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end of file +{"documenter":{"julia_version":"1.9.3","generation_timestamp":"2023-11-01T23:49:32","documenter_version":"1.1.2"}} \ No newline at end of file diff --git a/dev/examples/01_LSWT_SU3_FeI2.html b/dev/examples/01_LSWT_SU3_FeI2.html index 146876656..f102afd02 100644 --- a/dev/examples/01_LSWT_SU3_FeI2.html +++ b/dev/examples/01_LSWT_SU3_FeI2.html @@ -1,5 +1,5 @@ -1. Multi-flavor spin wave simulations of FeI₂ (Showcase) · Sunny documentation

Download this example as Jupyter notebook or Julia script.

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, GLMakie

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

Crystals

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

a = b = 4.05012  # Lattice constants for triangular lattice
+1. Multi-flavor spin wave simulations of FeI₂ (Showcase) · Sunny documentation

Download this example as Jupyter notebook or Julia script.

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, GLMakie

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

Crystals

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

a = b = 4.05012  # Lattice constants for triangular lattice
 c = 6.75214      # Spacing in the z-direction
 
 latvecs = lattice_vectors(a, b, c, 90, 90, 120) # A 3x3 matrix of lattice vectors that
@@ -100,7 +100,7 @@
                     0.0    J′2apm 0.0;
                     0.0    0.0    J′2azz], Bond(1,1,[1,2,1]))

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

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

Calculating structure factor intensities

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

Finding the ground state

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

randomize_spins!(sys)
-minimize_energy!(sys)
50

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

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

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

print_wrapped_intensities(sys)
Dominant wavevectors for spin sublattices:
+minimize_energy!(sys)
49

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

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

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

print_wrapped_intensities(sys)
Dominant wavevectors for spin sublattices:
 
     [-1/4, 1/4, 1/4]       37.33% weight
     [1/4, -1/4, -1/4]      37.33%
@@ -156,4 +156,4 @@
 fig = Figure()
 ax = Axis(fig[1,1]; xlabel="Momentum (r.l.u.)", ylabel="Energy (meV)", xticks, xticklabelrotation=π/6)
 heatmap!(ax, 1:size(is_averaged, 1), energies, is_averaged)
-fig
Example block output

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

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

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

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

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

disp, is = dssf(swt, path);

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

What's next?

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

The full SU(N) coherent state dynamics, with appropriate quantum correction factors, can be useful to model finite temperature scattering data. In particular, it captures certain anharmonic effects due to thermal fluctuations. 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.

+fig
Example block output

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

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

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

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

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

disp, is = dssf(swt, path);

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

What's next?

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

The full SU(N) coherent state dynamics, with appropriate quantum correction factors, can be useful to model finite temperature scattering data. In particular, it captures certain anharmonic effects due to thermal fluctuations. 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.

diff --git a/dev/examples/02_LSWT_CoRh2O4.html b/dev/examples/02_LSWT_CoRh2O4.html index f995c90ba..ff9e3a438 100644 --- a/dev/examples/02_LSWT_CoRh2O4.html +++ b/dev/examples/02_LSWT_CoRh2O4.html @@ -1,5 +1,5 @@ -2. Spin wave simulations of CoRh₂O₄ · Sunny documentation

Download this example as Jupyter notebook or Julia script.

2. Spin wave simulations of CoRh₂O₄

This tutorial illustrates the conventional spin wave theory of dipoles. We consider a simple model of the diamond-cubic crystal CoRh₂O₄, with parameters extracted from Ge et al., Phys. Rev. B 96, 064413.

using Sunny, GLMakie

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

a = 8.5031 # (Å)
+2. Spin wave simulations of CoRh₂O₄ · Sunny documentation

Download this example as Jupyter notebook or Julia script.

2. Spin wave simulations of CoRh₂O₄

This tutorial illustrates the conventional spin wave theory of dipoles. We consider a simple model of the diamond-cubic crystal CoRh₂O₄, with parameters extracted from Ge et al., Phys. Rev. B 96, 064413.

using Sunny, GLMakie

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

a = 8.5031 # (Å)
 latvecs = lattice_vectors(a, a, a, 90, 90, 90)
 cryst = Crystal(latvecs, [[0,0,0]], 227, setting="1")
Crystal
 Spacegroup 'F d -3 m' (227)
@@ -57,4 +57,4 @@
 fig = Figure()
 ax = Axis(fig[1,1]; xlabel="Q (Å⁻¹)", ylabel="ω (meV)")
 heatmap!(ax, radii, energies, output, colormap=:gnuplot2)
-fig
Example block output

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

+fig
Example block output

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

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Landau-Lifshitz dynamics of CoRh₂O₄ at finite T · Sunny documentation

Download this example as Jupyter notebook or Julia script.

