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1. Multi-flavor spin wave simulations of FeI₂ (Showcase)
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).
using Sunny, GLMakieIf 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
+Spacegroup '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
+Spacegroup '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]
+Importantly, 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)
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 7.0150136167509, 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
+Energy per site 0
+This system includes $4×4×4$ unit cells, i.e. 64 Fe atoms, each with spin $S=1$ and a $g$-factor of 2. Quantum mechanically, spin $S=1$ involves a superposition of $2S+1=3$ distinct angular momentum states. In :SUN mode, this superposition will be modeled explicitly using the formalism of SU(3) coherent states, which captures both dipolar and quadrupolar fluctuations. For the more traditional dipole dynamics, use :dipole mode instead.
Next we will use set_exchange! to assign interaction to bonds. Sunny will automatically propagate each interaction to all symmetry-equivalent bonds in the unit cell. The FeI₂ interactions below follow Bai et al.
J1pm = -0.236
+J1pmpm = -0.161
+J1zpm = -0.261
+J2pm = 0.026
+J3pm = 0.166
+J′0pm = 0.037
+J′1pm = 0.013
+J′2apm = 0.068
+
+J1zz = -0.236
+J2zz = 0.113
+J3zz = 0.211
+J′0zz = -0.036
+J′1zz = 0.051
+J′2azz = 0.073
+
+J1xx = J1pm + J1pmpm
+J1yy = J1pm - J1pmpm
+J1yz = J1zpm
+
+set_exchange!(sys, [J1xx 0.0 0.0;
+ 0.0 J1yy J1yz;
+ 0.0 J1yz J1zz], Bond(1,1,[1,0,0]))
+set_exchange!(sys, [J2pm 0.0 0.0;
+ 0.0 J2pm 0.0;
+ 0.0 0.0 J2zz], Bond(1,1,[1,2,0]))
+set_exchange!(sys, [J3pm 0.0 0.0;
+ 0.0 J3pm 0.0;
+ 0.0 0.0 J3zz], Bond(1,1,[2,0,0]))
+set_exchange!(sys, [J′0pm 0.0 0.0;
+ 0.0 J′0pm 0.0;
+ 0.0 0.0 J′0zz], Bond(1,1,[0,0,1]))
+set_exchange!(sys, [J′1pm 0.0 0.0;
+ 0.0 J′1pm 0.0;
+ 0.0 0.0 J′1zz], Bond(1,1,[1,0,1]))
+set_exchange!(sys, [J′2apm 0.0 0.0;
+ 0.0 J′2apm 0.0;
+ 0.0 0.0 J′2azz], Bond(1,1,[1,2,1]))The function set_onsite_coupling! assigns a single-ion anisotropy. The argument can be constructed using spin_matrices or stevens_matrices. Here we use Julia's anonymous function syntax to assign an easy-axis anisotropy along the direction $\hat{z}$.
D = 2.165
+set_onsite_coupling!(sys, S -> -D*S[3]^2, 1)Calculating structure factor intensities
In the remainder of this tutorial, we will examine Sunny's tools for calculating the dynamical structure factor using a multi-boson generalization of linear spin wave theory (LSWT). This theory describes non-interacting quasi-particle excitations that hybridize dipolar and quadrupolar modes.
Finding the ground state
Begin with a random configuration and use minimize_energy! to find a configuration of the SU(3) coherent states (i.e. spin dipoles and quadrupoles) that locally minimizes energy.
randomize_spins!(sys)
+minimize_energy!(sys)49A 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])
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)
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
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
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. See our generalized spin dynamics 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 dynamical SU(N) quench tutorial illustrates how a temperature quench can give rise to novel liquid phase of CP² skyrmions.
![](\"https://raw.githubusercontent.com/SunnySuite/Sunny.jl/main/docs/src/assets/FeI2_crystal.jpg\")
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Note that the order of the pair (i j) is significant if the exchange tensor contains antisymmetric Dzyaloshinskii–Moriya (DM) interactions.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"In the case of FeI₂, Bond(1, 1, [1,0,0]) is one of the 6 nearest-neighbor Fe-Fe bonds on a triangular lattice layer, and Bond(1, 1, [0,0,1]) is an Fe-Fe bond between layers.","category":"page"},{"location":"examples/fei2_tutorial.html#Building-a-spin-System","page":"Case Study: FeI₂","title":"Building a spin System","text":"","category":"section"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"In constructing a spin System, we must provide several additional details about the spins.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"sys = System(cryst, (4, 4, 4), [SpinInfo(1, S=1, g=2)], :SUN, seed=2)","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"This system includes 444 unit cells, i.e. 64 Fe atoms, each with spin S=1 and a g-factor of 2. Quantum mechanically, spin S=1 involves a superposition of 2S+1=3 distinct angular momentum states. In :SUN mode, this superposition will be modeled explicitly using the formalism of SU(3) coherent states, which captures both dipolar and quadrupolar fluctuations. For the more traditional dipole dynamics, use :dipole mode instead.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"Next we will use set_exchange! to assign interaction to bonds. Sunny will automatically propagate each interaction to all symmetry-equivalent bonds in the unit cell. The FeI₂ interactions below follow Bai et al.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"J1pm = -0.236\nJ1pmpm = -0.161\nJ1zpm = -0.261\nJ2pm = 0.026\nJ3pm = 0.166\nJ′0pm = 0.037\nJ′1pm = 0.013\nJ′2apm = 0.068\n\nJ1zz = -0.236\nJ2zz = 0.113\nJ3zz = 0.211\nJ′0zz = -0.036\nJ′1zz = 0.051\nJ′2azz = 0.073\n\nJ1xx = J1pm + J1pmpm\nJ1yy = J1pm - J1pmpm\nJ1yz = J1zpm\n\nset_exchange!(sys, [J1xx 0.0 0.0;\n 0.0 J1yy J1yz;\n 0.0 J1yz J1zz], Bond(1,1,[1,0,0]))\nset_exchange!(sys, [J2pm 0.0 0.0;\n 0.0 J2pm 0.0;\n 0.0 0.0 J2zz], Bond(1,1,[1,2,0]))\nset_exchange!(sys, [J3pm 0.0 0.0;\n 0.0 J3pm 0.0;\n 0.0 0.0 J3zz], Bond(1,1,[2,0,0]))\nset_exchange!(sys, [J′0pm 0.0 0.0;\n 0.0 J′0pm 0.0;\n 0.0 0.0 J′0zz], Bond(1,1,[0,0,1]))\nset_exchange!(sys, [J′1pm 0.0 0.0;\n 0.0 J′1pm 0.0;\n 0.0 0.0 J′1zz], Bond(1,1,[1,0,1]))\nset_exchange!(sys, [J′2apm 0.0 0.0;\n 0.0 J′2apm 0.0;\n 0.0 0.0 J′2azz], Bond(1,1,[1,2,1]))","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The function set_onsite_coupling! assigns a single-ion anisotropy. The argument can be constructed using spin_matrices or stevens_matrices. Here we use Julia's anonymous function syntax to assign an easy-axis anisotropy along the direction hatz.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"D = 2.165\nset_onsite_coupling!(sys, S -> -D*S[3]^2, 1)","category":"page"},{"location":"examples/fei2_tutorial.html#Calculating-structure-factor-intensities","page":"Case Study: FeI₂","title":"Calculating structure factor intensities","text":"","category":"section"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"In the remainder of this tutorial, we will examine Sunny's tools for calculating the dynamical structure factor using a multi-boson generalization of linear spin wave theory (LSWT). This theory describes non-interacting quasi-particle excitations that hybridize dipolar and quadrupolar modes.","category":"page"},{"location":"examples/fei2_tutorial.html#Finding-the-ground-state","page":"Case Study: FeI₂","title":"Finding the ground state","text":"","category":"section"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"Begin with a random configuration and use minimize_energy! to find a configuration of the SU(3) coherent states (i.e. spin dipoles and quadrupoles) that locally minimizes energy.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"randomize_spins!(sys)\nminimize_energy!(sys)","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"A positive number above indicates that the procedure has converged to a local energy minimum. The configuration, however, may still have defects. This can be checked by visualizing the spins, colored according to their z-components.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"plot_spins(sys; color=[s[3] for s in sys.dipoles])","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"A different understanding of the magnetic ordering can be obtained by moving to Fourier space. The 'instant' structure factor 𝒮(𝐪) is an experimental observable. To investigate 𝒮(𝐪) as true 3D data, Sunny provides instant_correlations and related functions. Here, however, we will use print_wrapped_intensities, which gives average intensities for the individual Bravais sublattices (in effect, all wavevectors are wrapped to the first Brillouin zone).","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"print_wrapped_intensities(sys)","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The result will likely be approximately consistent with the known zero-field energy-minimizing magnetic structure of FeI₂, which is single-Q (two-up, two-down antiferromagnetic order). Mathematically, spontaneous symmetry breaking should select one of Q = 0 -14 14, 14 0 14, or -141414, associated with the three-fold rotational symmetry of the crystal spacegroup. In nature, however, one will frequently encounter competing \"domains\" associated with the three possible orientations of the ground state.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"If the desired ground state is already known, as with FeI₂, it could be entered by hand using set_dipole!. Alternatively, in the case of FeI₂, we could repeatedly employ the above randomization and minimization procedure until a defect-free configuration is found. Some systems will have more complicated ground states, which can be much more challenging to find. For this, Sunny provides experimental support for powerful simulated annealing via parallel tempering, but that is outside the scope of this tutorial.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"Here, let's break the three-fold symmetry of FeI₂ by hand. Given one or more desired Q modes, Sunny can suggest a magnetic supercell with appropriate periodicity. Let's arbitrarily select one of the three possible ordering wavevectors, Q = 0 -14 14. Sunny suggests a corresponding magnetic supercell in units of the crystal lattice vectors.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"suggest_magnetic_supercell([[0, -1/4, 1/4]])","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The system returned by reshape_supercell is smaller, and is sheared relative to the original system. This makes it much easier to find the global energy minimum.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"sys_min = reshape_supercell(sys, [1 0 0; 0 2 1; 0 -2 1])\nrandomize_spins!(sys_min)\nminimize_energy!(sys_min);\nnothing #hide","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"Plot the system again, now including \"ghost\" spins out to 12Å","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"plot_spins(sys_min; color=[s[3] for s in sys_min.dipoles], ghost_radius=12)","category":"page"},{"location":"examples/fei2_tutorial.html#Linear-spin-wave-theory","page":"Case Study: FeI₂","title":"Linear spin wave theory","text":"","category":"section"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"Now that we have found the ground state for a magnetic supercell, we can immediately proceed to perform zero-temperature calculations using linear spin wave theory. We begin by instantiating a SpinWaveTheory type using the supercell.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"swt = SpinWaveTheory(sys_min)","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"Select a sequence of wavevectors that will define a piecewise linear interpolation in reciprocal lattice units (RLU).","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"q_points = [[0,0,0], [1,0,0], [0,1,0], [1/2,0,0], [0,1,0], [0,0,0]];\nnothing #hide","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The function reciprocal_space_path will linearly sample a path between the provided q-points with a given density. The xticks return value provides labels for use in plotting.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"density = 50\npath, xticks = reciprocal_space_path(cryst, q_points, density);\nnothing #hide","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The dispersion function defines the quasiparticle excitation energies ω_i(𝐪) for each point 𝐪 along the reciprocal space path.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"disp = dispersion(swt, path);\nnothing #hide","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"In addition to the band energies ω_i(𝐪), Sunny can calculate the inelastic neutron scattering intensity I_i(𝐪) for each band i according to an intensity_formula. We choose to apply a polarization correction (1 - 𝐪𝐪) by setting the mode argument to :perp. Selecting delta_function_kernel specifies that we want the energy and intensity of each band individually.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"formula = intensity_formula(swt, :perp; kernel=delta_function_kernel)","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The function intensities_bands uses linear spin wave theory to calculate both the dispersion and intensity data for the provided path.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"disp, intensity = intensities_bands(swt, path, formula);\nnothing #hide","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"These can be plotted in GLMakie.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"fig = Figure()\nax = Axis(fig[1,1]; xlabel=\"Momentum (r.l.u.)\", ylabel=\"Energy (meV)\", xticks, xticklabelrotation=π/6)\nylims!(ax, 0.0, 7.5)\nxlims!(ax, 1, size(disp, 1))\ncolorrange = extrema(intensity)\nfor i in axes(disp, 2)\n lines!(ax, 1:length(disp[:,i]), disp[:,i]; color=intensity[:,i], colorrange)\nend\nfig","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"To make comparisons with inelastic neutron scattering (INS) data, it is helpful to employ an empirical broadening kernel, e.g., a lorentzian.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"γ = 0.15 # width in meV\nbroadened_formula = intensity_formula(swt, :perp; kernel=lorentzian(γ))","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The intensities_broadened function requires an energy range in addition to the 𝐪-space path.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"energies = collect(0:0.01:10) # 0 < ω < 10 (meV).\nis1 = intensities_broadened(swt, path, energies, broadened_formula);\nnothing #hide","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"A real FeI₂ sample will exhibit competing magnetic domains associated with spontaneous symmetry breaking of the 6-fold rotational symmetry of the triangular lattice. Note that the wavevectors 𝐪 and -𝐪 are equivalent in the structure factor, which leaves three distinct domain orientations, which are related by 120° rotations about the z-axis. Rather than rotating the spin configuration directly, on can rotate the 𝐪-space path. Below, we use rotation_in_rlu to average the intensities over all three possible orientations.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"R = rotation_in_rlu(cryst, [0, 0, 1], 2π/3)\nis2 = intensities_broadened(swt, [R*q for q in path], energies, broadened_formula)\nis3 = intensities_broadened(swt, [R*R*q for q in path], energies, broadened_formula)\nis_averaged = (is1 + is2 + is3) / 3\n\nfig = Figure()\nax = Axis(fig[1,1]; xlabel=\"Momentum (r.l.u.)\", ylabel=\"Energy (meV)\", xticks, xticklabelrotation=π/6)\nheatmap!(ax, 1:size(is_averaged, 1), energies, is_averaged)\nfig","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"This result can be directly compared to experimental neutron scattering data from Bai et al.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"![](\"https://raw.githubusercontent.com/SunnySuite/Sunny.jl/main/docs/src/assets/FeI2_intensity.jpg\")
