It provides powerful tools to study equilibrium and non-equilibrium magnetic phenomena. In particular, it allows estimation of dynamical structure factor intensities, mathcalS(𝐪ω), to support quantitative modeling of experimental scattering data.","category":"page"},{"location":"index.html","page":"Overview","title":"Overview","text":"Features include:","category":"page"},{"location":"index.html","page":"Overview","title":"Overview","text":"Generalized spin dynamics using SU(N) coherent states.\nAbility to specify a crystal from a .cif file or its spacegroup symmetry.\nInteractive visualizations of the 3D crystals and magnetic ordering.\nSymmetry analysis to classify allowed interaction terms, and to propagate them by symmetry.\nSingle-ion anisotropy at arbitrary order, which can be specified using Stevens operators or as a polynomial of spin operators.\nMonte Carlo sampling of spin configurations in thermal equilibrium, and optimization tools.\nMeasurements of dynamical correlations. At low temperature, one can use linear spin wave theory and its multi-boson generalization. This generalizes to finite temperatures using the classical dynamics, which allows for strongly nonlinear effects.\nLong-range dipole-dipole interactions accelerated with the fast Fourier transform (FFT).\nSupport for comparison with experimental data: form factor, dipole factor, temperature-dependent classical-to-quantum factors, intensity binning, etc.","category":"page"}] +} diff --git a/previews/PR178/siteinfo.js b/previews/PR178/siteinfo.js new file mode 100644 index 000000000..dd1db184d --- /dev/null +++ b/previews/PR178/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "previews/PR178"; diff --git a/previews/PR178/structure-factor.html b/previews/PR178/structure-factor.html new file mode 100644 index 000000000..2125398db --- /dev/null +++ b/previews/PR178/structure-factor.html @@ -0,0 +1,9 @@ + +Structure Factor Calculations · Sunny documentation
      GitHub

      Structure Factor Calculations

      Overview

      The dynamical structure factor is of fundamental importance for characterizing a magnetic system, and facilitates quantitative comparison between theory and experimental scattering data.

      Consider, for example, a two-point dynamical spin correlation function, $⟨s^α(𝐱+Δ𝐱, t+Δt) s^β(𝐱, t)⟩$. Here $s^α(𝐱, t)$ represents the time dynamics of a spin dipole component $α$ at position $𝐱$, and brackets represent an average over equilibrium initial conditions and over $(𝐱, t)$. The dynamical structure factor is defined as the Fourier transform of this two-point correlation in both space and time, up to an overall scaling factor. Using the convolution theorem, the result is,

      \[𝒮^{αβ}(𝐪, ω) = \frac{1}{V} ⟨ŝ^α(𝐪, ω)^\ast ŝ^β(𝐪, ω) ⟩,\]

      with $V$ the system volume. We will restrict attention to lattice systems with periodic boundaries.

      Consider a crystal unit cell defined by three lattice vectors $𝐚_1, 𝐚_2, 𝐚_3$, and linear system sizes $L_1, L_2, L_3$ measured in unit cells. The allowed momentum vectors take on discrete values $𝐪 = \sum_{α=1}^{3} m_α 𝐛_α / L_α$, where $m_α$ are an integers and the reciprocal lattice vectors $𝐛_α$ are defined to satisfy $𝐚_α ⋅ 𝐛_β = 2π δ_{α,β}$. For a Bravais lattice, $𝐪$ will be periodic in the first Brillouin zone, i.e., under any shift $𝐪 → 𝐪 ± 𝐛_α$. More generally, consider a non-Bravais lattice such that each unit cell may contain multiple spins. By partitioning spins $s_j(𝐱,t)$ according to their sublattice index $j$, the relevant momenta $𝐪$ remain discretized as above, but now periodicity in the first Brillouin zone is lost. The structure factor may be written as a phase-average over the displacements between sublattices $𝐫_{j,k}$,

      \[𝒮^{αβ}(𝐪, ω) = ∑_{j,k} e^{i 𝐫_{j,k} ⋅ 𝐪} 𝒮̃^{αβ}_{j,k}(𝐪, ω) ⟩,\]

      From a theoretical perspective, the quantity

      \[𝒮̃^{αβ}_{j,k}(𝐪, ω) = \frac{1}{V} ⟨ŝ_j^α(𝐪, ω)^\ast ŝ_k^β(𝐪, ω)⟩\]

      is fundamental. For each sublattice $j$, the data $ŝ_j^α(𝐪, ω)$ can be efficiently obtained by fast Fourier tranformation of a real space configuration $s_j^α(𝐱, t)$. Internally, Sunny will calculate and store the discrete $𝒮̃^{αβ}_{j,k}(𝐪, ω)$ correlation data, and use this to construct $𝒮^{αβ}(𝐪,ω)$ intensities that can be compared with experiment.

      Calculating this structure factor involves several steps, with various possible settings. Sunny provides a number of tools to facilitate this calculation and to extract information from the results. These tools are briefly outlined below. Please see the Examples for a "real life" use case. Detailed function information is available in the Library API.

      Estimating stucture factors with classical dynamics

      Classical dynamics may be used to estimate structure factor data by analyzing the spin-spin correlations of dynamical trajectories. This is fundamentally a Monte Carlo approach, as the trajectories must be started from an initial spin configuration that is sampled at thermal equilibrium. (Note that it is not possible to estimate a true T=0 dynamical structure factor using this method, but the temperature may be very low.) Samples are accumulated into a SampledCorrelations, from which intensity information may be extracted. The user does not typically build their own SampledCorrelations but instead initializes one by calling either dynamical_correlations or instant_correlations, as described below.

      Estimating a dynamical structure factor: $𝒮(𝐪,ω)$

      A SampledCorrelations for estimating the dynamical structure factor, $𝒮^{αβ}(𝐪,ω)$, may be created by calling dynamical_correlations. This requires three keyword arguments. These will determine the dynamics used to calculate samples and, consequently, the $ω$ information that will be available.