3. Landau-Lifshitz dynamics of CoRh₂O₄ at finite T

using Sunny, GLMakie, Statistics

System construction

Construct the system as in the previous CoRh₂O₄ tutorial. After optimization, the system will be in an unfrustrated antiferromagnetic ground state.

a = 8.5031 # (Å)
+3. Landau-Lifshitz dynamics of CoRh₂O₄ at finite T · Sunny documentation

Download this example as Jupyter notebook or Julia script.

3. Landau-Lifshitz dynamics of CoRh₂O₄ at finite T

using Sunny, GLMakie, Statistics

System construction

Construct the system as in the previous CoRh₂O₄ tutorial. After optimization, the system will be in an unfrustrated antiferromagnetic ground state.

a = 8.5031 # (Å)
 latvecs = lattice_vectors(a, a, a, 90, 90, 90)
 cryst = Crystal(latvecs, [[0,0,0]], 227, setting="1")
 latsize = (2, 2, 2)
@@ -8,7 +8,7 @@
 sys = System(cryst, latsize, [SpinInfo(1; S, g=2)], :dipole; seed=0)
 set_exchange!(sys, J, Bond(1, 3, [0,0,0]))
 randomize_spins!(sys)
-minimize_energy!(sys)
28

Use resize_supercell to build a new system with a lattice of 10×10×10 unit cells. The desired Néel order is retained.

sys = resize_supercell(sys, (10, 10, 10))
+minimize_energy!(sys)
27

Use resize_supercell to build a new system with a lattice of 10×10×10 unit cells. The desired Néel order is retained.

sys = resize_supercell(sys, (10, 10, 10))
 @assert energy_per_site(sys) ≈ -2J*S^2

Use the stochastic Landau-Lifshitz dynamics to thermalize system into equilibrium at finite temperature. This is a Langevin equation, which includes damping and noise terms. The strength of the noise term is automatically fixed according to the damping time scale λ and the target temperature, according to a fluctuation-dissipation theorem.

Δt = 0.05/abs(J*S)   # Time step
 λ  = 0.1             # Dimensionless damping time-scale
 kT = 16 * meV_per_K  # 16K, a temperature slightly below ordering
@@ -17,8 +17,8 @@
     step!(sys, langevin)
     push!(energies, energy_per_site(sys))
 end
-plot(energies, color=:blue, figure=(resolution=(600,300),), axis=(xlabel="Time steps", ylabel="Energy (meV)"))
Example block output

Thermal fluctuations are apparent in the spin configuration.

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

Instantaneous structure factor

To visualize the instantaneous (equal-time) structure factor, create an object instant_correlations and use add_sample! to accumulated data for the equilibrated spin configuration.

sc = instant_correlations(sys)
+plot(energies, color=:blue, figure=(resolution=(600,300),), axis=(xlabel="Time steps", ylabel="Energy (meV)"))
Example block output

Thermal fluctuations are apparent in the spin configuration.

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

Instantaneous structure factor

To visualize the instantaneous (equal-time) structure factor, create an object instant_correlations and use add_sample! to accumulated data for the equilibrated spin configuration.

sc = instant_correlations(sys)
 add_sample!(sc, sys)    # Accumulate the newly sampled structure factor into `sf`

Collect 20 additional decorrelated samples. For each sample, about 100 Langevin time-steps is sufficient to collect approximately uncorrelated statistics.

for _ in 1:20
     for _ in 1:100
         step!(sys, langevin)
@@ -35,7 +35,7 @@
         ylabel="Momentum Transfer Qy (r.l.u)", ylabelsize=16,
         aspect=true,
     )
-)
Example block output

Dynamical structure factor

To collect statistics for the dynamical structure factor intensities $I(𝐪,ω)$ at finite temperature, use dynamical_correlations. Now, each call to add_sample! will run a classical spin dynamics trajectory. Longer-time trajectories will be required to achieve greater energy resolution, as controlled by . Here, we pick a moderate number of energies, nω = 50, which will make the simulation run quickly.

ωmax = 6.0  # Maximum  energy to resolve (meV)
+)
Example block output

Dynamical structure factor

To collect statistics for the dynamical structure factor intensities $I(𝐪,ω)$ at finite temperature, use dynamical_correlations. Now, each call to add_sample! will run a classical spin dynamics trajectory. Longer-time trajectories will be required to achieve greater energy resolution, as controlled by . Here, we pick a moderate number of energies, nω = 50, which will make the simulation run quickly.