","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"(The publication figure accidentally used a non-standard coordinate system to label the wave vectors.)","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"To get this agreement, the use of SU(3) coherent states is essential. In other words, we needed a theory of multi-flavored bosons. The lower band has large quadrupolar character, and arises from the strong easy-axis anisotropy of FeI₂. By setting mode = :SUN, the calculation captures this coupled dipole-quadrupole dynamics.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"An interesting exercise is to repeat the same study, but using mode = :dipole instead of :SUN. That alternative choice would constrain the coherent state dynamics to the space of dipoles only.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The full dynamical spin structure factor (DSSF) can be retrieved as a 33 matrix with the dssf function, for a given path of 𝐪-vectors.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"disp, is = dssf(swt, path);\nnothing #hide","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The first output disp is identical to that obtained from dispersion. The second output is contains a list of 33 matrix of intensities. For example, is[q,n][2,3] yields the (yz) component of the structure factor intensity for nth mode at the qth wavevector in the path.","category":"page"},{"location":"examples/fei2_tutorial.html#What's-next?","page":"Case Study: FeI₂","title":"What's next?","text":"","category":"section"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The multi-boson linear spin wave theory, applied above, can be understood as the quantization of a certain generalization of the Landau-Lifshitz spin dynamics. Rather than dipoles, this dynamics takes places on the space of SU(N) coherent states.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The full SU(N) coherent state dynamics, with appropriate quantum correction factors, can be useful to model finite temperature scattering data. In particular, it captures certain anharmonic effects due to thermal fluctuations. This is the subject of our FeI₂ at Finite Temperature tutorial.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"The classical dynamics is also a good starting point to study non-equilibrium phenomena. Empirical noise and damping terms can be used to model coupling to a thermal bath. This yields a Langevin dynamics of SU(N) coherent states. Our CP² Skyrmion Quench tutorial shows how this dynamics gives rise to the formation of novel topological defects in a temperature quench.","category":"page"},{"location":"examples/fei2_tutorial.html","page":"Case Study: FeI₂","title":"Case Study: FeI₂","text":"Relative to LSWT calculations, it can take much more time to estimate mathcalS(𝐪ω) intensities using classical dynamics simulation. See the SunnyTutorials notebooks for examples of \"production-scale\" simulations.","category":"page"},{"location":"structure-factor.html#Structure-Factor-Calculations","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"","category":"section"},{"location":"structure-factor.html#Overview","page":"Structure Factor Calculations","title":"Overview","text":"","category":"section"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"The dynamical structure factor is of fundamental importance for characterizing a magnetic system, and facilitates quantitative comparison between theory and experimental scattering data.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"Consider, for example, a two-point dynamical spin correlation function, s^α(𝐱+Δ𝐱 t+Δt) s^β(𝐱 t). Here s^α(𝐱 t) represents the time dynamics of a spin dipole component α at position 𝐱, and brackets represent an average over equilibrium initial conditions and over (𝐱 t). The dynamical structure factor is defined as the Fourier transform of this two-point correlation in both space and time, up to an overall scaling factor. Using the convolution theorem, the result is,","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"𝒮^αβ(𝐪 ω) = frac1V s^α(𝐪 ω)^ast s^β(𝐪 ω) ","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"with V the system volume. We will restrict attention to lattice systems with periodic boundaries.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"Consider a crystal unit cell defined by three lattice vectors 𝐚_1 𝐚_2 𝐚_3, and linear system sizes L_1 L_2 L_3 measured in unit cells. The allowed momentum vectors take on discrete values 𝐪 = sum_α=1^3 m_α 𝐛_α L_α, where m_α are an integers and the reciprocal lattice vectors 𝐛_α are defined to satisfy 𝐚_α 𝐛_β = 2π δ_αβ. For a Bravais lattice, 𝐪 will be periodic in the first Brillouin zone, i.e., under any shift 𝐪 𝐪 𝐛_α. More generally, consider a non-Bravais lattice such that each unit cell may contain multiple spins. By partitioning spins s_j(𝐱t) according to their sublattice index j, the relevant momenta 𝐪 remain discretized as above, but now periodicity in the first Brillouin zone is lost. The structure factor may be written as a phase-average over the displacements between sublattices 𝐫_jk,","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"𝒮^αβ(𝐪 ω) = _jk e^i 𝐫_jk 𝐪 𝒮^αβ_jk(𝐪 ω) ","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"From a theoretical perspective, the quantity","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"𝒮^αβ_jk(𝐪 ω) = frac1V s_j^α(𝐪 ω)^ast s_k^β(𝐪 ω)","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"is fundamental. For each sublattice j, the data s_j^α(𝐪 ω) can be efficiently obtained by fast Fourier tranformation of a real space configuration s_j^α(𝐱 t). Internally, Sunny will calculate and store the discrete 𝒮^αβ_jk(𝐪 ω) correlation data, and use this to construct 𝒮^αβ(𝐪ω) intensities that can be compared with experiment.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"Calculating this structure factor involves several steps, with various possible settings. Sunny provides a number of tools to facilitate this calculation and to extract information from the results. These tools are briefly outlined below. Please see the Examples for a \"real life\" use case. Detailed function information is available in the Library API.","category":"page"},{"location":"structure-factor.html#Estimating-stucture-factors-with-classical-dynamics","page":"Structure Factor Calculations","title":"Estimating stucture factors with classical dynamics","text":"","category":"section"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"Classical dynamics may be used to estimate structure factor data by analyzing the spin-spin correlations of dynamical trajectories. This is fundamentally a Monte Carlo approach, as the trajectories must be started from an initial spin configuration that is sampled at thermal equilibrium. (Note that it is not possible to estimate a true T=0 dynamical structure factor using this method, but the temperature may be very low.) Samples are accumulated into a SampledCorrelations, from which intensity information may be extracted. The user does not typically build their own SampledCorrelations but instead initializes one by calling either dynamical_correlations or instant_correlations, as described below.","category":"page"},{"location":"structure-factor.html#Estimating-a-dynamical-structure-factor:-𝒮(𝐪,ω)","page":"Structure Factor Calculations","title":"Estimating a dynamical structure factor: 𝒮(𝐪ω)","text":"","category":"section"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"A SampledCorrelations for estimating the dynamical structure factor, 𝒮^αβ(𝐪ω), may be created by calling dynamical_correlations. This requires three keyword arguments. These will determine the dynamics used to calculate samples and, consequently, the ω information that will be available. ","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"Δt: Determines the step size used for simulating the dynamics. A smaller number will require proportionally more calculation time. While a smaller Δt will enable the resolution of higher energies, Δt is typically selected to ensure numerical stability rather than to maximize the largest ω value. A safe choice is to use the smaller value of Δt = 0.1/(J* S^2) or Δt = 0.1/(D * S), where S is magnetic moment of the largest local spin (as specified in SpinInfo), J is the parameter governing the largest bilinear interaction (e.g. exchange), and D is the parameter governing the largest single-site term of the Hamiltonian (e.g., anisotropy or Zeeman term).\nωmax: Sets the maximum resolved energy. Note that this is not independent of Δt. If ωmax too large, Sunny will throw an error and ask you to choose a smaller Δt. \nnω: Determines the number of energy bins to resolve. A larger number will require more calculation time.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"A sample may be added by calling add_sample!(sc, sys). The input sys must be a spin configuration in good thermal equilibrium, e.g., using the continuous Langevin dynamics or using single spin flip trials with LocalSampler. The statistical quality of the 𝒮^αβ(𝐪ω) can be improved by repeatedly generating decorrelated spin configurations in sys and calling add_sample! on each configuration.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"The outline of typical use case might look like this:","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"# Make a `SampledCorrelations`\nsc = dynamical_correlations(sys; Δt=0.05, ωmax=10.0, nω=100) \n\n# Add samples\nfor _ in 1:nsamples\n decorrelate_system(sys) # Perform some type of Monte Carlo simulation\n add_sample!(sc, sys) # Use spins to calculate trajectory and accumulate new sample of 𝒮(𝐪,ω)\nend","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"The calculation may be configured in a number of ways; see the dynamical_correlations documentation for a list of all keywords.","category":"page"},{"location":"structure-factor.html#Estimating-an-instantaneous-(\"static\")-structure-factor:-𝒮(𝐪)","page":"Structure Factor Calculations","title":"Estimating an instantaneous (\"static\") structure factor: 𝒮(𝐪)","text":"","category":"section"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"Sunny provides two methods for calculating instantaneous, or static, structure factors: 𝒮^αβ(𝐪). The first involves calculating spatial spin-spin correlations at single time slices. The second involves calculating a dynamic structure factor first and integrating out the ω information. The advantage of the latter approach is that it enables application of an ω-dependent classical-to-quantum rescaling of structure factor intensities, a method that should be preferred whenever comparing results to experimental data or spin wave calculations. A disadvantage of this approach is that it is computationally more expensive. There are also many cases when it is not straightforward to calculate a meaningful dynamics, as when working with Ising spins. In this section we will discuss how to calculate instantaneous structure factors from static spin configurations. Information about calculating instantaneous data from a dynamical correlations can be found in the following section.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"The basic usage for the instantaneous case is very similar to the dynamic case, except one calls instant_correlations instead of dynamical_correlations to configure a SampledCorrelations. Note that there are no required keywords as there is no need to specify any dynamics. instant_correlations will return a SampledCorrelations containing no data. Samples may be added by calling add_sample!(sc, sys), where sc is the SampledCorrelations. When performing a finite-temperature calculation, it is important to ensure that the spin configuration in the sys represents a good equilibrium sample, as in the dynamical case. Note, however, that we recommend calculating instantaneous correlations at finite temperature calculations by using full dynamics (i.e., using dynamical_correlations) and then integrating out the energy axis. An approach to doing this is described in the next section.","category":"page"},{"location":"structure-factor.html#Extracting-information-from-sampled-correlation-data","page":"Structure Factor Calculations","title":"Extracting information from sampled correlation data","text":"","category":"section"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"The basic function for extracting information from a SampledCorrelations at a particular wave vector, 𝐪, is intensities_interpolated. It takes a SampledCorrelations, a list of wave vectors, and an intensity_formula. The intensity_formula specifies how to contract and correct correlation data to arrive at a physical intensity. A simple example is formula = intensity_formula(sc, :perp), which will instruct Sunny apply polarization corrections: sum_αβ(I-q_α q_β) 𝒮^αβ(𝐪ω). An intensity at the wave vector 𝐪 = (𝐛_2 + 𝐛_3)2 may then be retrieved with intensities_interpolated(sf, [[0.0, 0.5, 0.5]], formula) . intensities_interpolated returns a list of nω elements at each wavevector. The corresponding ω values can be retrieved by calling available_energies on sf.