      1. Δt: Determines the step size used for simulating the dynamics. A smaller number will require proportionally more calculation time. While a smaller Δt will enable the resolution of higher energies, Δt is typically selected to ensure numerical stability rather than to maximize the largest $ω$ value. A safe choice is to use the smaller value of Δt = 0.1/(J* S^2) or Δt = 0.1/(D * S), where S is magnetic moment of the largest local spin (as specified in SpinInfo), J is the parameter governing the largest bilinear interaction (e.g. exchange), and D is the parameter governing the largest single-site term of the Hamiltonian (e.g., anisotropy or Zeeman term).
      2. ωmax: Sets the maximum resolved energy. Note that this is not independent of Δt. If ωmax too large, Sunny will throw an error and ask you to choose a smaller Δt.
      3. nω: Determines the number of energy bins to resolve. A larger number will require more calculation time.

      A sample may be added by calling add_sample!(sc, sys). The input sys must be a spin configuration in good thermal equilibrium, e.g., using the continuous Langevin dynamics or using single spin flip trials with LocalSampler. The statistical quality of the $𝒮^{αβ}(𝐪,ω)$ can be improved by repeatedly generating decorrelated spin configurations in sys and calling add_sample! on each configuration.

      The outline of typical use case might look like this:

      # Make a `SampledCorrelations`
+sc = dynamical_correlations(sys; Δt=0.05, ωmax=10.0, nω=100) 
+
+# Add samples
+for _ in 1:nsamples
+   decorrelate_system(sys) # Perform some type of Monte Carlo simulation
+   add_sample!(sc, sys)    # Use spins to calculate trajectory and accumulate new sample of 𝒮(𝐪,ω)
+end

      The calculation may be configured in a number of ways; see the dynamical_correlations documentation for a list of all keywords.

      Estimating an instantaneous ("static") structure factor: $𝒮(𝐪)$

      Sunny provides two methods for calculating instantaneous, or static, structure factors: $𝒮^{αβ}(𝐪)$. The first involves calculating spatial spin-spin correlations at single time slices. The second involves calculating a dynamic structure factor first and integrating out the $ω$ information. The advantage of the latter approach is that it enables application of an $ω$-dependent classical-to-quantum rescaling of structure factor intensities, a method that should be preferred whenever comparing results to experimental data or spin wave calculations. A disadvantage of this approach is that it is computationally more expensive. There are also many cases when it is not straightforward to calculate a meaningful dynamics, as when working with Ising spins. In this section we will discuss how to calculate instantaneous structure factors from static spin configurations. Information about calculating instantaneous data from a dynamical correlations can be found in the following section.

      The basic usage for the instantaneous case is very similar to the dynamic case, except one calls instant_correlations instead of dynamical_correlations to configure a SampledCorrelations. Note that there are no required keywords as there is no need to specify any dynamics. instant_correlations will return a SampledCorrelations containing no data. Samples may be added by calling add_sample!(sc, sys), where sc is the SampledCorrelations. When performing a finite-temperature calculation, it is important to ensure that the spin configuration in the sys represents a good equilibrium sample, as in the dynamical case. Note, however, that we recommend calculating instantaneous correlations at finite temperature calculations by using full dynamics (i.e., using dynamical_correlations) and then integrating out the energy axis. An approach to doing this is described in the next section.

      Extracting information from sampled correlation data

      The basic function for extracting information from a SampledCorrelations at a particular wave vector, $𝐪$, is intensities_interpolated. It takes a SampledCorrelations, a list of wave vectors, and an intensity_formula. The intensity_formula specifies how to contract and correct correlation data to arrive at a physical intensity. A simple example is formula = intensity_formula(sc, :perp), which will instruct Sunny apply polarization corrections: $\sum_{αβ}(I-q_α q_β) 𝒮^{αβ}(𝐪,ω)$. An intensity at the wave vector $𝐪 = (𝐛_2 + 𝐛_3)/2$ may then be retrieved with intensities_interpolated(sf, [[0.0, 0.5, 0.5]], formula) . intensities_interpolated returns a list of nω elements at each wavevector. The corresponding $ω$ values can be retrieved by calling available_energies on sf.

      Since Sunny only calculates the structure factor on a finite lattice when performing classical simulations, it is important to realize that exact information is only available at a discrete set of wave vectors. Specifically, for each axis index $i$, we will get information at $q_i = \frac{n}{L_i}$, where $n$ runs from $(\frac{-L_i}{2}+1)$ to $\frac{L_i}{2}$ and $L_i$ is the linear dimension of the lattice used for the calculation. If you request a wave vector that does not fall into this set, Sunny will automatically round to the nearest $𝐪$ that is available. If intensities_interpolated is given the keyword argument interpolation=:linear, Sunny will use trilinear interpolation to determine a result at the requested wave vector.

      To retrieve the intensities at all wave vectors for which there is exact data, first call the function available_wave_vectors to generate a list of qs. This takes an optional keyword argument bzsize, which must be given a tuple of three integers specifying the number of Brillouin zones to calculate, e.g., bzsize=(2,2,2). The resulting list of wave vectors may then be passed to intensities_interpolated.

      Alternatively, intensities_binned can be used to place the exact data into histogram bins for comparison with experiment.

      The convenience function reciprocal_space_path returns a list of wavevectors sampled along a path that connects specified $𝐪$ points. This list can be used as an input to intensities. Another convenience method, reciprocal_space_shell will generate points on a sphere of a given radius. This is useful for powder averaging.

      A number of arguments for intensity_formula are available which modify the calculation of structure factor intensity. It is generally recommended to provide a value of kT corresponding to the temperature of sampled configurations. Given kT, Sunny will include an energy- and temperature-dependent classical-to-quantum rescaling of intensities in the formula.

