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+{"documenter":{"julia_version":"1.9.3","generation_timestamp":"2023-10-04T15:46:06","documenter_version":"1.1.0"}} \ No newline at end of file diff --git a/previews/PR176/anisotropy.html b/previews/PR176/anisotropy.html new file mode 100644 index 000000000..4ff8e4c25 --- /dev/null +++ b/previews/PR176/anisotropy.html @@ -0,0 +1,40 @@ + +Single-Ion Anisotropy · Sunny documentation

Single-Ion Anisotropy

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 accurate modeling of quantum spin Hamiltonians that include a strong single-ion anisotropy.

Defining on-site couplings

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_operators. For example, in the case of spin-$1/2$, this function returns the Pauli matrices divided by 2. The Stevens operators $\hat{\mathcal{O}}_{k,q}$ are polynomials of the spin operators, and are accessed using stevens_operators. 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 large_S_spin_operators or large_S_stevens_operators. 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 using large_S_stevens_operators. 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/PR176/assets/logo.png b/previews/PR176/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$ JUxlorentzian(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/PR176/assets/notebooks/fei2_tutorial.ipynb b/previews/PR176/assets/notebooks/fei2_tutorial.ipynb new file mode 100644 index 000000000..d1d1f4fba --- /dev/null +++ b/previews/PR176/assets/notebooks/fei2_tutorial.ipynb @@ -0,0 +1,779 @@ +{ + "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", + "operator. It can be constructed, e.g., from the matrices given by\n", + "`spin_operators` or `stevens_operators`. Here we construct an\n", + "easy-axis anisotropy along the direction $\\hat{z}$." + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "D = 2.165\n", + "S = spin_operators(sys, 1)\n", + "set_onsite_coupling!(sys, -D*S[3]^2, 1)" + ], + "metadata": {}, + "execution_count": null + }, + { + "cell_type": "markdown", + "source": [ + "Any anisotropy operator can be converted to a linear combination of Stevens\n", + "operators with `print_stevens_expansion`." + ], + "metadata": {} + }, + { + "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/PR176/assets/notebooks/ising2d.ipynb b/previews/PR176/assets/notebooks/ising2d.ipynb new file mode 100644 index 000000000..3d9a59ddd --- /dev/null +++ b/previews/PR176/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/PR176/assets/notebooks/out_of_equilibrium.ipynb b/previews/PR176/assets/notebooks/out_of_equilibrium.ipynb new file mode 100644 index 000000000..d0c8a2c63 --- /dev/null +++ b/previews/PR176/assets/notebooks/out_of_equilibrium.ipynb @@ -0,0 +1,279 @@ +{ + "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 the single-ion anisotropy," + ], + "metadata": {} + }, + { + "outputs": [], + "cell_type": "code", + "source": [ + "D = 19.0\n", + "Sz = spin_operators(sys, 1)[3]\n", + "set_onsite_coupling!(sys, D*Sz^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/PR176/assets/notebooks/powder_averaging.ipynb b/previews/PR176/assets/notebooks/powder_averaging.ipynb new file mode 100644 index 000000000..0f84c42b8 --- /dev/null +++ b/previews/PR176/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/PR176/assets/paraviewcolormap.png b/previews/PR176/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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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/PR176/assets/scripts/fei2_tutorial.jl b/previews/PR176/assets/scripts/fei2_tutorial.jl new file mode 100644 index 000000000..5d1b97d8a --- /dev/null +++ b/previews/PR176/assets/scripts/fei2_tutorial.jl @@ -0,0 +1,118 @@ +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 +S = spin_operators(sys, 1) +set_onsite_coupling!(sys, -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/PR176/assets/scripts/ising2d.jl b/previews/PR176/assets/scripts/ising2d.jl new file mode 100644 index 000000000..6ab158eb6 --- /dev/null +++ b/previews/PR176/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/PR176/assets/scripts/out_of_equilibrium.jl b/previews/PR176/assets/scripts/out_of_equilibrium.jl new file mode 100644 index 000000000..60ce32a36 --- /dev/null +++ b/previews/PR176/assets/scripts/out_of_equilibrium.jl @@ -0,0 +1,64 @@ +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 +Sz = spin_operators(sys, 1)[3] +set_onsite_coupling!(sys, D*Sz^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/PR176/assets/scripts/powder_averaging.jl b/previews/PR176/assets/scripts/powder_averaging.jl new file mode 100644 index 000000000..5e18d9af5 --- /dev/null +++ b/previews/PR176/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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html.theme--documenter-dark .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:hover,html.theme--documenter-dark .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:focus,html.theme--documenter-dark .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:focus:not(:active),html.theme--documenter-dark .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--documenter-dark .button.is-black:active,html.theme--documenter-dark .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--documenter-dark .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted:hover,html.theme--documenter-dark .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-outlined.is-hovered,html.theme--documenter-dark .