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"Since Sunny only calculates the structure factor on a finite lattice when performing classical simulations, it is important to realize that exact information is only available at a discrete set of wave vectors. Specifically, for each axis index i, we will get information at q_i = fracnL_i, where n runs from (frac-L_i2+1) to fracL_i2 and L_i is the linear dimension of the lattice used for the calculation. If you request a wave vector that does not fall into this set, Sunny will automatically round to the nearest 𝐪 that is available. If intensities_interpolated is given the keyword argument interpolation=:linear, Sunny will use trilinear interpolation to determine a result at the requested wave vector. ","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"To retrieve the intensities at all wave vectors for which there is exact data, first call the function available_wave_vectors to generate a list of qs. This takes an optional keyword argument bzsize, which must be given a tuple of three integers specifying the number of Brillouin zones to calculate, e.g., bzsize=(2,2,2). The resulting list of wave vectors may then be passed to intensities_interpolated.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"Alternatively, intensities_binned can be used to place the exact data into histogram bins for comparison with experiment.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"The convenience function reciprocal_space_path returns a list of wavevectors sampled along a path that connects specified 𝐪 points. This list can be used as an input to intensities. Another convenience method, reciprocal_space_shell will generate points on a sphere of a given radius. This is useful for powder averaging. ","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"A number of arguments for intensity_formula are available which modify the calculation of structure factor intensity. It is generally recommended to provide a value of kT corresponding to the temperature of sampled configurations. Given kT, Sunny will include an energy- and temperature-dependent classical-to-quantum rescaling of intensities in the formula.","category":"page"},{"location":"structure-factor.html","page":"Structure Factor Calculations","title":"Structure Factor Calculations","text":"To retrieve intensity data from a instantaneous structure factor, use instant_intensities_interpolated, which accepts similar arguments to intensities_interpolated. This function may also be used to calculate instantaneous information from a dynamical correlation data, i.e. from a SampledCorrelations created with dynamical_correlations. Note that it is important to supply a value to kT to reap the benefits of this approach over simply calculating a static structure factor at the outset. ","category":"page"},{"location":"writevtk.html#ParaView-Rendering","page":"ParaView Rendering","title":"ParaView Rendering","text":"","category":"section"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"The 4D correlation data produced by Sunny is too high-dimensional to visualize directly. This page describes how to export 3D slices of correlation data from Sunny to the Visual ToolKit (VTK) format, which is compatible with the ParaView visualization software. ParaView supports volumetric rendering:","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"![](\"./assets/paraviewrender.jpg\")
","category":"page"},{"location":"writevtk.html#Simulation-data","page":"ParaView Rendering","title":"Simulation data","text":"","category":"section"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"First, generate some correlation data in Sunny. We will use a 2D lattice, since the correlation data S(Q_xQ_yomega) is 3D and can be exported in its entirety. The following code sets up the system, thermalizes it, and records the correlation data in a SampledCorrelations called dsf.","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"using Sunny\n\n# Single layer 12x12 periodic square lattice\nlatsize = (12,12,1)\n\nlatvecs = lattice_vectors(8.,8.,12.,90,100,90)\npositions = [[0,0,0]]\ntypes = [\"Cu\"]\nformfactors = [FormFactor(\"Cu2\")]\nxtal = Crystal(latvecs, positions; types)\n\nsys = System(xtal, latsize, [SpinInfo(1, S=1/2, g=2)], :SUN; seed=1)\n\nJ = 10.\nset_exchange!(sys, J, Bond(1,1,[1,0,0]))\nset_exchange!(sys, J, Bond(1,1,[0,1,0]))\n\nΔt = 0.01\nkT = 0.5\nlangevin = Langevin(Δt; λ=0.5, kT)\nrandomize_spins!(sys)\nfor i in 1:10_000 # Long enough to reach equilibrium\n step!(sys, langevin)\nend \n\nωmax=10.\n\ndsf = dynamical_correlations(sys\n ;Δt=Δt\n ,nω=48\n ,ωmax=ωmax\n ,process_trajectory=:symmetrize)\n\nnsamples = 10\nfor _ in 1:nsamples\n for _ in 1:1000 \n step!(sys, langevin)\n end\n add_sample!(dsf, sys)\nend","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"The default histogram BinningParameters are already integrated over the z direction because the system is 2D:","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"unit_resolution_binning_parameters(dsf)","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"⊡ 12 bins from -0.042 to +0.958 along [+1.27 dx] (Δ = 0.065)\n⊡ 12 bins from -0.042 to +0.958 along [+1.27 dy] (Δ = 0.065)\n∫ Integrated from +0.000 to +0.000 along [-0.33 dx +1.88 dz] (Δ = 0.524)\n⊡ 48 bins from -0.107 to +10.134 along [+1.00 dE] (Δ = 0.213)","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"The histogram is very oblong; it's approximately 1x1x10. To make it a nicer shape, we will rescale the energy axis to be be fractions of ωmax:","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"params = unit_resolution_binning_parameters(dsf)\nscale_factor = ωmax\nparams.binend[4] /= scale_factor\nparams.binstart[4] /= scale_factor\nparams.binwidth[4] /= scale_factor\nparams.covectors[4,:] ./= scale_factor","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"Doing this changes the last axis of the histogram to fit in [0,1]:","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"⊡ 49 bins from -0.011 to +1.013 along [+0.10 dE] (Δ = 0.213)","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"Now that our histogram is a cube, we compute the intensity in the histogram bins using the usual intensities_binned:","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"formula = intensity_formula(dsf,:trace)\nsignal, counts = intensities_binned(dsf, params; formula)\nintensity = signal ./ counts","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"Now that we have our intensity data and the binning parameters, we can export to VTK format using export_vtk and move to ParaView for the visualization.","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"# Importing WriteVTK enables Sunny's export-to-VTK functions\nimport WriteVTK\n\n# [1,2,4] specifies that the (x,y,z) axes in ParaView are (Qx,Qy,ω)\nexport_vtk(\"square_lattice\", params, intensity; dims_kept = [1,2,4])\n# Writes a file square_lattice.vti in the current directory","category":"page"},{"location":"writevtk.html#Loading-in-ParaView","page":"ParaView Rendering","title":"Loading in ParaView","text":"","category":"section"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"In ParaView, use File > Open to open square_lattice.vti. This will add the file to the Pipeline Browser with a closed eye icon, indicating that the data is ready to be loaded.","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"In the Properties panel, both bin_centers and data will be selected for import by default. Uncheck bin_centers because we don't need that information for the visualization. Click the green Apply button to load the data.","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"By default, only the outline of the data is shown in the 3D viewport. Since we adjusted the energy axis, the outline is a 1x1x1 cube. Optionally enable the axes grid under \"View\", and customize using the adjacent edit button.","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"![](\"./assets/paraviewimport.png\")
","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"To enable the volumetric render:","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"Select \"Volume\" from the \"Representation\" drop-down menu under \"Display\".\nThe \"Coloring\" drop-down should automatically select data because it's the only data loaded.\nOpen the Color Map Editor to adjust the opacity of the fog, which may be too faint to see by default.","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"![](\"./assets/paraviewvolume.png\")
","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"Depending on your computer and your dataset size, the volumetric rendering may be slow, but our dataset is relatively small, so the render should be fast.","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"If nothing shows up at first, don't despair. Often, there are Bragg-like peaks in the correlation data which outshine everything else. To see this, enable Display Data Histogram in the Color Map Editor panel. To zoom in on the lower-intensity data, click and drag the right side handle of the opacity transfer function box to the middle a few times.","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"![](\"./assets/paraviewcolormap.png\")
","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"After suitable color mapping, the dispersion curve should become visible:","category":"page"},{"location":"writevtk.html","page":"ParaView Rendering","title":"ParaView Rendering","text":"![](\"https://raw.githubusercontent.com/SunnySuite/Sunny.jl/main/docs/src/assets/CoRh2O4_intensity.jpg\")
","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"EditURL = \"../../../examples/out_of_equilibrium.jl\"","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"Download this example as Jupyter notebook or Julia script.","category":"page"},{"location":"examples/out_of_equilibrium.html#CP-Skyrmion-Quench","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"","category":"section"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"This example demonstrates Sunny's ability to simulate the out-of-equilibrium dynamics of generalized spin systems. We will implement the model Hamiltonian of Zhang et al., Nature Communications 14, 3626 (2023), which supports a novel type of topological defect, a CP² skyrmion, that involves both the dipolar and quadrupolar parts of a quantum spin.","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"Beginning from an initial high-temperature state, a disordered gas of CP² skyrmions can be formed by rapidly quenching to low temperature. To model the coupled dynamics of dipoles and quadrupoles, Sunny uses a recently developed generalization of the Landau-Lifshitz spin dynamics, Dahlbom et al., Phys. Rev. B 106, 235154 (2022).","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"using Sunny, GLMakie","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"The Hamiltonian we will implement,","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"mathcalH = sum_langle ij rangle J_ij( hatS_i^x hatS_j^x + hatS_i^y hatS_j^y + DeltahatS_i^z hatS_j^z) - hsum_ihatS_i^z + Dsum_i(hatS_i^z)^2","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"contains competing ferromagnetic nearest-neightbor and antiferromagnetic next-nearest-neighbor exchange terms on a triangular lattice. Both exchanges exhibit anisotropy on the z-term. Additionally, there is an external magnetic field, h, and easy-plane single-ion anisotropy, D 0. We begin by implementing the Crystal.","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"lat_vecs = lattice_vectors(1.0, 1.0, 2.0, 90, 90, 120)\nbasis_vecs = [[0,0,0]]\ncryst = Crystal(lat_vecs, basis_vecs)","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"The crystal is then used to create a spin System. All parameters in this model system are dimensionless, so we select \"theory\" units and set the g-factor to one.","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"L = 40\ndims = (L, L, 1)\nsys = System(cryst, dims, [SpinInfo(1, S=1, g=1)], :SUN; seed=101, units=Units.theory)","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"We proceed to implement each term of the Hamiltonian, selecting our parameters so that the system occupies a region of the phase diagram that supports skyrmions. The exchange interactions are set as follows.","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"J1 = -1 # Nearest-neighbor ferromagnetic\nJ2 = (2.0/(1+√5)) # Tune competing exchange to set skyrmion scale length\nΔ = 2.6 # Exchange anisotropy\n\nex1 = J1 * [1.0 0.0 0.0;\n 0.0 1.0 0.0;\n 0.0 0.0 Δ]\nex2 = J2 * [1.0 0.0 0.0;\n 0.0 1.0 0.0;\n 0.0 0.0 Δ]\nset_exchange!(sys, ex1, Bond(1, 1, [1, 0, 0]))\nset_exchange!(sys, ex2, Bond(1, 1, [1, 2, 0]))","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"Next we add the external field,","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"h = 15.5\nfield = set_external_field!(sys, [0.0 0.0 h])","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"and finally an easy-plane single-ion anisotropy,","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"D = 19.0\nset_onsite_coupling!(sys, S -> D*S[3]^2, 1)","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"Initialize system to an infinite temperature (fully randomized) initial condition.","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"randomize_spins!(sys)","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"We are now ready to simulate the quenching process using a generalized Langevin spin dynamics. If we were working with spin dipoles only, then Langevin dynamics would be the usual Landau-Lifshitz spin dynamics, augmented with damping and noise terms. In the present study, we are instead working with quantum spin-1 (an (N=3)-level system that includes both dipoles and quadrupoles). Here, Langevin captures the coupled dipole-quadrupole dynamics using the formalism of SU(N) coherent states.","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"Selecting kT = 0 in the Langevin dynamics will effective disable the noise term. Then the parameter λ effectively determines the damping time-scale.","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"Δt = 0.2/D # Integration time step (inverse meV). Typically this will be\n # inversely proportional to the largest energy scale in the\n # system. We can use a fairly large time-step here because\n # accuracy isn't critical.\nkT = 0 # Target equilibrium temperature (meV)\nλ = 0.1 # Magnitude of coupling to thermal bath (dimensionless)\nintegrator = Langevin(Δt; kT, λ)","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"Finally we run the dynamics. We will record the state of the system at three different times during the quenching process by copying the coherents field of the System.","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"τs = [4., 16, 256] # Times to record snapshots\nframes = [] # Empty array to store snapshots\nfor i in eachindex(τs)\n dur = i == 1 ? τs[1] : τs[i] - τs[i-1] # Determine the length of time to simulate\n numsteps = round(Int, dur/Δt)\n for _ in 1:numsteps # Perform the integration\n step!