      To retrieve intensity data from a instantaneous structure factor, use instant_intensities_interpolated, which accepts similar arguments to intensities_interpolated. This function may also be used to calculate instantaneous information from a dynamical correlation data, i.e. from a SampledCorrelations created with dynamical_correlations. Note that it is important to supply a value to kT to reap the benefits of this approach over simply calculating a static structure factor at the outset.

      diff --git a/previews/PR178/versions.html b/previews/PR178/versions.html new file mode 100644 index 000000000..fc66146a0 --- /dev/null +++ b/previews/PR178/versions.html @@ -0,0 +1,2 @@ + +Version History · Sunny documentation
      GitHub

      Version History

      v0.5.6

      v0.5.5

      • reshape_supercell now allows reshaping to multiples of the primitive unit cell, which can speed up certain calculations. This is illustrated in the CoRh₂O₄ powder averaging tutorial.
      • resize_supercell now allows all resizings.
      • Added energy_per_site.
      • set_spiral_order_on_sublattice! cannot work on reshaped systems.
      • Various bug fixes. In particular, an intensity_formula with :full will now uniformly calculate a 3x3 matrix of complex numbers.

      v0.5.4

      • Various enhancements to view_crystal. Atoms are now labeled by index, and bonds support interactive inspection (GLMakie only). Font sizes work correctly on Makie v0.20-beta. If using Makie v0.19 on a high-resolution display, pass rescale=1.5 to enlarge font sizes.
      • The function suggest_magnetic_supercell now requires only a list of wavevectors, and will return a $3×3$ matrix that can be programmatically passed to reshape_supercell. The new tolerance parameter tol allows suggest_magnetic_supercell to approximate incommensurate wavevectors with nearby commensurate ones.
      • New functions set_spiral_order! and set_spiral_order_on_sublattice! can be used to initialize a spiral, single-$Q$ order.
      • Sunny now retains all constant energy shifts that have been introduced by anisotropy operators.
      • Fix export_vtk functionality.

      v0.5.3

      • Add large_S_spin_operators and large_S_stevens_operators to support single-ion anisotropies in dipole mode without renormalization. Set large_S=true in set_exchange! to avoid renormalization of biquadratics.
      • view_crystal has been rewritten in Makie.
      • plot_spins now expects ghost_radius in physical length units.
      • SpinWaveTheory will (currently) error if provided a system with enable_dipole_dipole!.

      v0.5.2

      • Form factors for 5d transition ions.
      • Download links for notebooks and scripts on each doc example
      • Various bug fixes.

      v0.5.1

      • Fix binning edge cases.
      • plot_spins accepts resolution argument.

      v0.5.0

      New features.

      Support for Linear Spin Wave Theory in :dipole and :SUN modes. (Thanks Hao Zhang!)

      New function minimize_energy! to efficiently find an optimal configuration of spin dipoles or SU(N) coherent states.

      Major refactors and enhancements to intensity calculations. This new interface allows unification between LSWT and classical spin dynamics calculations. This interface allows: Custom observables as local quantum operators, better support for linebroadening, and automatic binning to facilitate comparison with experimental data. See intensity_formula for documentation. Use load_nxs to load experimental neutron scattering data.

      Breaking changes.

      Require Julia 1.9.

      Replace set_anisotropy! with a new function set_onsite_coupling! (and similarly set_onsite_coupling_at!). The latter expects an explicit matrix representation for the local Hamiltonian. This can be constructed, e.g., as a linear combination of stevens_operators, or as a polynomial of spin_operators. To understand the mapping between these two, the new function print_stevens_expansion acts on an arbitrary local operator.

      Remove set_biquadratic!. Instead, use an optional keyword argument biquad to set_exchange!.

      Rename DynamicStructureFactor to dynamical_correlations. Similarly, replace InstantStructureFactor with instant_correlations. The return type has been renamed SampledCorrelations to emphasize that the object may be holding thermodynamic samples, which are collected using add_sample!.

      Remove intensities function. Instead, use one of intensities_interpolated or intensities_binned. These will require an intensity_formula, which defines a calculator (e.g., LSWT).

      Rename connected_path to reciprocal_space_path, which now returns an xticks object that can be used in plotting. Replace spherical_shell with reciprocal_space_shell that functions similarly.

      Rename polarize_spin! to set_dipole! for consistency with set_coherent!. The behavior of the former function is unchanged: the spin at a given site will still be polarized along the provided direction.

      Rename all_sites to eachsite consistent with Julia convention for iterators.

      Rename reshape_geometry to reshape_supercell, which is the fundamental reshaping function. Rename resize_periodically to resize_supercell.

      The constructor SpinInfo now requires a $g$-factor or tensor as a named argument.

      The constructor FormFactor no longer accepts an atom index. Instead, the form factors are associated with site-symmetry classes in order of appearance.

      v0.4.3

      Experimental support for linear SpinWaveTheory, implemented in SU(N) mode. This module may evolve rapidly.

      Implement renormalization of single-ion anisotropy and biquadratic interactions when in :dipole mode. This makes the model more faithful to the quantum mechanical Hamiltonian, but is also a breaking change.

      Various improvements and bugfixes for to_inhomogeneous. Setting inhomogeneous interactions via set_exchange_at! should now infer the correct bond offset direction, or will report an ambiguity error. Ambiguities can be resolved by passing an explicit offset.

      The function remove_periodicity! disables periodicity along specified dimensions.

      Rename StaticStructureFactor to InstantStructureFactor.

      v0.4.2

      Introduce LocalSampler, a framework for MCMC sampling with local spin updates.

      Rename print_dominant_wavevectors to print_wrapped_intensities to reduce confusion with the physical instantaneous intensities.

      The function spherical_shell now takes a radius in physical units of inverse Å.

      New exported functions global_position, magnetic_moment, all_sites.

      Remove all uses of Base.deepcopy which resolves crashes.

      v0.4.1

      The function to_inhomogeneous creates a system that supports inhomogeneous interactions, which can be set using set_exchange_at!, etc.

      set_biquadratic! replaces set_exchange_with_biquadratic!.

      v0.4.0

      This update includes many breaking changes, and is missing some features of 0.3.0.

