build based on 20ac2cc

previews/PR176/.documenter-siteinfo.json

@@ -1 +1 @@
-{"documenter":{"julia_version":"1.9.3","generation_timestamp":"2023-10-07T14:53:21","documenter_version":"1.1.0"}}
+{"documenter":{"julia_version":"1.9.3","generation_timestamp":"2023-10-07T15:54:48","documenter_version":"1.1.0"}}

previews/PR176/anisotropy.html

@@ -37,4 +37,4 @@
 \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*}\]</p><p>Stevens operators <span>$\hat{\mathcal{O}}_{k,q}$</span> for odd <span>$k$</span> 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).</p><p>For each <span>$k$</span> value, the collection of operators <span>$\{\hat{\mathcal{O}}_{k,q&#39;}\}$</span> for <span>$q&#39; = -k, \dots, k$</span> is an irreducible representation of the group of rotations O(3). In particular, a physical rotation will transform <span>$\hat{\mathcal{O}}_{k,q}$</span> into a linear combination of <span>$\hat{\mathcal{O}}_{k,q&#39;}$</span> where <span>$q&#39;$</span> varies but <span>$k$</span> remains fixed. </p><p>In taking the large-<span>$S$</span> limit, each dipole operator is replaced by its expectation value <span>$\mathbf{s} = \langle \hat{\mathbf{S}} \rangle$</span>, and only leading-order terms are retained. The operator <span>$\hat{\mathcal{O}}_{k,q}$</span> becomes a homogeneous polynomial <span>$O_{k,q}(\mathbf{s})$</span> of order <span>$k$</span> in the spin components. One can see these polynomials using <a href="library.html#Sunny.large_S_stevens_operators"><code>large_S_stevens_operators</code></a>. Due to the normalization constraint, each dipole can be expressed in polar angles, <span>$(\theta, \phi)$</span>. Then the Stevens functions <span>$O_{k,q}(\mathbf{s})$</span> correspond to the spherical harmonic functions <span>$Y_l^m(\theta, \phi)$</span> where <span>$l=k$</span> and <span>$m=q$</span>, and modulo <span>$k$</span> and <span>$q$</span>-dependent rescaling factors.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="structure-factor.html">« Structure Factor Calculations</a><a class="docs-footer-nextpage" href="library.html">Library API »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.0 on <span class="colophon-date" title="Saturday 7 October 2023 14:53">Saturday 7 October 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\end{align*}\]</p><p>Stevens operators <span>$\hat{\mathcal{O}}_{k,q}$</span> for odd <span>$k$</span> 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).</p><p>For each <span>$k$</span> value, the collection of operators <span>$\{\hat{\mathcal{O}}_{k,q&#39;}\}$</span> for <span>$q&#39; = -k, \dots, k$</span> is an irreducible representation of the group of rotations O(3). In particular, a physical rotation will transform <span>$\hat{\mathcal{O}}_{k,q}$</span> into a linear combination of <span>$\hat{\mathcal{O}}_{k,q&#39;}$</span> where <span>$q&#39;$</span> varies but <span>$k$</span> remains fixed. </p><p>In taking the large-<span>$S$</span> limit, each dipole operator is replaced by its expectation value <span>$\mathbf{s} = \langle \hat{\mathbf{S}} \rangle$</span>, and only leading-order terms are retained. The operator <span>$\hat{\mathcal{O}}_{k,q}$</span> becomes a homogeneous polynomial <span>$O_{k,q}(\mathbf{s})$</span> of order <span>$k$</span> in the spin components. One can see these polynomials using <a href="library.html#Sunny.large_S_stevens_operators"><code>large_S_stevens_operators</code></a>. Due to the normalization constraint, each dipole can be expressed in polar angles, <span>$(\theta, \phi)$</span>. Then the Stevens functions <span>$O_{k,q}(\mathbf{s})$</span> correspond to the spherical harmonic functions <span>$Y_l^m(\theta, \phi)$</span> where <span>$l=k$</span> and <span>$m=q$</span>, and modulo <span>$k$</span> and <span>$q$</span>-dependent rescaling factors.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="structure-factor.html">« Structure Factor Calculations</a><a class="docs-footer-nextpage" href="library.html">Library API »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.0 on <span class="colophon-date" title="Saturday 7 October 2023 15:54">Saturday 7 October 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