(sys, integrator)\n end\n push!(frames, copy(sys.coherents)) # Save a snapshot spin configuration\nend","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"To visualize the state of the system contained in each snapshot, we will calculate and plot the skyrmion density on each plaquette of our lattice. The function plot_triangular_plaquettes is not part of the core Sunny package, but rather something you could define yourself. We are using the definition in plotting2d.jl from the Sunny examples/extra directory.","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"include(pkgdir(Sunny, \"examples\", \"extra\", \"plotting2d.jl\"))\n\nfunction sun_berry_curvature(z₁, z₂, z₃)\n z₁, z₂, z₃ = normalize.((z₁, z₂, z₃))\n n₁ = z₁ ⋅ z₂\n n₂ = z₂ ⋅ z₃\n n₃ = z₃ ⋅ z₁\n return angle(n₁ * n₂ * n₃)\nend\n\nplot_triangular_plaquettes(sun_berry_curvature, frames; resolution=(1800,600),\n offset_spacing=10, texts = [\"\\tt = \"*string(τ) for τ in τs], text_offset = (0.0, 6.0)\n)","category":"page"},{"location":"examples/out_of_equilibrium.html","page":"CP² Skyrmion Quench","title":"CP² Skyrmion Quench","text":"The times are given in hbarJ_1. The white background corresponds to a quantum paramagnetic state, where the local spin exhibits a strong quadrupole moment and little or no dipole moment. Observe that the process has generated a number of well-formed skyrmions of both positive (red) and negative (blue) charge in addition to a number of other metastable spin configurations. A full-sized version of this figure is available in Dahlbom et al..","category":"page"},{"location":"library.html#Library-API","page":"Library API","title":"Library API","text":"","category":"section"},{"location":"library.html","page":"Library API","title":"Library API","text":"This page describes the public types and functions exported by Sunny. This documentation can be also be accessed using the Julia help system (enter ? at the Julia command prompt).","category":"page"},{"location":"library.html","page":"Library API","title":"Library API","text":"","category":"page"},{"location":"library.html","page":"Library API","title":"Library API","text":"Sunny.plot_spins\nSunny.view_crystal\nSunny.export_vtk","category":"page"},{"location":"library.html","page":"Library API","title":"Library API","text":"Modules = [Sunny]\nPrivate = false","category":"page"},{"location":"library.html#Sunny.Site","page":"Library API","title":"Sunny.Site","text":"(cell1, cell2, cell3, i) :: Site\n\nFour indices identifying a single site in a System. The first three indices select the lattice cell and the last selects the sublattice (i.e., the atom within the unit cell).\n\nThis object can be used to index dipoles and coherents fields of a System. A Site is also required to specify inhomogeneous interactions via functions such as set_external_field_at! or set_exchange_at!.\n\nNote that the definition of a cell may change when a system is reshaped. In this case, it is convenient to construct the Site using position_to_site, which always takes a position in fractional coordinates of the original lattice vectors.\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.Units","page":"Library API","title":"Sunny.Units","text":"Units.meV\nUnits.theory\n\nThe unit system is implicitly determined by the definition of two physical constants: the vacuum permeability μ₀ and the Bohr magneton μ_B. Temperatures are effectively measured in units of energy (k_B = 1) and time is effectively measured in units of inverse energy (ħ = 1). The default unit system, Units.meV, employs (meV, Å, tesla). Select alternatively Units.theory for a units system defined so that μ₀ = μ_B = 1.\n\nSee also meV_per_K\n\n\n\n\n\n","category":"constant"},{"location":"library.html#Sunny.meV_per_K","page":"Library API","title":"Sunny.meV_per_K","text":"meV_per_K = 0.086173332621451774\n\nA physical constant. Useful for converting kelvin into the default energy units, meV.\n\n\n\n\n\n","category":"constant"},{"location":"library.html#Sunny.BinningParameters","page":"Library API","title":"Sunny.BinningParameters","text":"BinningParameters(binstart,binend,binwidth;covectors = I(4))\nBinningParameters(binstart,binend;numbins,covectors = I(4))\n\nDescribes a 4D parallelepided histogram in a format compatible with experimental Inelasitic Neutron Scattering data. See generate_mantid_script_from_binning_parameters to convert BinningParameters to a format understandable by the Mantid software, or load_nxs to load BinningParameters from a Mantid .nxs file.\n\nThe coordinates of the histogram axes are specified by multiplication of (q,ω) with each row of the covectors matrix, with q given in [R.L.U.]. Since the default covectors matrix is the identity matrix, the default axes are (qx,qy,qz,ω) in absolute units.\n\nThe convention for the binning scheme is that:\n\nThe left edge of the first bin starts at binstart\nThe bin width is binwidth\nThe last bin contains binend\nThere are no \"partial bins;\" the last bin may contain values greater than binend. C.f. count_bins.\n\nA value can be binned by computing its bin index:\n\ncoords = covectors * value\nbin_ix = 1 .+ floor.(Int64,(coords .- binstart) ./ binwidth)\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.Bond","page":"Library API","title":"Sunny.Bond","text":"Bond(i, j, n)\n\nRepresents a bond between atom indices i and j. n is a vector of three integers specifying unit cell displacement in terms of lattice vectors.\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.Crystal","page":"Library API","title":"Sunny.Crystal","text":"An object describing a crystallographic unit cell and its space group symmetry. Constructors are as follows:\n\nCrystal(filename; symprec=1e-5)\n\nReads the crystal from a .cif file located at the path filename. The optional parameter symprec controls the precision tolerance for spacegroup symmetries.\n\nCrystal(latvecs, positions; types=nothing, symprec=1e-5)\n\nConstructs a crystal from the complete list of atom positions positions, with coordinates (between 0 and 1) in units of lattice vectors latvecs. Spacegroup symmetry information is automatically inferred. The optional parameter types is a list of strings, one for each atom, and can be used to break symmetry-equivalence between atoms.\n\nCrystal(latvecs, positions, spacegroup_number; types=nothing, setting=nothing, symprec=1e-5)\n\nBuilds a crystal by applying symmetry operators for a given international spacegroup number. For certain spacegroups, there are multiple possible unit cell settings; in this case, a warning message will be printed, and a list of crystals will be returned, one for every possible setting. Alternatively, the optional setting string will disambiguate between unit cell conventions.\n\nCurrently, crystals built using only the spacegroup number will be missing some symmetry information. It is generally preferred to build a crystal from a .cif file or from the full specification of the unit cell.\n\nExamples\n\n# Read a Crystal from a .cif file\nCrystal(\"filename.cif\")\n\n# Build an FCC crystal using the primitive unit cell. The spacegroup number\n# 225 is inferred.\nlatvecs = [1 1 0;\n 1 0 1;\n 0 1 1] / 2\npositions = [[0, 0, 0]]\nCrystal(latvecs, positions)\n\n# Build a CsCl crystal (two cubic sublattices). By providing distinct type\n# strings, the spacegroup number 221 is inferred.\nlatvecs = lattice_vectors(1, 1, 1, 90, 90, 90)\npositions = [[0,0,0], [0.5,0.5,0.5]]\ntypes = [\"Na\", \"Cl\"]\ncryst = Crystal(latvecs, positions; types)\n\n# Build a diamond cubic crystal from its spacegroup number 227. This\n# spacegroup has two possible settings (\"1\" or \"2\"), which determine an\n# overall unit cell translation.\nlatvecs = lattice_vectors(1, 1, 1, 90, 90, 90)\npositions = [[1, 1, 1] / 4]\ncryst = Crystal(latvecs, positions, 227; setting=\"1\")\n\nSee also lattice_vectors.\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.FormFactor-Tuple{String}","page":"Library API","title":"Sunny.FormFactor","text":"FormFactor(ion::String; g_lande=2)\n\nThe magnetic form factor for a given magnetic ion and charge state. When passed to an intensity_formula, determines a 𝐪-dependent scaling of the structure factor.\n\nThe parameter ion must be one of the following strings:\n\nAm2, Am3, Am4, Am5, Am6, Am7, Au1, Au2, Au3, Au4, Au5, Ce2, Co0, Co1, Co2, Co3,\nCo4, Cr0, Cr1, Cr2, Cr3, Cr4, Cu0, Cu1, Cu2, Cu3, Cu4, Dy2, Dy3, Er2, Er3, Eu2,\nEu3, Fe0, Fe1, Fe2, Fe3, Fe4, Gd2, Gd3, Hf2, Hf3, Ho2, Ho3, Ir0a, Ir0b, Ir0c,\nIr1a, Ir1b, Ir2, Ir3, Ir4, Ir5, Ir6, Mn0, Mn1, Mn2, Mn3, Mn4, Mo0, Mo1, Nb0,\nNb1, Nd2, Nd3, Ni0, Ni1, Ni2, Ni3, Ni4, Np3, Np4, Np5, Np6, Os0a, Os0b, Os0c,\nOs1a, Os1b, Os2, Os3, Os4, Os5, Os6, Os7, Pd0, Pd1, Pr3, Pt1, Pt2, Pt3, Pt4,\nPt5, Pt6, Pu3, Pu4, Pu5, Pu6, Re0a, Re0b, Re0c, Re1a, Re1b, Re2, Re3, Re4, Re5,\nRe6, Rh0, Rh1, Ru0, Ru1, Sc0, Sc1, Sc2, Sm2, Sm3, Ta2, Ta3, Ta4, Tb2, Tb3, Tc0,\nTc1, Ti0, Ti1, Ti2, Ti3, Tm2, Tm3, U3, U4, U5, V0, V1, V2, V3, V4, W0a, W0b,\nW0c, W1a, W1b, W2c, W3, W4, W5, Y0, Yb2, Yb3, Zr0, Zr1\n\nThe trailing number denotes ionization state. For example, \"Fe0\" denotes a neutral iron atom, while \"Fe2\" denotes Fe²⁺. If multiple electronic configurations are possible, they will be distinguished by a trailing letter (a, b, ...). Omitting this letter will print an informative error,\n\nFormFactor(\"Ir0\")\n\nERROR: Disambiguate form factor according to electronic configuration:\n \"Ir0a\" -- 6s⁰5d⁹\n \"Ir0b\" -- 6s¹5d⁸\n \"Ir0c\" -- 6s²5d⁷\n\nThe form factor is approximated as\n\nF(s) = j_0(s) + frac2-gg j_2(s) s^2,\n\ninvolving the Landé g-factor. The j_l(s) are radial integrals associated with the lth Bessel function of the magnetic dipole, where s = k4π, and k is the magnitude of momentum transfer. \n\nThe radial integrals have been calculated using Hartree-Fock for transition metals, or Dirac-Fock for the rare earths and actinide series [1–3]. Sunny uses approximate fits as a sum of Gaussians,\n\nj_0(s) = A e^-as^2 + B e^-bs^2 + C e^-cs^2 + D e^-ds^2 + E \nj_l(s) = (A e^-as^2 + B e^-bs^2 + C e^-cs^2 + D e^-ds^2 + E) s^2\n\nReferences:\n\nhttps://www.ill.eu/sites/ccsl/ffacts/ffachtml.html\nJ. Brown, The Neutron Data Booklet, 2nd ed., Sec. 2.5 Magnetic Form Factors (2003)\nK. Kobayashi, T. Nagao, M. Ito, Acta Cryst. A, 67 pp 473–480 (2011)\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.ImplicitMidpoint","page":"Library API","title":"Sunny.ImplicitMidpoint","text":"ImplicitMidpoint(Δt::Float64; atol=1e-12) where N\n\nEnergy-conserving spin dynamics. One call to the step! function will advance a System by Δt units of time.\n\nUses the spherical midpoint integration scheme for dipole systems and the Schrödinger midpoint integration scheme for SU(N) spin systems. Both integration schemes are symplectic, and therefore avoid energy drift over long periods of simulation time.\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.Langevin","page":"Library API","title":"Sunny.Langevin","text":"Langevin(Δt::Float64; λ::Float64, kT::Float64)\n\nSpin dynamics with coupling to a Langevin thermostat, which includes damping and noise terms. One call to the step! function will advance a System by Δt units of time.\n\nAssuming ergodicity, the Langevin dynamics will sample from thermal equilibrium for the target temperature kT. The empirical parameter λ determines the strength of the coupling to the thermal bath. In other words, 1/λ is the decorrelation time-scale. If λ = 0, then the spin dynamics coincides with ImplicitMidpoint.\n\nAn alternative approach to sampling is LocalSampler, which may be preferred when the allowed spin values become effective discrete (e.g. Ising spins).\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.LocalSampler","page":"Library API","title":"Sunny.LocalSampler","text":"LocalSampler(; kT, nsweeps=1.0, propose=propose_uniform)\n\nMonte Carlo simulation involving Metropolis updates to individual spins. One call to the step! function will perform nsweeps of MCMC sampling for a provided System. The default value of 1.0 means that step! performs, on average, one trial update per spin.\n\nAssuming ergodicity, the LocalSampler will sample from thermal equilibrium for the target temperature kT. \n\nThe trial spin updates are sampled using the propose function. Built-in options include propose_uniform, propose_flip, and propose_delta. Multiple proposals can be mixed with the macro @mix_proposals.\n\nThe returned object stores fields ΔE and Δs, which represent the cumulative change to the net energy and dipole, respectively.\n\nAn alternative approach to sampling is Langevin, which may be preferred for simulating continuous spins, especially in the presence of long-range dipole-dipole interactions (cf. enable_dipole_dipole!).\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.SampledCorrelations","page":"Library API","title":"Sunny.SampledCorrelations","text":"SampledCorrelations\n\nBasic data type for storing sampled correlation data. A SampleCorrelations is initialized by calling either dynamical_correlations or instant_correlations.\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.SpinInfo","page":"Library API","title":"Sunny.SpinInfo","text":"SpinInfo(atom::Int; S, g=2)\n\nCharacterizes the spin at a given atom index within the crystal unit cell. S is an integer multiple of 1/2 and gives the spin angular momentum in units of ħ. g is the g-factor or tensor, such that an angular momentum dipole s produces a magnetic moment g s in units of the Bohr magneton.\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.SpinWaveTheory","page":"Library API","title":"Sunny.SpinWaveTheory","text":"SpinWaveTheory(sys, energy_ϵ::Float64=1e-8)\n\nConstructs an object to perform linear spin wave theory. Use it with dispersion and dssf functions.\n\nThe optional parameter energy_ϵ adds a small positive shift to the diagonal of the dynamical matrix D to avoid numerical issues with zero-energy quasi-particle modes.\n\n\n\n\n\n","category":"type"},{"location":"library.html#Sunny.System-Tuple{Crystal, Tuple{Int64, Int64, Int64}, Vector{SpinInfo}, Symbol}","page":"Library API","title":"Sunny.System","text":"System(crystal::Crystal, latsize, infos, mode; units=Units.meV, seed::Int)\n\nConstruct a System of spins for a given Crystal symmetry. The latsize parameter determines the number of unit cells in each lattice vector direction. The infos parameter is a list of SpinInfo objects, which determine the magnitude S and g-tensor of each spin.