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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.button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined{background-color:transparent;border-color:#ad8100;box-shadow:none;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ad8100 #ad8100 !important}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-warning.is-light{background-color:#fffaeb;color:#d19c00}html.theme--documenter-dark .button.is-warning.is-light:hover,html.theme--documenter-dark .button.is-warning.is-light.is-hovered{background-color:#fff7de;border-color:transparent;color:#d19c00}html.theme--documenter-dark .button.is-warning.is-light:active,html.theme--documenter-dark .button.is-warning.is-light.is-active{background-color:#fff3d1;border-color:transparent;color:#d19c00}html.theme--documenter-dark .button.is-danger{background-color:#9e1b0d;border-color:transparent;color:#fff}html.theme--documenter-dark 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.button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#9e1b0d;box-shadow:none;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fdeeec;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#fce3e0;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#fcd8d5;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .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 .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--documenter-dark .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--documenter-dark .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--documenter-dark .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--documenter-dark .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--documenter-dark .content li+li{margin-top:0.25em}html.theme--documenter-dark .content p:not(:last-child),html.theme--documenter-dark .content dl:not(:last-child),html.theme--documenter-dark .content ol:not(:last-child),html.theme--documenter-dark .content ul:not(:last-child),html.theme--documenter-dark .content blockquote:not(:last-child),html.theme--documenter-dark .content pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content 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 .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--documenter-dark .icon.is-medium{height:2rem;width:2rem}html.theme--documenter-dark .icon.is-large{height:3rem;width:3rem}html.theme--documenter-dark .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--documenter-dark .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--documenter-dark .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--documenter-dark .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--documenter-dark div.icon-text{display:flex}html.theme--documenter-dark .image,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--documenter-dark .image img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--documenter-dark .image img.is-rounded,html.theme--documenter-dark #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 .docs-sidebar .docs-logo>img.is-5by4 .has-ratio,html.theme--documenter-dark .image.is-4by3 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by3 img,html.theme--documenter-dark .image.is-4by3 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by3 .has-ratio,html.theme--documenter-dark .image.is-3by2 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by2 img,html.theme--documenter-dark .image.is-3by2 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by2 .has-ratio,html.theme--documenter-dark .image.is-5by3 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--documenter-dark .image.is-5by3 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--documenter-dark .image.is-16by9 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-16by9 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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/PR176/assets/themeswap.js b/previews/PR176/assets/themeswap.js new file mode 100644 index 000000000..9f5eebe6a --- /dev/null +++ b/previews/PR176/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/PR176/assets/warner.js b/previews/PR176/assets/warner.js new file mode 100644 index 000000000..3f6f5d008 --- /dev/null +++ b/previews/PR176/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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z6nB6=GJ2HjrXse3;H9TO=`C;>awARf6+>YJ52W)@5TA`LxXqCvg&8;jy1F)d{!vTz zTod%X{oCL?hh6jId0NUm5*;7EG@j>Pr7e;z4jj#%T^xJe)zF|PNH$7*8Xn%)G1Im9 z><9Opxuyi+rcM0~t~|m54?&PCnL@MK428c(kk%2rRXz6l)eCq$EfwPQoEl>7eY32S zM$8dqWNc9c2P9Xm(|Jh(Ohm?xr?`hinY3Guvj88LjBGm4awI){1-4TniCTy3$4A;I!zkJnK0*jlVs4n1Ti<=T5(8F~tar9V+SZGlo=+P84xjlRby5F^ am$uhj*~6_7bakX19(%&}iFF}~xqks*_W`v4 literal 0 HcmV?d00001 diff --git a/previews/PR176/examples/fei2_classical.html b/previews/PR176/examples/fei2_classical.html new file mode 100644 index 000000000..d460e83ec --- /dev/null +++ b/previews/PR176/examples/fei2_classical.html @@ -0,0 +1,177 @@ + +FeI₂ at Finite Temperature · Sunny documentation

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 (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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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 operator. It can be constructed, e.g., from the matrices given by spin_operators or stevens_operators. Here we construct an easy-axis anisotropy along the direction $\hat{z}$.

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

Any anisotropy operator can be converted to a linear combination of Stevens operators with print_stevens_expansion.

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)
49

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/PR176/examples/out_of_equilibrium.html @@ -0,0 +1,55 @@ + +CP² Skyrmion Quench · Sunny documentation

    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 the single-ion anisotropy,

    D = 19.0
    +Sz = spin_operators(sys, 1)[3]
    +set_onsite_coupling!(sys, D*Sz^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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    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.