      Creating a spin System

      Rename SpinSystem to System. Its constructor now has the form,

      System(crystal, latsize, infos, mode)

      The parameter infos is now a list of SpinInfo objects. Each defines spin angular momentum $S = \frac{1}{2}, 1, \frac{3}{2}, …$, and an optional $g$-factor or tensor.

      The parameter mode is one of :SUN or :dipole.

      Setting interactions

      Interactions are now added mutably to an existing System using the following functions: set_external_field!, set_exchange!, set_onsite_coupling!, enable_dipole_dipole!.

      As a convenience, one can use dmvec(D) to convert a DM vector to a $3×3$ antisymmetric exchange matrix.

      Fully general single-ion anisotropy is now possible. The function set_onsite_coupling! expects the single ion anisotropy to be expressed as a polynomial in symbolic spin operators 𝒮, or as a linear combination of symbolic Stevens operators 𝒪. For example, an easy axis anisotropy in the direction n may be written D*(𝒮⋅n)^2.

      Stevens operators 𝒪[k,q] admit polynomial expression in spin operators 𝒮[α]. Conversely, a polynomial of spin operators can be expressed as a linear combination of Stevens operators. To see this expansion use print_anisotropy_as_stevens.

      Inhomogeneous field

      An external field can be applied to a single site with set_external_field_at!.

      Structure factor rewrite

      The calculation of structure factors has been completely rewritten. For the new interface, see the FeI₂ at Finite Temperature page.

      Various

      • The "Sampler" interface is in flux. Langevin replaces both LangevinHeunP and LangevinSampler. Local spin-flip Monte Carlo sampling methods are temporarily broken.

      • repeat_periodically replaces extend_periodically.

      Additional related functions include resize_periodically and reshape_geometry, the latter being fundamental.

      The new function includes the list of symmetry-allowed single ion anisotropies in addition to exchange interactions.

      • When reading CIF files, the field _atom_site_label is now used in place of the field _atom_site_type_symbol.

      This is required for correctness. The field _atom_site_label is guaranteed to be present, and is guaranteed to be a distinct label for each symmetry-inequivalent site. Code that explicitly referred to site labels (e.g. in calls to subcrystal) will need to be updated to use the new label.

      diff --git a/previews/PR178/writevtk.html b/previews/PR178/writevtk.html new file mode 100644 index 000000000..d7016ca10 --- /dev/null +++ b/previews/PR178/writevtk.html @@ -0,0 +1,64 @@ + +ParaView Rendering · Sunny documentation
      GitHub

      ParaView Rendering

      The 4D correlation data produced by Sunny is too high-dimensional to visualize directly. This page describes how to export 3D slices of correlation data from Sunny to the Visual ToolKit (VTK) format, which is compatible with the ParaView visualization software. ParaView supports volumetric rendering:

      Simulation data

      First, generate some correlation data in Sunny. We will use a 2D lattice, since the correlation data $S(Q_x,Q_y,\omega)$ is 3D and can be exported in its entirety. The following code sets up the system, thermalizes it, and records the correlation data in a SampledCorrelations called dsf.

      using Sunny
+
+# Single layer 12x12 periodic square lattice
+latsize = (12,12,1)
+
+latvecs = lattice_vectors(8.,8.,12.,90,100,90)
+positions = [[0,0,0]]
+types = ["Cu"]
+formfactors = [FormFactor("Cu2")]
+xtal = Crystal(latvecs, positions; types)
+
+sys = System(xtal, latsize, [SpinInfo(1, S=1/2, g=2)], :SUN; seed=1)
+
+J = 10.
+set_exchange!(sys, J, Bond(1,1,[1,0,0]))
+set_exchange!(sys, J, Bond(1,1,[0,1,0]))
+
+Δt = 0.01
+kT = 0.5
+langevin = Langevin(Δt; λ=0.5, kT)
+randomize_spins!(sys)
+for i in 1:10_000 # Long enough to reach equilibrium
+    step!(sys, langevin)
+end 
+
+ωmax=10.
+
+dsf = dynamical_correlations(sys
+                             ;Δt=Δt
+                             ,nω=48
+                             ,ωmax=ωmax
+                             ,process_trajectory=:symmetrize)
+
+nsamples = 10
+for _ in 1:nsamples
+    for _ in 1:1000 
+        step!(sys, langevin)
+    end
+    add_sample!(dsf, sys)
+end

      The default histogram BinningParameters are already integrated over the $z$ direction because the system is 2D:

      unit_resolution_binning_parameters(dsf)
      ⊡    12 bins from -0.042 to +0.958 along [+1.27 dx] (Δ = 0.065)
+⊡    12 bins from -0.042 to +0.958 along [+1.27 dy] (Δ = 0.065)
+∫ Integrated from +0.000 to +0.000 along [-0.33 dx +1.88 dz] (Δ = 0.524)
+⊡    48 bins from -0.107 to +10.134 along [+1.00 dE] (Δ = 0.213)

      The histogram is very oblong; it's approximately 1x1x10. To make it a nicer shape, we will rescale the energy axis to be be fractions of ωmax:

      params = unit_resolution_binning_parameters(dsf)
+scale_factor = ωmax
+params.binend[4] /= scale_factor
+params.binstart[4] /= scale_factor
+params.binwidth[4] /= scale_factor
+params.covectors[4,:] ./= scale_factor

      Doing this changes the last axis of the histogram to fit in [0,1]:

      ⊡    49 bins from -0.011 to +1.013 along [+0.10 dE] (Δ = 0.213)

      Now that our histogram is a cube, we compute the intensity in the histogram bins using the usual intensities_binned:

      formula = intensity_formula(dsf,:trace)
+signal, counts = intensities_binned(dsf, params; formula)
+intensity = signal ./ counts

      Now that we have our intensity data and the binning parameters, we can export to VTK format using export_vtk and move to ParaView for the visualization.

      # Importing WriteVTK enables Sunny's export-to-VTK functions
+import WriteVTK
+
+# [1,2,4] specifies that the (x,y,z) axes in ParaView are (Qx,Qy,ω)
+export_vtk("square_lattice", params, intensity; dims_kept = [1,2,4])
+# Writes a file square_lattice.vti in the current directory

      Loading in ParaView

      In ParaView, use File > Open to open square_lattice.vti. This will add the file to the Pipeline Browser with a closed eye icon, indicating that the data is ready to be loaded.