\n\nThe two primary options for mode are :SUN and :dipole. In the former, each spin-S degree of freedom is described as an SU(N) coherent state, i.e. a quantum superposition of N = 2S + 1 levels. This formalism can be useful to capture multipolar spin fluctuations or local entanglement effects. \n\nMode :dipole projects the SU(N) dynamics onto the restricted space of pure dipoles. In practice this means that Sunny will simulate Landau-Lifshitz dynamics, but single-ion anisotropy and biquadratic exchange interactions will be renormalized to improve accuracy. To disable this renormalization, use the mode :dipole_large_S which applies the S classical limit. For details, see the documentation page: Interaction Strength Renormalization.\n\nThe default units system of (meV, Å, tesla) can be overridden by with the units parameter; see Units. \n\nAn optional seed may be provided to achieve reproducible random number generation.\n\nAll spins are initially polarized in the z-direction.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.add_sample!-Tuple{SampledCorrelations, System}","page":"Library API","title":"Sunny.add_sample!","text":"add_sample!(sc::SampledCorrelations, sys::System)\n\nadd_trajectory uses the spin configuration contained in the System to generate a correlation data and accumulate it into sc. For static structure factors, this involves analyzing the spin-spin correlations of the spin configuration provided. For a dynamic structure factor, a trajectory is calculated using the given spin configuration as an initial condition. The spin-spin correlations are then calculating in time and accumulated into sc. \n\nThis function will change the state of sys when calculating dynamical structure factor data. To preserve the initial state of sys, it must be saved separately prior to calling add_sample!. Alternatively, the initial spin configuration may be copied into a new System and this new System can be passed to add_sample!.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.available_energies-Tuple{SampledCorrelations}","page":"Library API","title":"Sunny.available_energies","text":"available_energies(sc::SampledCorrelations; negative_energies=false)\n\nReturn the ω values for the energy index of a SampledCorrelations. By default, only returns values for non-negative energies, which corresponds to the default output of intensities. Set negative_energies to true to retrieve all ω values.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.available_wave_vectors-Tuple{SampledCorrelations}","page":"Library API","title":"Sunny.available_wave_vectors","text":"available_wave_vectors(sc::SampledCorrelations; bzsize=(1,1,1))\n\nReturns all wave vectors for which sc contains exact values. bsize specifies the number of Brillouin zones to be included.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.axes_bincenters-Tuple{Any, Any, Any}","page":"Library API","title":"Sunny.axes_bincenters","text":"axes_bincenters(params::BinningParameters)\n\nReturns tick marks which label the bins of the histogram described by BinningParameters by their bin centers.\n\nThe following alternative syntax can be used to compute bin centers for a single axis:\n\naxes_bincenters(binstart,binend,binwidth)\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.broaden_energy-Tuple{SampledCorrelations, Any, Function}","page":"Library API","title":"Sunny.broaden_energy","text":"broaden_energy(sc::SampledCorrelations, vals, kernel::Function; negative_energies=false)\n\nPerforms a real-space convolution along the energy axis of an array of intensities. Assumes the format of the intensities array corresponds to what would be returned by intensities_interpolated. kernel must be a function that takes two numbers: kernel(ω, ω₀), where ω is a frequency, and ω₀ is the center frequency of the kernel. Sunny provides lorentzian for the most common use case:\n\nnewvals = broaden_energy(sc, vals, (ω, ω₀) -> lorentzian(ω-ω₀, 0.2))\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.count_bins-Tuple{Any, Any, Any}","page":"Library API","title":"Sunny.count_bins","text":"count_bins(binstart,binend,binwidth)\n\nReturns the number of bins in the binning scheme implied by binstart, binend, and binwidth. To count the bins in a BinningParameters, use params.numbins.\n\nThis function defines how partial bins are handled, so it should be used preferentially over computing the number of bins manually.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.dispersion-Tuple{SpinWaveTheory, Any}","page":"Library API","title":"Sunny.dispersion","text":"dispersion(swt::SpinWaveTheory, qs)\n\nComputes the spin excitation energy dispersion relations given a SpinWaveTheory and an array of wave vectors qs. Each element q of qs must be a 3-vector in units of reciprocal lattice units. I.e., qᵢ is given in 2πaᵢ with aᵢ the lattice constant of the original chemical lattice.\n\nThe first indices of the returned array correspond to those of qs. A final index, corresponding to mode, is added to these. Each entry of the array is an energy.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.dmvec-Tuple{Any}","page":"Library API","title":"Sunny.dmvec","text":"dmvec(D)\n\nAntisymmetric matrix representation of the Dzyaloshinskii-Moriya pseudo-vector,\n\n [ 0 D[3] -D[2]\n -D[3] 0 D[1]\n D[2] -D[1] 0 ]\n\nUseful in the context of set_exchange!.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.dssf-Tuple{SpinWaveTheory, Any}","page":"Library API","title":"Sunny.dssf","text":"dssf(swt::SpinWaveTheory, qs)\n\nGiven a SpinWaveTheory object, computes the dynamical spin structure factor,\n\n 𝒮^αβ(𝐤 ω) = 1(2πN)dt _𝐫 expi(ωt - 𝐤𝐫) S^α(𝐫 t)S^β(0 0)\n\nusing the result from linear spin-wave theory,\n\n 𝒮^αβ(𝐤 ω) = _n A_n^αβ(𝐤)^2 δω-ω_n(𝐤)\n\nqs is an array of wave vectors of arbitrary dimension. Each element q of qs must be a 3-vector in reciprocal lattice units (RLU), i.e., in the basis of reciprocal lattice vectors.\n\nThe first indices of the returned array correspond to those of qs. A final index, corresponding to mode, is added to these. Each entry of this array is a tensor (3×3 matrix) corresponding to the indices α and β.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.dynamical_correlations-Union{Tuple{System{N}}, Tuple{N}} where N","page":"Library API","title":"Sunny.dynamical_correlations","text":"dynamical_correlations(sys::System; Δt, nω, ωmax, \n process_trajectory=:none, observables=nothing, correlations=nothing)\n\nCreates an empty SampledCorrelations object for calculating and storing dynamical structure factor intensities 𝒮(𝐪ω). Call add_sample! to accumulate data for the given configuration of a spin system. Internally, this will run a dynamical trajectory and measure time correlations. The 𝒮(𝐪ω) data can be retrieved by calling intensities_interpolated. Alternatively, instant_intensities_interpolated will integrate out ω to obtain 𝒮(𝐪), optionally applying classical-to-quantum correction factors.\n\nThree keywords are required to specify the dynamics used for the trajectory calculation.\n\nΔt: The time step used for calculating the trajectory from which dynamic spin-spin correlations are calculated. The trajectories are calculated with an ImplicitMidpoint integrator.\nωmax: The maximum energy, ω, that will be resolved.\nnω: The number of energy bins to calculated between 0 and ωmax.\n\nAdditional keyword options are the following:\n\nobservables: Allows the user to specify custom observables. The observables must be given as a list of complex N×N matrices or LinearMaps. It's recommended to name each observable, for example: observables = [:A => a_observable_matrix, :B => b_map, ...]. By default, Sunny uses the 3 components of the dipole, :Sx, :Sy and :Sz.\ncorrelations: Specify which correlation functions are calculated, i.e. which matrix elements αβ of 𝒮^αβ(qω) are calculated and stored. Specified with a vector of tuples. By default Sunny records all auto- and cross-correlations generated by all observables. To retain only the xx and xy correlations, one would set correlations=[(:Sx,:Sx), (:Sx,:Sy)] or correlations=[(1,1),(1,2)].\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.eachsite-Tuple{System}","page":"Library API","title":"Sunny.eachsite","text":"eachsite(sys::System)\n\nAn iterator over all Sites in the system. \n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.enable_dipole_dipole!-Union{Tuple{System{N}}, Tuple{N}} where N","page":"Library API","title":"Sunny.enable_dipole_dipole!","text":"enable_dipole_dipole!(sys::System)\n\nEnables long-range dipole-dipole interactions,\n\n -(μ_04π) _ij (3 (𝐌_j𝐫_ij)(𝐌_i𝐫_ij) - 𝐌_i𝐌_j) 𝐫_ij^3\n\nwhere the sum is over all pairs of spins (singly counted), including periodic images, regularized using the Ewald summation convention. The magnetic moments are 𝐌_i = μ_B g 𝐒_i where g is the g-factor or g-tensor, and 𝐒_i is the spin angular momentum dipole in units of ħ. The Bohr magneton μ_B and vacuum permeability μ_0 are physical constants, with numerical values determined by the unit system.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.energy-Union{Tuple{System{N}}, Tuple{N}} where N","page":"Library API","title":"Sunny.energy","text":"energy(sys::System)\n\nThe total system energy. See also energy_per_site.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.energy_per_site-Union{Tuple{System{N}}, Tuple{N}} where N","page":"Library API","title":"Sunny.energy_per_site","text":"energy_per_site(sys::System)\n\nThe total system energy divided by the number of sites.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.generate_mantid_script_from_binning_parameters-Tuple{Any}","page":"Library API","title":"Sunny.generate_mantid_script_from_binning_parameters","text":"generate_mantid_script_from_binning_parameters(params::BinningParameters)\n\nGenerate a Mantid script which bins data according to the given BinningParameters.\n\nwarning: Units\nTake care to ensure the units are correct (R.L.U. or absolute). You may want to call Sunny.bin_rlu_as_absolute_units! or Sunny.bin_absolute_units_as_rlu! first.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.global_position-Tuple{System, Any}","page":"Library API","title":"Sunny.global_position","text":"global_position(sys::System, site::Site)\n\nPosition of a Site in global coordinates.\n\nTo precompute a full list of positions, one can use eachsite as below:\n\npos = [global_position(sys, site) for site in eachsite(sys)]\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.instant_correlations-Tuple{System}","page":"Library API","title":"Sunny.instant_correlations","text":"instant_correlations(sys::System; process_trajectory=:none, observables=nothing, correlations=nothing)\n\nCreates an empty SampledCorrelations object for calculating and storing instantaneous structure factor intensities 𝒮(𝐪). Call add_sample! to accumulate data for the given configuration of a spin system. Call instant_intensities_interpolated to retrieve averaged 𝒮(𝐪) data.\n\nImportant note: When dealing with continuous (non-Ising) spins, consider creating using dynamical_correlations instead of instant_correlations. The former will provide full 𝒮(𝐪ω) data, from which 𝒮(𝐪) can be obtained by integrating out ω. During this integration step, Sunny can incorporate temperature- and ω-dependent classical-to-quantum correction factors to produce more accurate 𝒮(𝐪) estimates. See instant_intensities_interpolated for more information.\n\nThe following optional keywords are available:\n\nobservables: Allows the user to specify custom observables. The observables must be given as a list of complex N×N matrices or LinearMaps. It's recommended to name each observable, for example: observables = [:A => a_observable_matrix, :B => b_map, ...]. By default, Sunny uses the 3 components of the dipole, :Sx, :Sy and :Sz.\ncorrelations: Specify which correlation functions are calculated, i.e. which matrix elements αβ of 𝒮^αβ(qω) are calculated and stored. Specified with a vector of tuples. By default Sunny records all auto- and cross-correlations generated by all observables. To retain only the xx and xy correlations, one would set correlations=[(:Sx,:Sx), (:Sx,:Sy)] or correlations=[(1,1),(1,2)].\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.instant_intensities_interpolated-Tuple{SampledCorrelations, Any, Any}","page":"Library API","title":"Sunny.instant_intensities_interpolated","text":"instant_intensities_interpolated(sc::SampledCorrelations, qs; kwargs...)\n\nReturn 𝒮(𝐪) intensities at wave vectors qs. The functionality is very similar to intensities_interpolated, except the returned array has dimensions identical to qs. If called on a SampledCorrelations with dynamical information, i.e., 𝒮(𝐪ω), the ω information is integrated out.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.integrate_axes!-Tuple{BinningParameters}","page":"Library API","title":"Sunny.integrate_axes!","text":"integrate_axes!(params::BinningParameters; axes)\n\nIntegrate over one or more axes of the histogram by setting the number of bins in that axis to 1. Examples:\n\nintegrate_axes!(params; axes = [2,3])\nintegrate_axes!(params; axes = 2)\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.integrated_lorentzian-Tuple{Float64}","page":"Library API","title":"Sunny.integrated_lorentzian","text":"integrated_lorentzian(η)\n\nReturns x mapsto atan(xη)π for use with intensities_binned.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.intensities_bands-Tuple{SpinWaveTheory, Any, Sunny.SpinWaveIntensityFormula}","page":"Library API","title":"Sunny.intensities_bands","text":"dispersion, intensities = intensities_bands(swt::SpinWaveTheory, ks, formula::SpinWaveIntensityFormula)\n\nComputes the scattering intensities at each energy band for each momentum transfer k in ks, according to Linear Spin Wave Theory and the given intensity formula. The formula must have a delta-function kernel, e.g.:\n\nformula = intensity_formula(swt, :perp, formula; kernel = delta_function_kernel)\n\nor else the bands will be broadened, and their intensity can not be computed.\n\nThe outputs will be arrays with indices identical to ks, with the last index giving the band index. dispersions reports the energy of each band, while intensities reports the scattering intensity.