previews/PR176/examples/fei2_classical.html

@@ -174,4 +174,4 @@
     )
 )
 Colorbar(hm.figure[1,2], hm.plot)
-hm</code></pre><img src="fei2_classical-a20d933e.png" alt="Example block output"/></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="powder_averaging.html">« Powder Averaged CoRh₂O₄</a><a class="docs-footer-nextpage" href="ising2d.html">Classical Ising model »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.0 on <span class="colophon-date" title="Saturday 7 October 2023 14:53">Saturday 7 October 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+hm</code></pre><img src="fei2_classical-a20d933e.png" alt="Example block output"/></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="powder_averaging.html">« Powder Averaged CoRh₂O₄</a><a class="docs-footer-nextpage" href="ising2d.html">Classical Ising model »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.0 on <span class="colophon-date" title="Saturday 7 October 2023 15:54">Saturday 7 October 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

previews/PR176/examples/fei2_tutorial.html

@@ -156,4 +156,4 @@
 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</code></pre><img src="fei2_tutorial-1fdd91bd.png" alt="Example block output"/><p>This result can be directly compared to experimental neutron scattering data from <a href="https://doi.org/10.1038/s41567-020-01110-1">Bai et al.</a></p><img src="https://raw.githubusercontent.com/SunnySuite/Sunny.jl/main/docs/src/assets/FeI2_intensity.jpg"><p>(The publication figure accidentally used a non-standard coordinate system to label the wave vectors.)</p><p>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 <code>mode = :SUN</code>, the calculation captures this coupled dipole-quadrupole dynamics.</p><p>An interesting exercise is to repeat the same study, but using <code>mode = :dipole</code> instead of <code>:SUN</code>. That alternative choice would constrain the coherent state dynamics to the space of dipoles only.</p><p>The full dynamical spin structure factor (DSSF) can be retrieved as a <span>$3×3$</span> matrix with the <a href="../library.html#Sunny.dssf-Tuple{SpinWaveTheory, Any}"><code>dssf</code></a> function, for a given path of <span>$𝐪$</span>-vectors.</p><pre><code class="language-julia hljs">disp, is = dssf(swt, path);</code></pre><p>The first output <code>disp</code> is identical to that obtained from <code>dispersion</code>. The second output <code>is</code> contains a list of <span>$3×3$</span> matrix of intensities. For example, <code>is[q,n][2,3]</code> yields the <span>$(ŷ,ẑ)$</span> component of the structure factor intensity for <code>nth</code> mode at the <code>q</code>th wavevector in the <code>path</code>.</p><h2 id="What&#39;s-next?"><a class="docs-heading-anchor" href="#What&#39;s-next?">What&#39;s next?</a><a id="What&#39;s-next?-1"></a><a class="docs-heading-anchor-permalink" href="#What&#39;s-next?" title="Permalink"></a></h2><p>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 <a href="https://arxiv.org/abs/2106.14125">SU(<em>N</em>) coherent states</a>.</p><p>The full SU(<em>N</em>) 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 <a href="fei2_classical.html#FeI-at-Finite-Temperature">FeI₂ at Finite Temperature</a> tutorial.</p><p>The classical dynamics is also a good starting point to study non-equilibrium phenomena. Empirical noise and damping terms can be used to model <a href="https://arxiv.org/abs/2209.01265">coupling to a thermal bath</a>. This yields a Langevin dynamics of SU(<em>N</em>) coherent states. Our <a href="out_of_equilibrium.html#CP-Skyrmion-Quench">CP² Skyrmion Quench</a> tutorial shows how this dynamics gives rise to the formation of novel topological defects in a temperature quench.