      In the Properties panel, both bin_centers and data will be selected for import by default. Uncheck bin_centers because we don't need that information for the visualization. Click the green Apply button to load the data.

      By default, only the outline of the data is shown in the 3D viewport. Since we adjusted the energy axis, the outline is a 1x1x1 cube. Optionally enable the axes grid under "View", and customize using the adjacent edit button.

      To enable the volumetric render:

      1. Select "Volume" from the "Representation" drop-down menu under "Display".
      2. The "Coloring" drop-down should automatically select data because it's the only data loaded.
      3. Open the Color Map Editor to adjust the opacity of the fog, which may be too faint to see by default.

      Depending on your computer and your dataset size, the volumetric rendering may be slow, but our dataset is relatively small, so the render should be fast.

      If nothing shows up at first, don't despair. Often, there are Bragg-like peaks in the correlation data which outshine everything else. To see this, enable Display Data Histogram in the Color Map Editor panel. To zoom in on the lower-intensity data, click and drag the right side handle of the opacity transfer function box to the middle a few times.

      After suitable color mapping, the dispersion curve should become visible:

      Experiment data

      Note that since only the data and binning parameters are required for exporting to VTK, experiment data can be exported in the same way. For example, to visualize $S(Q_x,Q_y,Q_z)$, do this:

      # Load 4D histogram data from Mantid
+params, signal = load_nxs("experiment_data.nxs")
+
+# Integrate out the energy axis so we are 3D
+integrate_axes!(params; axes = 4)
+signal = sum(signal; dims = 4)
+
+# Export to ParaView
+export_vtk("experiment_data_as_vtk", params, signal)
      GitHub

      Download this example as Jupyter notebook or Julia script.

      Kagome Antiferromagnet

      This is a Sunny port of SpinW Tutorial 8, authored by Bjorn Fak and Sandor Toth. The goal is to calculate the linear spin wave theory spectrum for the $\sqrt{3} \times \sqrt{3}$ order of a Kagome antiferromagnet.

      Load Packages

      using Sunny, GLMakie

      Build a Crystal with $P\overline{3}$ space group and Cr⁺ ions on each site.

      a = 1
+latvecs = lattice_vectors(a, a, 10a, 90, 90, 120)
+crystal = Crystal(latvecs, [[1/2,0,0]], 147)
      Crystal
+HM symbol 'P -3' (147)
+Lattice params a=1, b=1, c=10, α=90°, β=90°, γ=120°
+Cell volume 8.66
+:
+   1. [1/2, 0, 0]
+   2. [0, 1/2, 0]
+   3. [1/2, 1/2, 0]
+

      Build a System with antiferrogmanetic nearest neighbor exchange $J=1$.

      S = 1
+sys = System(crystal, (3,3,1), [SpinInfo(1; S, g=2)], :dipole)
+J = 1.0
+set_exchange!(sys, J, Bond(2,3,[0,0,0]))

      Initialize to an energy minimizing magnetic structure, for which nearest-neighbor spins are at 120° angles.

      q = -[1/3, 1/3, 0]
+axis = [0,0,1]
+set_spiral_order_on_sublattice!(sys, 1; q, axis, S0=[cos(0),sin(0),0])
+set_spiral_order_on_sublattice!(sys, 2; q, axis, S0=[cos(0),sin(0),0])
+set_spiral_order_on_sublattice!(sys, 3; q, axis, S0=[cos(2π/3),sin(2π/3),0])
+plot_spins(sys; dims=2)
      Example block output

      Check energy. Each site participates in 4 bonds with energy $JS^2\cos(2π/3)$. Factor of 1/2 avoids double counting.

      @assert energy_per_site(sys) ≈ (4/2)*J*S^2*cos(2π/3)

      Define a path in reciprocal space.

      points_rlu = [[-1/2, 0, 0], [0, 0, 0], [1/2, 1/2, 0]]
+density = 100
+path, xticks = reciprocal_space_path(crystal, points_rlu, density);

      Calculate discrete intensities

      swt = SpinWaveTheory(sys)
+formula = intensity_formula(swt, :perp; kernel=delta_function_kernel)
+disp, intensity = intensities_bands(swt, path, formula);

      Plot over a restricted color range from [0,1e-2]. Note that the intensities of the flat band at zero-energy are off-scale.

      fig = Figure()
+ax = Axis(fig[1,1]; xlabel="Momentum (r.l.u.)", ylabel="Energy (meV)", xticks, xticklabelrotation=π/6)
+ylims!(ax, -1e-1, 2.3)
+for i in axes(disp)[2]
+    lines!(ax, 1:length(disp[:,i]), disp[:,i]; color=intensity[:,i], colorrange=(0,1e-2))
+end
+fig
      Example block output
    GitHub

    Download this example as Jupyter notebook or Julia script.

    FeI₂ at Finite Temperature

    using Sunny, LinearAlgebra, GLMakie

    In our previous Case Study: FeI₂, we used linear spin wave theory (LSWT) to calculate the dynamical structure factor. Here, we perform a similar calculation using classical spin dynamics. Because we are interested in the coupled dynamics of spin dipoles and quadrupoles, we employ a classical dynamics of SU(3) coherent states that generalizes the Landau-Lifshitz equation.

    Compared to LSWT, simulations using classical dynamics are much slower, and are limited in $k$-space resolution. However, they make it is possible to capture nonlinear effects associated with finite temperature fluctuations. Classical dynamics are also appealing for studying out-of-equilibrium systems (e.g., relaxation of spin glasses), or systems with quenched inhomogeneities that require large simulation volumes.