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.intensities_binned-Tuple{SampledCorrelations, BinningParameters, Sunny.ClassicalIntensityFormula}","page":"Library API","title":"Sunny.intensities_binned","text":"intensity, counts = intensities_binned(sc::SampledCorrelations, params::BinningParameters, formula; integrated_kernel)\n\nGiven correlation data contained in a SampledCorrelations and BinningParameters describing the shape of a histogram, compute the intensity and normalization for each histogram bin using a given intensity_formula.\n\nThe BinningParameters are expected to accept (q,ω) in R.L.U. for the (possibly reshaped) crystal associated with sc.\n\nThis is an alternative to intensities_interpolated which bins the scattering intensities into a histogram instead of interpolating between them at specified qs values. See unit_resolution_binning_parameters for a reasonable default choice of BinningParameters which roughly emulates intensities_interpolated with interpolation = :round.\n\nIf a function integrated_kernel(Δω) is passed, it will be used as the CDF of a kernel function for energy broadening. For example, integrated_kernel = Δω -> atan(Δω/η)/pi (c.f. integrated_lorentzian implements Lorentzian broadening with parameter η. Energy-dependent energy broadening can be achieved by providing an integrated_kernel(ω,Δω) whose first argument is the energy transfer ω.\n\nCurrently, energy broadening is only supported if the BinningParameters are such that the first three axes are purely spatial and the last (energy) axis is [0,0,0,1].\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.intensities_broadened-Tuple{SpinWaveTheory, Any, Any, Any}","page":"Library API","title":"Sunny.intensities_broadened","text":"intensities_broadened(swt::SpinWaveTheory, ks, ωvals, formula)\n\nComputes the scattering intensities at each (Q,ω) according to Linear Spin Wave Theory and the given intensity formula. The required formula must have a non-delta-function kernel, e.g.:\n\nformula = intensity_formula(swt, :perp; kernel = lorentzian(0.05))\n\nor else the intensity at ωvals which are not exactly on the dispersion curve can not be calculated.\n\nThe intensity is computed at each wave vector in ks and each energy in ωvals. The output will be an array with indices identical to ks, with the last index matching ωvals.\n\nNote that ks is an array of wave vectors of arbitrary dimension. Each element k of ks must be a 3-wavevector in absolute units.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.intensities_interpolated-Tuple{SampledCorrelations, Any, Sunny.ClassicalIntensityFormula}","page":"Library API","title":"Sunny.intensities_interpolated","text":"intensities_interpolated(sc::SampledCorrelations, qs, formula:ClassicalIntensityFormula; interpolation=nothing, negative_energies=false)\n\nThe basic function for retrieving 𝒮(𝐪ω) information from a SampledCorrelations. Maps an array of wave vectors qs to an array of structure factor intensities, including an additional energy index. The values of ω associated with the energy index can be retrieved by calling available_energies. The three coordinates of each wave vector are measured in reciprocal lattice units, i.e., multiples of the reciprocal lattice vectors.\n\ninterpolation: Since 𝒮(𝐪 ω) is calculated on a finite lattice, data is only available at discrete wave vectors. By default, Sunny will round a requested q to the nearest available wave vector. Linear interpolation can be applied by setting interpolation=:linear.\nnegative_energies: If set to true, Sunny will return the periodic extension of the energy axis. Most users will not want this.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.intensity_formula-Tuple{Function, SampledCorrelations, AbstractVector{Int64}}","page":"Library API","title":"Sunny.intensity_formula","text":"formula = intensity_formula(sc::SampledCorrelations)\n\nEstablish a formula for computing the intensity of the discrete scattering modes (q,ω) using the correlation data 𝒮^αβ(qω) stored in the SampledCorrelations. The formula returned from intensity_formula can be passed to intensities_interpolated or intensities_binned.\n\nintensity_formula(sc,...; kT = Inf, formfactors = ...)\n\nThere are keyword arguments providing temperature and form factor corrections:\n\nkT: If a temperature is provided, the intensities will be rescaled by a temperature- and ω-dependent classical-to-quantum factor. kT should be specified when making comparisons with spin wave calculations or experimental data. If kT is not specified, infinite temperature (no correction) is assumed.\nformfactors: To apply form factor corrections, provide this keyword with a list of FormFactors, one for each symmetry-distinct site in the crystal. The order of FormFactors must correspond to the order of site symmetry classes, e.g., as they appear when printed in display(crystal).\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.intensity_formula-Tuple{Function, SampledCorrelations, Any}","page":"Library API","title":"Sunny.intensity_formula","text":"A custom intensity formula can be specifed by providing a function intensity = f(q,ω,correlations) and specifying which correlations it requires:\n\nintensity_formula(f,sc::SampledCorrelations, required_correlations; kwargs...)\n\nThe function is intended to be specified using do notation. For example, this custom formula sums the off-diagonal correlations:\n\nrequired = [(:Sx,:Sy),(:Sy,:Sz),(:Sx,:Sz)]\nintensity_formula(sc,required,return_type = ComplexF64) do k, ω, off_diagonal_correlations\n sum(off_diagonal_correlations)\nend\n\nIf your custom formula returns a type other than Float64, use the return_type keyword argument to flag this.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.intensity_formula-Tuple{Function, SpinWaveTheory, AbstractVector{Int64}}","page":"Library API","title":"Sunny.intensity_formula","text":"formula = intensity_formula(swt::SpinWaveTheory; kernel = ...)\n\nEstablish a formula for computing the scattering intensity by diagonalizing the hamiltonian H(q) using Linear Spin Wave Theory.\n\nIf kernel = delta_function_kernel, then the resulting formula can be used with intensities_bands.\n\nIf kernel is an energy broadening kernel function, then the resulting formula can be used with intensities_broadened. Energy broadening kernel functions can either be a function of Δω only, e.g.:\n\nkernel = Δω -> ...\n\nor a function of both the energy transfer ω and of Δω, e.g.:\n\nkernel = (ω,Δω) -> ...\n\nThe integral of a properly normalized kernel function over all Δω is one.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.intensity_formula-Tuple{SpinWaveTheory, Symbol}","page":"Library API","title":"Sunny.intensity_formula","text":"intensity_formula([swt or sc], contraction_mode::Symbol)\n\nSunny has several built-in formulas that can be selected by setting contraction_mode to one of these values:\n\n:trace (default), which yields operatornametr 𝒮(qω) = _α 𝒮^αα(qω)\n:perp, which contracts 𝒮^αβ(qω) with the dipole factor δ_αβ - q_αq_β, returning the unpolarized intensity.\n:full, which will return all elements 𝒮^αβ(𝐪ω) without contraction.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.lattice_params-Tuple{StaticArraysCore.SMatrix{3, 3, Float64, 9}}","page":"Library API","title":"Sunny.lattice_params","text":"lattice_params(latvecs::Mat3)\n\nCompute the lattice parameters (a b c α β γ) for the three lattice vectors provided as columns of latvecs. The inverse mapping is lattice_vectors.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.lattice_vectors-NTuple{6, Any}","page":"Library API","title":"Sunny.lattice_vectors","text":"lattice_vectors(a, b, c, α, β, γ)\n\nReturn the lattice vectors, as columns of the 33 output matrix, that correspond to the conventional unit cell defined by the lattice constants (a b c) and the angles (α β γ) in degrees. The inverse mapping is lattice_params.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.load_nxs-Tuple{Any}","page":"Library API","title":"Sunny.load_nxs","text":"params, signal = load_nxs(filename)\n\nGiven the name of a Mantid-exported MDHistoWorkspace file, load the BinningParameters and the signal from that file.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.lorentzian-Tuple{Any, Any}","page":"Library API","title":"Sunny.lorentzian","text":"lorentzian(x, η)\n\nReturns η(π(x^2 + η^2)).\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.magnetic_moment-Tuple{System, Any}","page":"Library API","title":"Sunny.magnetic_moment","text":"magnetic_moment(sys::System, site::Site)\n\nGet the magnetic moment for a Site. This is the spin dipole multiplied by the Bohr magneton and the local g-tensor.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.merge_correlations-Union{Tuple{Array{SampledCorrelations{N}, 1}}, Tuple{N}} where N","page":"Library API","title":"Sunny.merge_correlations","text":"merge_correlations(scs::Vector{SampledCorrelations)\n\nAccumulate a list of SampledCorrelations into a single, summary SampledCorrelations. Useful for reducing the results of parallel computations.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.minimize_energy!-Union{Tuple{System{N}}, Tuple{N}} where N","page":"Library API","title":"Sunny.minimize_energy!","text":"minimize_energy!(sys::System{N}; maxiters=100, subiters=20,\n method=Optim.ConjugateGradient(), kwargs...) where N\n\nOptimizes the spin configuration in sys to minimize energy. A total of maxiters iterations will be attempted, with restarts after every subiters iterations. The remaining kwargs will be forwarded to the optimize method of the Optim.jl package.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.polarize_spins!-Union{Tuple{N}, Tuple{System{N}, Any}} where N","page":"Library API","title":"Sunny.polarize_spins!","text":"polarize_spins!(sys::System, dir)\n\nPolarize all spins in the system along the direction dir.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.position_to_site-Tuple{System, Any}","page":"Library API","title":"Sunny.position_to_site","text":"position_to_site(sys::System, r)\n\nConverts a position r to four indices of a Site. The coordinates of r are given in units of the lattice vectors for the original crystal. This function can be useful for working with systems that have been reshaped using reshape_supercell.\n\nExample\n\n# Find the `site` at the center of a unit cell which is displaced by four\n# multiples of the first lattice vector\nsite = position_to_site(sys, [4.5, 0.5, 0.5])\n\n# Print the dipole at this site\nprintln(sys.dipoles[site])\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.powder_average_binned-Tuple{SampledCorrelations, Any, Sunny.ClassicalIntensityFormula}","page":"Library API","title":"Sunny.powder_average_binned","text":"powder_average_binned(sc::SampledCorrelations, radial_binning_parameters; formula\n ω_binning_parameters, integrated_kernel = nothing, bzsize = nothing)\n\nThis function emulates the experimental situation of \"powder averaging,\" where only the magnitude (and not the direction) of the momentum transfer is resolvable. The intensities are binned similarly to intensities_binned, but the histogram x-axis is |k| in absolute units, which is a nonlinear function of kx,ky,kz. The y-axis is energy.\n\nRadial binning parameters are specified as tuples (start,end,bin_width), e.g. radial_binning_parameters = (0,6π,6π/55).\n\nEnergy broadening is supported in the same way as intensities_binned, and this function accepts the same kind of intensity_formula.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.primitive_cell_shape-Tuple{Crystal}","page":"Library API","title":"Sunny.primitive_cell_shape","text":"primitive_cell_shape(cryst::Crystal)\n\nReturns the shape of the primitive cell as a 3×3 matrix, in fractional coordinates of the conventional lattice vectors. May be useful for constructing inputs to reshape_supercell.\n\nExamples\n\n# Valid if `cryst` has not been reshaped\n@assert cryst.prim_latvecs ≈ cryst.latvecs * primitive_cell_shape(cryst)\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.print_bond-Tuple{Crystal, Bond}","page":"Library API","title":"Sunny.print_bond","text":"print_bond(cryst::Crystal, bond::Bond; b_ref::Bond)\n\nPrints symmetry information for bond bond. A symmetry-equivalent reference bond b_ref can optionally be provided to fix the meaning of the coefficients A, B, ...\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.print_site-Tuple{Any, Any}","page":"Library API","title":"Sunny.print_site","text":"print_site(cryst, i; R=I)\n\nPrint symmetry information for the site i, including allowed g-tensor and allowed anisotropy operator. An optional rotation matrix R can be provided to define the reference frame for expression of the anisotropy.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.print_stevens_expansion-Tuple{AbstractMatrix}","page":"Library API","title":"Sunny.print_stevens_expansion","text":"function print_stevens_expansion(op)\n\nPrints a local Hermitian operator as a linear combination of Stevens operators. The operator op may be a finite-dimensional matrix or an abstract spin polynomial in the large-S limit.\n\nExamples\n\nS = spin_matrices(2)\nprint_stevens_expansion(S[1]^4 + S[2]^4 + S[3]^4)\n# Prints: (1/20)𝒪₄₀ + (1/4)𝒪₄₄ + 102/5\n\nS = spin_matrices(Inf)\nprint_stevens_expansion(S[1]^4 + S[2]^4 + S[3]^4)\n# Prints: (1/20)𝒪₄₀ + (1/4)𝒪₄₄ + (3/5)𝒮⁴\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.print_suggested_frame-Tuple{Crystal, Int64}","page":"Library API","title":"Sunny.print_suggested_frame","text":"print_suggested_frame(cryst, i; digits=4)\n\nPrint a suggested reference frame, as a rotation matrix R, that can be used as input to print_site(). The purpose is to simplify the description of allowed anisotropies.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.print_symmetry_table-Tuple{Crystal, Any}","page":"Library API","title":"Sunny.print_symmetry_table","text":"print_symmetry_table(cryst::Crystal, max_dist)\n\nPrint symmetry information for all equivalence classes of sites and bonds, up to a maximum bond distance of max_dist. Equivalent to calling print_bond(cryst, b) for every bond b in reference_bonds(cryst, max_dist), where Bond(i, i, [0,0,0]) refers to a single site i.