</p><p>Relative to LSWT calculations, it can take much more time to estimate <span>$\mathcal{S}(𝐪,ω)$</span> intensities using classical dynamics simulation. See the <a href="https://nbviewer.org/github/SunnySuite/SunnyTutorials/tree/main/Tutorials/">SunnyTutorials notebooks</a> for examples of &quot;production-scale&quot; simulations.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../index.html">« Overview</a><a class="docs-footer-nextpage" href="out_of_equilibrium.html">CP² Skyrmion Quench »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.0 on <span class="colophon-date" title="Saturday 7 October 2023 14:53">Saturday 7 October 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+fig</code></pre><img src="fei2_tutorial-1fdd91bd.png" alt="Example block output"/><p>This result can be directly compared to experimental neutron scattering data from <a href="https://doi.org/10.1038/s41567-020-01110-1">Bai et al.</a></p><img src="https://raw.githubusercontent.com/SunnySuite/Sunny.jl/main/docs/src/assets/FeI2_intensity.jpg"><p>(The publication figure accidentally used a non-standard coordinate system to label the wave vectors.)</p><p>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 <code>mode = :SUN</code>, the calculation captures this coupled dipole-quadrupole dynamics.</p><p>An interesting exercise is to repeat the same study, but using <code>mode = :dipole</code> instead of <code>:SUN</code>. That alternative choice would constrain the coherent state dynamics to the space of dipoles only.</p><p>The full dynamical spin structure factor (DSSF) can be retrieved as a <span>$3×3$</span> matrix with the <a href="../library.html#Sunny.dssf-Tuple{SpinWaveTheory, Any}"><code>dssf</code></a> function, for a given path of <span>$𝐪$</span>-vectors.</p><pre><code class="language-julia hljs">disp, is = dssf(swt, path);</code></pre><p>The first output <code>disp</code> is identical to that obtained from <code>dispersion</code>. The second output <code>is</code> contains a list of <span>$3×3$</span> matrix of intensities. For example, <code>is[q,n][2,3]</code> yields the <span>$(ŷ,ẑ)$</span> component of the structure factor intensity for <code>nth</code> mode at the <code>q</code>th wavevector in the <code>path</code>.</p><h2 id="What&#39;s-next?"><a class="docs-heading-anchor" href="#What&#39;s-next?">What&#39;s next?</a><a id="What&#39;s-next?-1"></a><a class="docs-heading-anchor-permalink" href="#What&#39;s-next?" title="Permalink"></a></h2><p>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 <a href="https://arxiv.org/abs/2106.14125">SU(<em>N</em>) coherent states</a>.</p><p>The full SU(<em>N</em>) 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 <a href="fei2_classical.html#FeI-at-Finite-Temperature">FeI₂ at Finite Temperature</a> tutorial.</p><p>The classical dynamics is also a good starting point to study non-equilibrium phenomena. Empirical noise and damping terms can be used to model <a href="https://arxiv.org/abs/2209.01265">coupling to a thermal bath</a>. This yields a Langevin dynamics of SU(<em>N</em>) coherent states. Our <a href="out_of_equilibrium.html#CP-Skyrmion-Quench">CP² Skyrmion Quench</a> tutorial shows how this dynamics gives rise to the formation of novel topological defects in a temperature quench.</p><p>Relative to LSWT calculations, it can take much more time to estimate <span>$\mathcal{S}(𝐪,ω)$</span> intensities using classical dynamics simulation. See the <a href="https://nbviewer.org/github/SunnySuite/SunnyTutorials/tree/main/Tutorials/">SunnyTutorials notebooks</a> for examples of &quot;production-scale&quot; simulations.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../index.html">« Overview</a><a class="docs-footer-nextpage" href="out_of_equilibrium.html">CP² Skyrmion Quench »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.0 on <span class="colophon-date" title="Saturday 7 October 2023 15:54">Saturday 7 October 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>