    In this tutorial, we show how to study the finite temperature dynamics of FeI₂ using the classical approach. It is important to stress that the estimation of $S(𝐪,ω)$ with classical dynamics is fundamentally a Monte Carlo calculation: sample spin configurations are drawn from thermal equilibrium and used as initial conditions for generating dissipationless trajectories. The correlations of these trajectories are then averaged and used to calculate scattering intensities. It is therefore important to ensure that the initial spin configurations are sampled appropriately and that sufficient statistics are collected. We will demonstrate one approach here.

    As an overview, we will:

    1. Identify the ground state
    2. Measure correlation data describing the excitations around that ground state
    3. Use the correlation data to compute scattering intensities

    As the implementation of the FeI₂ model is already covered in detail in the LSWT tutorial, we will not repeat it below. Instead, we will assume that you already have defined a sys in the same way with lattice dimensions $4×4×4$.

    sys
    System [SU(3)]
+Lattice: (4, 4, 4)×1
+

    Finding a ground state

    Sunny uses the Langevin dynamics of SU(N) coherent states to sample spin configurations from the thermal equlibrium. One first constructs a Langevin integrator. This requires a time step, temperature, and a phenomenological damping parameter $λ$ that sets the coupling to the thermal bath.

    Δt = 0.05/D    # Should be inversely proportional to the largest energy scale
+               # in the system. For FeI2, this is the easy-axis anisotropy,
+               # `D = 2.165` (meV). The prefactor 0.05 is relatively small,
+               # and achieves high accuracy.
+kT = 0.2       # Temperature of the thermal bath (meV).
+λ = 0.1        # This value is typically good for Monte Carlo sampling,
+               # independent of system details.
+
+langevin = Langevin(Δt; kT, λ);

    Langevin dynamics can be used to search for a magnetically ordered state. For this, the temperature kT must be below the ordering temperature, but large enough that the dynamical sampling procedure can overcome local energy barriers and eliminate defects.

    randomize_spins!(sys)
+for _ in 1:20_000
+    step!(sys, langevin)
+end

    Although thermal fluctuations are present, the correct antiferromagnetic order (2 up, 2 down) is apparent.

    plot_spins(sys; color=[s[3] for s in sys.dipoles])
    Example block output

    For other systems, it can be much harder to find the magnetic ordering in an unbiased way, and more complicated sampling procedures may be necessary.

    Calculating Thermal-Averaged Correlations $\langle S^{\alpha\beta}(𝐪,ω)\rangle$

    Our aim is to study the classical spin dynamics for states sampled in thermal equilibrium. To minimize finite size effects, and achieve sufficient momentum space resolution, we should significantly enlarge the system volume. The function resize_supercell takes new dimensions as multiples of the unit cell lattice vectors.

    sys_large = resize_supercell(sys, (16,16,4)) # 16x16x4 copies of the original unit cell
+plot_spins(sys_large; color=[s[3] for s in sys_large.dipoles])
    Example block output

    Now we will re-thermalize the system to a configuration just above the ordering temperature. Sunny expects energies in meV by default, so we use meV_per_K to convert from kelvin.

    kT = 3.5 * meV_per_K     # 3.5K ≈ 0.30 meV
+langevin.kT = kT
+for _ in 1:10_000
+    step!(sys_large, langevin)
+end

    The next step is to collect correlation data $S^{\alpha\beta}$. This will involve sampling spin configurations from thermal equilibrium, and then integrating the Hamiltonian dynamics of SU(N) coherent states to collect Fourier-space information about normal modes. Quantization of these modes yields the magnons, and the associated dynamical spin-spin correlations can be compared with neutron scattering intensities $S^{\alpha\beta}(q,\omega)$. Because this a real-space calculation, data is only available for discrete $q$ modes (the resolution scales like inverse system size).

    To store the correlation data, we initialize a SampledCorrelations object by calling dynamical_correlations. It requires three keyword arguments: an integration step size, a target number of ωs to retain, and a maximum energy ω to resolve. For the time step, twice the value used for the Langevin integrator is usually a good choice.

    sc = dynamical_correlations(sys_large; Δt=2Δt, nω=120, ωmax=7.5)
    SampledCorrelations (33.615 MiB)
+[S(q,ω) | nω = 120, Δω = 0.0632 | 0 sample]
+Lattice: (16, 16, 4)×1
+6 correlations in SU(3) mode:
+╔ ⬤ ⬤ ⬤ Sx
+║ ⋅ ⬤ ⬤ Sy
+╚ ⋅ ⋅ ⬤ Sz
+

    The function add_sample! will collect data by running a dynamical trajectory starting from the current system configuration.

    add_sample!(sc, sys_large)        # Accumulate the sample into `sc`

    To collect additional data, it is required to re-sample the spin configuration from the thermal distribution. For efficiency, the dynamics should be run long enough that consecutive samples are uncorrelated.

    for _ in 1:2
+    for _ in 1:1000               # Enough steps to decorrelate spins
+        step!(sys_large, langevin)
+    end
+    add_sample!(sc, sys_large)    # Accumulate the sample into `sc`
+end

    Now, sc has more samples included:

    sc
    SampledCorrelations (33.615 MiB)
+[S(q,ω) | nω = 120, Δω = 0.0632 | 3 samples]
+Lattice: (16, 16, 4)×1
+6 correlations in SU(3) mode:
+╔ ⬤ ⬤ ⬤ Sx
+║ ⋅ ⬤ ⬤ Sy
+╚ ⋅ ⋅ ⬤ Sz
+

    Computing Scattering Intensities

    With the thermally-averaged correlation data $\langle S^{\alpha\beta}(q,\omega)\rangle$ in hand, we now need to specify how to extract a scattering intensity from this information. This is done by constructing an intensity_formula. By way of example, we will use a formula which computes the trace of the structure factor and applies a classical-to-quantum temperature-dependent rescaling kT.

    formula = intensity_formula(sc, :trace; kT)
    Classical Scattering Intensity Formula
+At discrete scattering modes S = S[ix_q,ix_ω], use:
+
+  Intensity[ix_q,ix_ω] = Tr S
+
+No form factors specified
+Temperature corrected (kT = 0.3016066641750812) ✓
+