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.print_wrapped_intensities-Union{Tuple{System{N}}, Tuple{N}} where N","page":"Library API","title":"Sunny.print_wrapped_intensities","text":"print_wrapped_intensities(sys::System; nmax=10)\n\nFor Bravais lattices: Prints up to nmax wavevectors according to their instantaneous (static) structure factor intensities, listed in descending order. For non-Bravais lattices: Performs the same analysis for each spin sublattice independently; the output weights are naïvely averaged over sublattices, without incorporating phase shift information. This procedure therefore wraps all wavevectors into the first Brillouin zone. Each wavevector coordinate is given between -12 and 12 in reciprocal lattice units (RLU). The output from this function will typically be used as input to suggest_magnetic_supercell.\n\nBecause this function does not incorporate phase information in its averaging over sublattices, the printed weights are not directly comparable with experiment. For that purpose, use instant_correlations instead.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.propose_delta-Tuple{Any}","page":"Library API","title":"Sunny.propose_delta","text":"propose_delta(magnitude)\n\nGenerate a proposal function that adds a Gaussian perturbation to the existing spin state. In :dipole mode, the procedure is to first introduce a random three-vector perturbation 𝐬 = 𝐬 + 𝐬 ξ and then return the properly normalized spin 𝐬 (𝐬𝐬). Each component of the random vector ξ is Gaussian distributed with a standard deviation of magnitude; the latter is dimensionless and typically smaller than one. \n\nIn :SUN mode, the procedure is analogous, but now involving Gaussian perturbations to each of the N complex components of an SU(N) coherent state.\n\nIn the limit of very large magnitude, this function coincides with propose_uniform.\n\nFor use with LocalSampler.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.propose_flip-Union{Tuple{N}, Tuple{System{N}, Any}} where N","page":"Library API","title":"Sunny.propose_flip","text":"propose_flip\n\nFunction to propose pure spin flip updates in the context of a LocalSampler. Dipoles are flipped as 𝐬 -𝐬. SU(N) coherent states are flipped using the time-reversal operator.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.propose_uniform","page":"Library API","title":"Sunny.propose_uniform","text":"propose_uniform\n\nFunction to propose a uniformly random spin update in the context of a LocalSampler. In :dipole mode, the result is a random three-vector with appropriate normalization. In :SUN mode, the result is a random SU(N) coherent state with appropriate normalization.\n\n\n\n\n\n","category":"function"},{"location":"library.html#Sunny.randomize_spins!-Union{Tuple{System{N}}, Tuple{N}} where N","page":"Library API","title":"Sunny.randomize_spins!","text":"randomize_spins!(sys::System)\n\nRandomizes all spins under appropriate the uniform distribution.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.reciprocal_space_path-Tuple{Crystal, Any, Any}","page":"Library API","title":"Sunny.reciprocal_space_path","text":"reciprocal_space_path(cryst::Crystal, qs, density)\n\nReturns a pair (path, xticks). The path return value is a list of wavevectors that samples linearly between the provided wavevectors qs. The xticks return value can be used to label the special 𝐪 values on the x-axis of a plot.\n\nSpecial note about units: the wavevectors qs must be provided in reciprocal lattice units (RLU) for the given crystal, but the sampling density must be specified in the global frame. Specifically, the density is given as number of sample points per unit of radian inverse length, where the unit of length is the same as that used to specify the lattice vectors of the Crystal. The path will therefore include more samples between q-points that are further apart in absolute Fourier distance.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.reciprocal_space_path_bins-Tuple{Any, Any, Any, Vararg{Any}}","page":"Library API","title":"Sunny.reciprocal_space_path_bins","text":"reciprocal_space_path_bins(sc,qs,density,args...;kwargs...)\n\nTakes a list of wave vectors, qs in R.L.U., and builds a series of histogram BinningParameters whose first axis traces a path through the provided points. The second and third axes are integrated over according to the args and kwargs, which are passed through to slice_2D_binning_parameters.\n\nAlso returned is a list of marker indices corresponding to the input points, and a list of ranges giving the indices of each histogram x-axis within a concatenated histogram. The density parameter is given in samples per reciprocal lattice unit (R.L.U.).\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.reciprocal_space_shell-Tuple{Crystal, Any, Any}","page":"Library API","title":"Sunny.reciprocal_space_shell","text":"reciprocal_space_shell(cryst::Crystal, radius, n)\n\nSample n points on the reciprocal space sphere with a given radius (units of inverse length).\n\nExamples\n\n# Sample wavevectors on the sphere at fixed density\nreciprocal_space_shell(cryst, r, 4π*r^2*density)\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.reference_bonds-Tuple{Crystal, Float64}","page":"Library API","title":"Sunny.reference_bonds","text":"reference_bonds(cryst::Crystal, max_dist)\n\nReturns a full list of bonds, one for each symmetry equivalence class, up to distance max_dist. The reference bond b for each equivalence class is selected according to a scoring system that prioritizes simplification of the elements in basis_for_symmetry_allowed_couplings(cryst, b).\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.remove_periodicity!-Union{Tuple{N}, Tuple{System{N}, Any}} where N","page":"Library API","title":"Sunny.remove_periodicity!","text":"remove_periodicity!(sys::System, dims)\n\nRemove periodic interactions along the dimensions where dims is true. The system must support inhomogeneous interactions via to_inhomogeneous.\n\nExample\n\n# Remove periodic boundaries along the 1st and 3rd dimensions\nremove_periodicity!(sys::System, (true, false, true))\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.repeat_periodically-Union{Tuple{N}, Tuple{System{N}, Tuple{Int64, Int64, Int64}}} where N","page":"Library API","title":"Sunny.repeat_periodically","text":"repeat_periodically(sys::System{N}, counts::NTuple{3,Int}) where N\n\nCreates a System identical to sys but repeated a given number of times in each dimension, specified by the tuple counts.\n\nSee also reshape_supercell.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.reshape_supercell-Union{Tuple{N}, Tuple{System{N}, Any}} where N","page":"Library API","title":"Sunny.reshape_supercell","text":"reshape_supercell(sys::System, shape)\n\nMaps an existing System to a new one that has the shape and periodicity of a requested supercell. The columns of the 33 integer matrix shape represent the supercell lattice vectors measured in units of the original crystal lattice vectors.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.resize_supercell-Union{Tuple{N}, Tuple{System{N}, Tuple{Int64, Int64, Int64}}} where N","page":"Library API","title":"Sunny.resize_supercell","text":"resize_supercell(sys::System{N}, latsize::NTuple{3,Int}) where N\n\nCreates a System with a given number of conventional unit cells in each lattice vector direction. Interactions and other settings will be inherited from sys.\n\nConvenience function for:\n\nreshape_supercell(sys, [latsize[1] 0 0; 0 latsize[2] 0; 0 0 latsize[3]])\n\nSee also reshape_supercell.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.rotate_operator-Tuple{LinearAlgebra.Hermitian{ComplexF64, Matrix{ComplexF64}}, Any}","page":"Library API","title":"Sunny.rotate_operator","text":"rotate_operator(A, R)\n\nRotates the local quantum operator A according to the 33 rotation matrix R.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.rotation_in_rlu-Tuple{Crystal, Any, Any}","page":"Library API","title":"Sunny.rotation_in_rlu","text":"rotation_in_rlu(cryst::Crystal, axis, angle)\n\nReturns a 33 matrix that rotates wavevectors in reciprocal lattice units (RLU). The axis vector is a real-space direction in absolute units (but arbitrary magnitude), and the angle is in radians.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_coherent!-Union{Tuple{N}, Tuple{System{N}, Any, Any}} where N","page":"Library API","title":"Sunny.set_coherent!","text":"set_coherent!(sys::System, Z, site::Site)\n\nSet a coherent spin state at a Site using the N complex amplitudes in Z.\n\nFor a standard SpinInfo, these amplitudes will be interpreted in the eigenbasis of 𝒮ᶻ. That is, Z[1] represents the amplitude for the basis state fully polarized along the z-direction, and subsequent components represent states with decreasing angular momentum along this axis (m = S S-1 -S).\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_dipole!-Union{Tuple{N}, Tuple{System{N}, Any, Any}} where N","page":"Library API","title":"Sunny.set_dipole!","text":"set_dipole!(sys::System, dir, site::Site)\n\nPolarize the spin at a Site along the direction dir.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_exchange!-Union{Tuple{N}, Tuple{System{N}, Any, Bond}} where N","page":"Library API","title":"Sunny.set_exchange!","text":"set_exchange!(sys::System, J, bond::Bond)\n\nSets a 3×3 spin-exchange matrix J along bond, yielding a pairwise interaction energy 𝐒_iJ 𝐒_j. This interaction will be propagated to equivalent bonds in consistency with crystal symmetry. Any previous interactions on these bonds will be overwritten. The parameter bond has the form Bond(i, j, offset), where i and j are atom indices within the unit cell, and offset is a displacement in unit cells.\n\nThe parameter J may be scalar or matrix-valued. As a convenience, dmvec(D) can be used to construct the antisymmetric part of the exchange, where D is the Dzyaloshinskii-Moriya pseudo-vector. The resulting interaction will be 𝐃(𝐒_i𝐒_j).\n\nFor more general interactions, such as biquadratic, use set_pair_coupling! instead.\n\nExamples\n\n# An explicit exchange matrix\nJ1 = [2 3 0;\n -3 2 0;\n 0 0 2]\nset_exchange!(sys, J1, bond)\n\n# An equivalent Heisenberg + DM exchange \nJ2 = 2*I + dmvec([0,0,3])\nset_exchange!(sys, J2, bond)\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_exchange_at!-Union{Tuple{N}, Tuple{System{N}, Any, Union{NTuple{4, Int64}, CartesianIndex{4}}, Union{NTuple{4, Int64}, CartesianIndex{4}}}} where N","page":"Library API","title":"Sunny.set_exchange_at!","text":"set_exchange_at!(sys::System, J, site1::Site, site2::Site; offset=nothing)\n\nSets an exchange interaction `𝐒_i⋅J 𝐒_j along the single bond connecting two Sites, ignoring crystal symmetry. Any previous coupling on this bond will be overwritten. The system must support inhomogeneous interactions via to_inhomogeneous.\n\nIf the system is relatively small, the direction of the bond can be ambiguous due to possible periodic wrapping. Resolve this ambiguity by passing an explicit offset vector, in multiples of unit cells.\n\nFor more general interactions, such as biquadratic, use set_pair_coupling_at! instead.\n\nSee also set_exchange!.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_external_field!-Tuple{System, Any}","page":"Library API","title":"Sunny.set_external_field!","text":"set_external_field!(sys::System, B::Vec3)\n\nSets the external field B that couples to all spins.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_external_field_at!-Tuple{System, Any, Any}","page":"Library API","title":"Sunny.set_external_field_at!","text":"set_external_field_at!(sys::System, B::Vec3, site::Site)\n\nSets a Zeeman coupling between a field B and a single spin. Site includes a unit cell and a sublattice index.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_onsite_coupling!-Tuple{System, Any, Int64}","page":"Library API","title":"Sunny.set_onsite_coupling!","text":"set_onsite_coupling!(sys::System, op, i::Int)\n\nSet the single-ion anisotropy for the ith atom of every unit cell, as well as all symmetry-equivalent atoms. The operator op may be provided as an abstract function of the local spin operators, as a polynomial of spin_matrices, or as a linear combination of stevens_matrices.\n\nExamples\n\n# An easy axis anisotropy in the z-direction\nset_onsite_coupling!(sys, S -> -D*S[3]^3, i)\n\n# The unique quartic single-ion anisotropy for a site with cubic point group\n# symmetry\nset_onsite_coupling!(sys, S -> 20*(S[1]^4 + S[2]^4 + S[3]^4), i)\n\n# An equivalent expression of this quartic anisotropy, up to a constant shift\nO = stevens_matrices(spin_label(sys, i))\nset_onsite_coupling!(sys, O[4,0] + 5*O[4,4], i)\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_onsite_coupling_at!-Tuple{System, Any, Union{NTuple{4, Int64}, CartesianIndex{4}}}","page":"Library API","title":"Sunny.set_onsite_coupling_at!","text":"set_onsite_coupling_at!(sys::System, op, site::Site)\n\nSets the single-ion anisotropy operator op for a single Site, ignoring crystal symmetry. The system must support inhomogeneous interactions via to_inhomogeneous.\n\nSee also set_onsite_coupling!.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_pair_coupling!-Union{Tuple{N}, Tuple{System{N}, AbstractMatrix, Any}} where N","page":"Library API","title":"Sunny.set_pair_coupling!","text":"set_pair_coupling!(sys::System, op, bond)\n\nSets an arbitrary coupling op along bond. This coupling will be propagated to equivalent bonds in consistency with crystal symmetry. Any previous interactions on these bonds will be overwritten. The parameter bond has the form Bond(i, j, offset), where i and j are atom indices within the unit cell, and offset is a displacement in unit cells. The operator op may be provided as an anonymous function that accepts two spin dipole operators, or as a matrix that acts in the tensor product space of the two sites.\n\nExamples\n\n# Bilinear+biquadratic exchange involving 3×3 matrices J1 and J2\nset_pair_coupling!(sys, (Si, Sj) -> Si'*J1*Sj + (Si'*J2*Sj)^2, bond)\n\n# Equivalent expression using an appropriate fixed matrix representation\nS = spin_matrices(1/2)\nSi, Sj = to_product_space(S, S)\nset_pair_coupling!(sys, Si'*J1*Sj + (Si'*J2*Sj)^2, bond)\n\nSee also spin_matrices, to_product_space.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_pair_coupling_at!-Union{Tuple{N}, Tuple{System{N}, AbstractMatrix, Union{NTuple{4, Int64}, CartesianIndex{4}}, Union{NTuple{4, Int64}, CartesianIndex{4}}}} where N","page":"Library API","title":"Sunny.set_pair_coupling_at!","text":"set_pair_coupling_at!(sys::System, op, bond)\n\nSets an arbitrary coupling along the single bond connecting two Sites, ignoring crystal symmetry. Any previous coupling on this bond will be overwritten. The operator op may be provided as an anonymous function that accepts two spin dipole operators, or as a matrix that acts in the tensor product space of the two sites. The documentation for set_pair_coupling! provides examples constructing op.