    Recall that $\langle S^{\alpha\beta}(q,\omega)\rangle$ is only available at certain discrete $q$ values, due to the finite lattice size. There are two basic approaches to handling this discreteness. The first approach is to interpolate between the available data using intensities_interpolated. For example, we can plot single-$q$ slices at (0,0,0) and (π,π,π) using this method:

    qs = [[0, 0, 0], [0.5, 0.5, 0.5]]
+is = intensities_interpolated(sc, qs, formula; interpolation = :round)
+
+ωs = available_energies(sc)
+fig = lines(ωs, is[1,:]; axis=(xlabel="meV", ylabel="Intensity"), label="(0,0,0)")
+lines!(ωs, is[2,:]; label="(π,π,π)")
+axislegend()
+fig
    Example block output

    The resolution in energy can be improved by increasing nω (and decreasing Δt), and the general accuracy can be improved by collecting additional samples from the thermal equilibrium.

    For real calculations, one often wants to apply further corrections and more accurate formulas. Here, we apply FormFactor corrections appropriate for Fe2 magnetic ions, and a dipole polarization correction :perp.

    formfactors = [FormFactor("Fe2"; g_lande=3/2)]
+new_formula = intensity_formula(sc, :perp; kT, formfactors = formfactors)
    Classical Scattering Intensity Formula
+At discrete scattering modes S = S[ix_q,ix_ω], use:
+
+  Intensity[ix_q,ix_ω] = ∑_ij (I - Q⊗Q){i,j} S{i,j}
+                         
+                         (i,j = Sx,Sy,Sz)
+
+Form factors included in S ✓
+Temperature corrected (kT = 0.3016066641750812) ✓
+

    Frequently, one wants to extract energy intensities along lines that connect special wave vectors–a so-called "spaghetti plot". The function reciprocal_space_path creates an appropriate horizontal axis for this plot by linearly sampling between provided $q$-points with a given sample density. The number of sample points between two wavevectors q1 and q2 is given by dist*density where dist = norm(cryst.recipvecs * (q1 - q2)) is measured in the global frame.

    points = [[0,   0, 0],  # List of wave vectors that define a path
+          [1,   0, 0],
+          [0,   1, 0],
+          [1/2, 0, 0],
+          [0,   1, 0],
+          [0,   0, 0]]
+density = 40
+path, xticks = reciprocal_space_path(cryst, points, density);

    Again using intensities_interpolated, we can evaluate the (interpolated) intensity at each point on the path. Since scattering intensities are only available at a certain discrete $(Q,\omega)$ points, the intensity on the path can be calculated by interpolating between these discrete points:

    is_interpolated = intensities_interpolated(sc, path, new_formula;
+    interpolation = :linear,       # Interpolate between available wave vectors
+);
+# Add artificial broadening
+is_interpolated_broadened = broaden_energy(sc, is, (ω, ω₀)->lorentzian(ω-ω₀, 0.05));

    The second approach to handle the discreteness of the data is to bin the intensity at the discrete points into the bins of a histogram. First, the five sub-histograms are set up using reciprocal_space_path_bins in analogy to reciprocal_space_path.

    cut_width = 0.3
+density = 15
+paramsList, markers, ranges = reciprocal_space_path_bins(sc,points,density,cut_width);

    Then, the intensity data is computed using intensities_binned for each sub-histogram:

    total_bins = ranges[end][end]
+energy_bins = paramsList[1].numbins[4]
+is_binned = zeros(Float64,total_bins,energy_bins)
+integrated_kernel = integrated_lorentzian(0.05) # Lorentzian broadening
+for k in eachindex(paramsList)
+    bin_data, counts = intensities_binned(sc,paramsList[k], new_formula;
+        integrated_kernel = integrated_kernel
+    )
+    is_binned[ranges[k],:] = bin_data[:,1,1,:] ./ counts[:,1,1,:]
+end

    The graph produced by interpolating (top) is similar to the one produced by binning (bottom):

    fig = Figure()
+ax_top = Axis(fig[1,1],ylabel = "meV",xticklabelrotation=π/8,xticklabelsize=12;xticks)
+ax_bottom = Axis(fig[2,1],ylabel = "meV",xticks = (markers, string.(points)),xticklabelrotation=π/8,xticklabelsize=12)
+
+heatmap!(ax_top,1:size(is_interpolated,1), ωs, is_interpolated;
+    colorrange=(0.0,0.07),
+)
+
+heatmap!(ax_bottom,1:size(is_binned,1), ωs, is_binned;
+    colorrange=(0.0,0.05),
+)
+
+fig
    Example block output

    Note that we have clipped the colors in order to make the higher-energy excitations more visible.

    Unconventional RLU Systems and Constant Energy Cuts

    Often it is useful to plot cuts across multiple wave vectors but at a single energy. We'll pick an energy,

    ωidx = 60
+target_ω = ωs[ωidx]
    3.7312072301840273

    and take a constant-energy cut at that energy. The most straightforward way is to make a plot whose axes are aligned with the conventional reciprocal lattice of the crystal. This is accomplished using unit_resolution_binning_parameters:

    params = unit_resolution_binning_parameters(sc)
+params.binstart[1:2] .= -1 # Expand plot range slightly
+
+# Set energy integration range
+omega_width = 0.3
+params.binstart[4] = target_ω - (omega_width/2)
+params.binend[4] = target_ω # `binend` should be inside (e.g. at the center) of the range
+params.binwidth[4] = omega_width
+
+integrate_axes!(params, axes = 3) # Integrate out z direction entirely
+
+params
    Binning Parameters
+⊡    31 bins from -1.000 to +0.969 along [+1.00 dx] (Δ = 0.064)
+⊡    31 bins from -1.000 to +0.969 along [+1.00 dy] (Δ = 0.064)
+∫ Integrated from -0.125 to +1.625 along [+1.00 dz] (Δ = 1.750)
+∫ Integrated from +3.581 to +3.881 along [+1.00 dE] (Δ = 0.300)
+

    In each of the following plots, black dashed lines represent (direct) lattice vectors. Since these plots are in reciprocal space, direct lattice vectors are represented as covectors (i.e. coordinate grids) instead of as arrows.

    is, counts = intensities_binned(sc,params,new_formula)
+
+fig = Figure()
+ax = Axis(fig[1,1];
+    title="Δω=0.3 meV (Binned)", aspect=true,
+    xlabel = "[H, 0, 0]",
+    ylabel = "[0, K, 0]"
+)
+bcs = axes_bincenters(params)
+hm = heatmap!(ax,bcs[1],bcs[2],is[:,:,1,1] ./ counts[:,:,1,1])
+add_lines!(ax,params)
+Colorbar(fig[1,2], hm);
+fig
    Example block output

    In the above plot, the dashed-line (direct) lattice vectors are clearly orthogonal. However, we know that in real space, the lattice vectors $a$ and $b$ are not orthogonal, but rather point along the edges of a hexagon (see lower left corner):



    Thus, plotting the direct lattice vectors as orthogonal (even in reciprocal space) is somewhat misleading. Worse yet, the [H,0,0] by [0,K,0] plot apparently loses the 6-fold symmetry of the crystal! Lastly, if one works out the components of the real-space metric with respect to the axes of the plot, one finds that there are non-zero off-diagonal entries,

    latvecs = sys.crystal.latvecs
+metric = latvecs' * I(3) * latvecs
    3×3 Matrix{Float64}:
+ 16.4035   -8.20174   0.0
+ -8.20174  16.4035    0.0
+  0.0       0.0      45.5914

    so real-space rotations and angles map into reciprocal space rotations angles in a complicated way.

    To resolve these important issues, we want to use axes which are orthogonal (i.e. they diagonalize the metric and solve all of the problems just mentioned). The canonical choice is to use the combination $\frac{1}{2}a + b$ of lattice vectors (equiv. $a^* - \frac{1}{2}b^*$), which is orthogonal to $a$:

    (latvecs * [1/2,1,0]) ⋅ (latvecs * [1,0,0]) == 0
    true

    This new vector $\frac{1}{2}a+b$ is visibly orthogonal to $a$ in real space:

    Example block output

    To use "projection onto the new vector" as a histogram axis, only a single change is needed to the binning parameters. The second covector (previously $b$) must be swapped out for $\frac{1}{2}a + b$ (recall that reciprocal space covectors, such as those used in BinningParameters correspond to direct space vectors).

    params.covectors[2,1:3] = [1/2,1,0] # [1/2,1,0] times [a;b;c] is (a/2 + b)
    Binning Parameters
+⊡    31 bins from -1.000 to +0.969 along [+1.00 dx] (Δ = 0.064)
+⊡    31 bins from -1.000 to +0.969 along [+0.50 dx +1.00 dy] (Δ = 0.057)
+∫ Integrated from -0.125 to +1.625 along [+1.00 dz] (Δ = 1.750)
+∫ Integrated from +3.581 to +3.881 along [+1.00 dE] (Δ = 0.300)
+

    The second axis of the histogram now agrees with what is conventionally labelled as [H,-H/2,0].

    Length of the new vector

    Note that, although $\frac{1}{2}a+b$ is orthogonal to $a$, it is not the same length as $a$. Instead, it is sqrt(3/4) times as long. Note the unsymmetrical axes labels in the plots that follow as a direct result of this!

    # Zoom out horizontal axis
+params.binstart[1], params.binend[1] = -2, 2
+
+# Adjust vertical axis bounds to account for
+# length of a/2 + b
+params.binstart[2], params.binend[2] = -2 * sqrt(3/4), 2 * sqrt(3/4)
+
+# Re-compute in the new coordinate system
+is, counts = intensities_binned(sc,params,new_formula)
+
+add_lines!(ax_right,params)

    For comparison purposes, we will make the same plot using intensities_interpolated to emulate zero-width bins. This time, it's more convenient to think in terms of reciprocal vectors $a^*$ and $b^*$. Now, our coordinate transformation consists of establishing a new, orthogonal basis to specify our wave vectors: $a^* - \frac{1}{2}b^*$, $b^*$ and $c^*$. Writing this in matrix form allows us to sample a rectilinear grid of wave vectors in this frame. Finally, we'll convert these back into the original RLU system for input into Sunny.

    # New basis matrix
+A = [1    0 0
+     -1/2 1 0
+     0    0 1]
+
+# Define our grid of wave vectors
+npoints = 60
+as = range(-2, 2, npoints)
+bs = range(-3/√3, 3/√3, npoints)
+qs_ortho = [[a, b, 0] for a in as, b in bs]
+
+# Convert to original RLU system for input to Sunny
+qs = [A * q for q in qs_ortho]
+
+# Use interpolation to get intensities
+is = intensities_interpolated(sc, qs, new_formula; interpolation=:linear)
+
+add_lines!(ax_left,params)
+fig
    Example block output

    Now, not only are the dashed-line lattice vectors no longer misleadingly orthogonal, but the six-fold symmetry has been restored as well! Further, the metric has been diagonalized:

    metric = (latvecs * inv(A'))' * I(3) * (latvecs * inv(A'))
    3×3 Matrix{Float64}:
+ 16.4035   0.0      0.0
+  0.0     12.3026   0.0
+  0.0      0.0     45.5914

    Finally, we note that instantaneous structure factor data, $𝒮(𝐪)$, can be obtained from a dynamic structure factor with instant_intensities_interpolated. Here we'll reuse the grid of wave vectors we generated above.

    is_static = instant_intensities_interpolated(sc, qs, new_formula; interpolation = :linear)
+
+hm = heatmap(as, bs, is_static;
+    axis=(
+        title="Instantaneous Structure Factor",
+        xlabel = "[H, -1/2H, 0]",
+        ylabel = "[0, K, 0]",
+        aspect=true
+    )
+)
+Colorbar(hm.figure[1,2], hm.plot)
+hm
    Example block output

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