\n\nThe system must support inhomogeneous interactions via to_inhomogeneous.\n\nIf the system is relatively small, the direction of the bond can be ambiguous due to possible periodic wrapping. Resolve this ambiguity by passing an explicit offset vector, in multiples of unit cells.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_spiral_order!-Tuple{Any}","page":"Library API","title":"Sunny.set_spiral_order!","text":"set_spiral_order!(sys; q, axis, S0)\n\nInitializes the system with a spiral order described by the wavevector q, an axis of rotation axis, and an initial dipole direction S0 at the real-space origin. The wavevector is expected in repicrocal lattice units (RLU), while the direction vectors axis and S0 are expected in global Cartesian coordinates.\n\nExample\n\n# Spiral order for a wavevector propagating in the direction of the first\n# reciprocal lattice vector (i.e., orthogonal to the lattice vectors ``𝐚_2``\n# and ``𝐚_3``), repeating with a period of 10 lattice constants, and spiraling\n# about the ``ẑ``-axis. The spin at the origin will point in the direction\n# ``𝐒_0 = ŷ + ẑ``. Here, ``(x̂, ŷ, ẑ)`` are the axes of Cartesian coordinate\n# system in the global frame.\nset_spiral_order!(sys; q=[1/10, 0, 0], axis=[0, 0, 1], S0=[0, 1, 1])\n\nSee also set_spiral_order_on_sublattice!.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_spiral_order_on_sublattice!-Tuple{Any, Any}","page":"Library API","title":"Sunny.set_spiral_order_on_sublattice!","text":"set_spiral_order_on_sublattice!(sys, i; q, axis, S0)\n\nInitializes sublattice i with a spiral order described by the wavevector q, an axis of rotation axis, and an initial dipole direction S0. The phase is selected such that the spin at sys.dipole[1,1,1,i] will point in the direction of S0. The wavevector is expected in repicrocal lattice units (RLU), while the direction vectors axis and S0 are expected in global Cartesian coordinates.\n\nThis function is not available for systems with reshaped unit cells.\n\nSee also set_spiral_order!.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.set_vacancy_at!-Union{Tuple{N}, Tuple{System{N}, Any}} where N","page":"Library API","title":"Sunny.set_vacancy_at!","text":"set_vacancy_at!(sys::System, site::Site)\n\nMake a single site nonmagnetic. Site includes a unit cell and a sublattice index.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.slice_2D_binning_parameters-Tuple{Vector{Float64}, Any, Any, Int64, Any}","page":"Library API","title":"Sunny.slice_2D_binning_parameters","text":"slice_2D_binning_parameter(sc::SampledCorrelations, cut_from_q, cut_to_q, cut_bins::Int64, cut_width::Float64; plane_normal = [0,0,1],cut_height = cutwidth)\n\nCreates BinningParameters which make a cut along one dimension of Q-space.\n\nThe x-axis of the resulting histogram consists of cut_bins-many bins ranging from cut_from_q to cut_to_q. The width of the bins in the transverse direciton is controlled by cut_width and cut_height.\n\nThe binning in the transverse directions is defined in the following way, which sets their normalization and orthogonality properties:\n\ncut_covector = normalize(cut_to_q - cut_from_q)\ntransverse_covector = normalize(plane_normal × cut_covector)\ncotransverse_covector = normalize(transverse_covector × cut_covector)\n\nIn other words, the axes are orthonormal with respect to the Euclidean metric.\n\nIf the cut is too narrow, there will be very few scattering vectors per bin, or the number per bin will vary substantially along the cut. If the output appears under-resolved, try increasing cut_width.\n\nThe four axes of the resulting histogram are:\n\nAlong the cut\nFist transverse Q direction\nSecond transverse Q direction\nEnergy\n\nThis function can be used without reference to a SampledCorrelations using this alternate syntax to manually specify the bin centers for the energy axis:\n\nslice_2D_binning_parameter(ω_bincenters, cut_from, cut_to,...)\n\nwhere ω_bincenters specifies the energy axis, and both cut_from and cut_to are arbitrary covectors, in any units.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.spin_label-Tuple{System, Int64}","page":"Library API","title":"Sunny.spin_label","text":"spin_label(sys::System, i::Int)\n\nIf atom i carries a single spin-S moment, then returns the half-integer label S. Otherwise, throws an error.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.spin_matrices-Tuple{Any}","page":"Library API","title":"Sunny.spin_matrices","text":"spin_matrices(S)\n\nReturns a triple of NN spin matrices, where N = 2S+1. These are the generators of SU(2) in the spin-S representation.\n\nIf S == Inf, then the return values are abstract symbols denoting infinite-dimensional matrices that commute. These can be useful for repeating historical studies, or modeling micromagnetic systems. A technical discussion appears in the Sunny documentation page: Interaction Strength Renormalization.\n\nExample\n\nS = spin_matrices(3/2)\n@assert S'*S ≈ (3/2)*(3/2+1)*I\n@assert S[1]*S[2] - S[2]*S[1] ≈ im*S[3]\n\nS = spin_matrices(Inf)\n@assert S[1]*S[2] - S[2]*S[1] == 0\n\nSee also print_stevens_expansion.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.step!","page":"Library API","title":"Sunny.step!","text":"step!(sys::System, dynamics)\n\nAdvance the spin configuration one dynamical time-step. The dynamics object may be a continuous spin dynamics, such as Langevin or ImplicitMidpoint, or it may be a discrete Monte Carlo sampling scheme such as LocalSampler.\n\n\n\n\n\n","category":"function"},{"location":"library.html#Sunny.stevens_matrices-Tuple{Any}","page":"Library API","title":"Sunny.stevens_matrices","text":"stevens_matrices(S)\n\nReturns a generator of Stevens operators in the spin-S representation. The return value O can be indexed as O[k,q], where 0 k 6 labels an irrep of SO(3) and -k q k. This will produce an NN matrix where ``N = 2S\n\n1``. Linear combinations of Stevens operators can be used as a \"physical\n\nbasis\" for decomposing local observables. To see this decomposition, use print_stevens_expansion.\n\nIf S == Inf, then symbolic operators will be returned. In this infinite dimensional limit, the Stevens operators become homogeneous polynomials of commuting spin operators.\n\nExample\n\nO = stevens_matrices(2)\nS = spin_matrices(2)\n\nA = (1/20)O[4,0] + (1/4)O[4,4] + (102/5)I\nB = S[1]^4 + S[2]^4 + S[3]^4\n@assert A ≈ B\n\nSee also spin_matrices and Interaction Strength Renormalization.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.subcrystal-Union{Tuple{N}, Tuple{Crystal, Vararg{String, N}}} where N","page":"Library API","title":"Sunny.subcrystal","text":"subcrystal(cryst, types) :: Crystal\n\nFilters sublattices of a Crystal by atom types, keeping the space group unchanged.\n\nsubcrystal(cryst, classes) :: Crystal\n\nFilters sublattices of Crystal by equivalence classes, keeping the space group unchanged.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.suggest_magnetic_supercell-Tuple{Any}","page":"Library API","title":"Sunny.suggest_magnetic_supercell","text":"suggest_magnetic_supercell(qs; tol=1e-12, maxsize=100)\n\nSuggests a magnetic supercell, in units of the crystal lattice vectors, that is consistent with periodicity of the wavevectors qs in RLU. If the wavevectors are incommensurate (with respect to the maximum supercell size maxsize), one can select a larger error tolerance tol to find a supercell that is almost commensurate.\n\nPrints a 33 matrix of integers that is suitable for use in reshape_supercell.\n\nExamples\n\n# A magnetic supercell for a single-Q structure. Will print\nq1 = [0, -1/4, 1/4]\nsuggest_magnetic_supercell([q1]) # [1 0 0; 0 2 1; 0 -2 1]\n\n# A larger magnetic supercell for a double-Q structure\nq2 = [1/4, 0, 1/4]\nsuggest_magnetic_supercell([q1, q2]) # [1 2 2; -1 2 -2; -1 2 2]\n\n# If given incommensurate wavevectors, find an approximate supercell that\n# is exactly commensurate for nearby wavevectors.\nsuggest_magnetic_supercell([[0, 0, 1/√5], [0, 0, 1/√7]]; tol=1e-2)\n\n# This prints [1 0 0; 0 1 0; 0 0 16], which becomes commensurate under the\n# approximations `1/√5 ≈ 7/16` and `1/√7 ≈ 3/8`.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.symmetry_equivalent_bonds-Tuple{System, Bond}","page":"Library API","title":"Sunny.symmetry_equivalent_bonds","text":"symmetry_equivalent_bonds(sys::System, bond::Bond)\n\nGiven a Bond for the original (unreshaped) crystal, return all symmetry equivalent bonds in the System. Each returned bond is represented as a pair of Sites, which may be used as input to set_exchange_at!. Reverse bonds are not included (no double counting).\n\nExample\n\nfor (site1, site2, offset) in symmetry_equivalent_bonds(sys, bond)\n @assert site1 < site2\n set_exchange_at!(sys, J, site1, site2; offset)\nend\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.to_inhomogeneous-Union{Tuple{System{N}}, Tuple{N}} where N","page":"Library API","title":"Sunny.to_inhomogeneous","text":"to_inhomogeneous(sys::System)\n\nReturns a copy of the system that allows for inhomogeneous interactions, which can be set using set_onsite_coupling_at!, set_exchange_at!, and set_vacancy_at!.\n\nInhomogeneous systems do not support symmetry-propagation of interactions or system reshaping.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.to_product_space-Tuple{Any, Any, Vararg{Any}}","page":"Library API","title":"Sunny.to_product_space","text":"to_product_space(A, B, Cs...)\n\nGiven lists of operators acting on local Hilbert spaces individually, return the corresponding operators that act on the tensor product space. In typical usage, the inputs will represent local physical observables and the outputs will be used to define quantum couplings.\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.unit_resolution_binning_parameters-Tuple{Any, Any, Vararg{Any}}","page":"Library API","title":"Sunny.unit_resolution_binning_parameters","text":"unit_resolution_binning_parameters(sc::SampledCorrelations)\n\nCreate BinningParameters which place one histogram bin centered at each possible (q,ω) scattering vector of the crystal. This is the finest possible binning without creating bins with zero scattering vectors in them.\n\nThis function can be used without reference to a SampledCorrelations using an alternate syntax to manually specify the bin centers for the energy axis and the lattice size:\n\nunit_resolution_binning_parameters(ω_bincenters,latsize,[reciprocal lattice vectors])\n\nThe last argument may be a 3x3 matrix specifying the reciprocal lattice vectors, or a Crystal.\n\nLastly, binning parameters for a single axis may be specifed by their bin centers:\n\n(binstart,binend,binwidth) = unit_resolution_binning_parameters(bincenters::Vector{Float64})\n\n\n\n\n\n","category":"method"},{"location":"library.html#Sunny.@mix_proposals-Tuple","page":"Library API","title":"Sunny.@mix_proposals","text":"@mix_proposals weight1 propose1 weight2 propose2 ...\n\nMacro to generate a proposal function that randomly selects among the provided functions according to the provided probability weights. For use with LocalSampler.\n\nExample\n\n# A proposal function that proposes a spin flip 40% of the time, and a\n# Gaussian perturbation 60% of the time.\n@mix_proposals 0.4 propose_flip 0.6 propose_delta(0.2)\n\n\n\n\n\n","category":"macro"},{"location":"library.html#Optional-Makie-extensions","page":"Library API","title":"Optional Makie extensions","text":"","category":"section"},{"location":"library.html","page":"Library API","title":"Library API","text":"The following will be enabled through a package extension if either GLMakie or WGLMakie is loaded.","category":"page"},{"location":"library.html","page":"Library API","title":"Library API","text":"plot_spins\nview_crystal","category":"page"},{"location":"library.html#Sunny.plot_spins","page":"Library API","title":"Sunny.plot_spins","text":"plot_spins(sys::System; arrowscale=1.0, color=:red, colorfn=nothing,\n colormap=:viridis, colorrange=nothing, show_cell=true, orthographic=false,\n ghost_radius=0, dims=3\n\nPlot the spin configuration defined by sys. Optional parameters are:\n\narrowscale: Scale all arrows by dimensionless factor.\ncolor: Arrow colors. May be symbolic or numeric. If scalar, will be shared among all sites.\ncolorfn: Function that dynamically maps from a site index to a numeric color value. Useful for animations.\ncolormap, colorrange: Used to populate colors from numbers following Makie conventions.\nshow_cell: Show original crystallographic unit cell.\northographic: Use camera with orthographic projection.\nghost_radius: Show translucent periodic images up to a given distance (length units).\ndims: Spatial dimensions of system (1, 2, or 3).\n\nCalling notify on the return value will animate the figure.\n\n\n\n\n\n","category":"function"},{"location":"library.html#Sunny.view_crystal","page":"Library API","title":"Sunny.view_crystal","text":"view_crystal(crystal::Crystal, max_dist::Real; show_axis=true, orthographic=false)\n\nAn interactive crystal viewer, with bonds up to max_dist.\n\n\n\n\n\n","category":"function"},{"location":"library.html#Optional-WriteVTK-extensions","page":"Library API","title":"Optional WriteVTK extensions","text":"","category":"section"},{"location":"library.html","page":"Library API","title":"Library API","text":"The following will be enabled through a package extension if WriteVTK is loaded.","category":"page"},{"location":"library.html","page":"Library API","title":"Library API","text":"export_vtk","category":"page"},{"location":"library.html#Sunny.export_vtk","page":"Library API","title":"Sunny.export_vtk","text":"export_vtk(filename,params::BinningParameters,data)\n\nExport a VTK-compatible file to filename (do not include file extension when specifying the file name) which contains the data as VTK Cell Data on a grid parameterized by params.\n\nAt least one axis of the BinningParameters must be integrated over, since VTK does not support 4D data. See integrate_axes!.\n\n\n\n\n\n","category":"function"},{"location":"examples/contributed/MgCr2O4-tutorial.html#MgCr2O4-at-Finite-Temperature","page":"MgCr2O4 at Finite Temperature","title":"MgCr2O4 at Finite Temperature","text":"","category":"section"},{"location":"examples/contributed/MgCr2O4-tutorial.html","page":"MgCr2O4 at Finite Temperature","title":"MgCr2O4 at Finite Temperature","text":"Author: Martin Mourigal 