ωmax = 6.0  # Maximum  energy to resolve (meV)
 nω = 50     # Number of energies to resolve
 sc = dynamical_correlations(sys; Δt, nω, ωmax, process_trajectory=:symmetrize)
SampledCorrelations (616.354 MiB)
 [S(q,ω) | nω = 50, Δω = 0.1333 | 0 sample]
@@ -71,7 +71,7 @@
         xticklabelrotation=π/5,
         aspect = 1.4,
     )
-)
Example block output

Powder averaged intensity

Define spherical shells in reciprocal space via their radii, in absolute units of 1/Å. For each shell, calculate and average the intensities at 100 $𝐪$-points, sampled approximately uniformly.

radii = 0:0.05:3.5 # (1/Å)
+)
Example block output

Powder averaged intensity

Define spherical shells in reciprocal space via their radii, in absolute units of 1/Å. For each shell, calculate and average the intensities at 100 $𝐪$-points, sampled approximately uniformly.

radii = 0:0.05:3.5 # (1/Å)
 output = zeros(Float64, length(radii), length(ωs))
 for (i, radius) in enumerate(radii)
     pts = reciprocal_space_shell(sc.crystal, radius, 100)
@@ -85,4 +85,4 @@
         aspect = 1.4,
     ),
     colorrange = (0, 20.0)
-)
Example block output
+)
Example block output
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z6nB6=GJ2HjrXse3;H9TO=`C;>awARf6+>YJ52W)@5TA`LxXqCvg&8;jy1F)d{!vTz zTod%X{oCL?hh6jId0NUm5*;7EG@j>Pr7e;z4jj#%T^xJe)zF|PNH$7*8Xn%)G1Im9 z><9Opxuyi+rcM0~t~|m54?&PCnL@MK428c(kk%2rRXz6l)eCq$EfwPQoEl>7eY32S zM$8dqWNc9c2P9Xm(|Jh(Ohm?xr?`hinY3Guvj88LjBGm4awI){1-4TniCTy3$4A;I!zkJnK0*jlVs4n1Ti<=T5(8F~tar9V+SZGlo=+P84xjlRby5F^ am$uhj*~6_7bakX19(%&}iFF}~xqks*_W`v4 literal 0 HcmV?d00001 diff --git a/dev/examples/04_GSD_FeI2.html b/dev/examples/04_GSD_FeI2.html index bcea29c36..1c190aa62 100644 --- a/dev/examples/04_GSD_FeI2.html +++ b/dev/examples/04_GSD_FeI2.html @@ -1,5 +1,5 @@ -4. Generalized spin dynamics of FeI₂ at finite T · Sunny documentation

    Download this example as Jupyter notebook or Julia script.

    4. Generalized spin dynamics of FeI₂ at finite T

    using Sunny, LinearAlgebra, GLMakie

    In the previous FeI₂ tutorial, we used multi-flavor spin wave theory to calculate the dynamical structure factor. Here, we perform a similar calculation using classical spin dynamics at finite temperature. Because we are interested in the coupled dynamics of spin dipoles and quadrupoles, we employ a classical dynamics of SU(3) coherent states that generalizes the Landau-Lifshitz equation.

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

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

    As an overview, we will:

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

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

    sys
    System [SU(3)]
    +4. Generalized spin dynamics of FeI₂ at finite T · Sunny documentation

    Download this example as Jupyter notebook or Julia script.

    4. Generalized spin dynamics of FeI₂ at finite T

    using Sunny, LinearAlgebra, GLMakie

    In the previous FeI₂ tutorial, we used multi-flavor spin wave theory to calculate the dynamical structure factor. Here, we perform a similar calculation using classical spin dynamics at finite temperature. Because we are interested in the coupled dynamics of spin dipoles and quadrupoles, we employ a classical dynamics of SU(3) coherent states that generalizes the Landau-Lifshitz equation.

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

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

    As an overview, we will:

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

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

    sys
    System [SU(3)]
     Lattice (4×4×4)×1
     Energy per site -353/250
     

    Finding a ground state

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

    Δt = 0.05/D    # Should be inversely proportional to the largest energy scale
    @@ -13,8 +13,8 @@
     langevin = Langevin(Δt; kT, λ);

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

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

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

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

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

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

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

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

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

    kT = 3.5 * meV_per_K     # 3.5K ≈ 0.30 meV
    +end

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

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

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

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

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

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

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

    kT = 3.5 * meV_per_K     # 3.5K ≈ 0.30 meV
     langevin.kT = kT
     for _ in 1:10_000
         step!(sys_large, langevin)
    @@ -51,7 +51,7 @@
     fig = lines(ωs, is[1,:]; axis=(xlabel="meV", ylabel="Intensity"), label="(0,0,0)")
     lines!(ωs, is[2,:]; label="(π,π,π)")
     axislegend()
    -fig
    Example block output

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

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

    formfactors = [FormFactor("Fe2"; g_lande=3/2)]
    +fig
    Example block output

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

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

    formfactors = [FormFactor("Fe2"; g_lande=3/2)]
     new_formula = intensity_formula(sc, :perp; kT, formfactors = formfactors)
    Classical Scattering Intensity Formula
     At discrete scattering modes S = S[ix_q,ix_ω], use:
     
    @@ -95,7 +95,7 @@
         colorrange=(0.0,0.05),
     )
     
    -fig
    Example block output

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

    Unconventional RLU Systems and Constant Energy Cuts

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

    ωidx = 60
    +fig
    Example block output

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

    Unconventional RLU Systems and Constant Energy Cuts

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

    ωidx = 60
     target_ω = ωs[ωidx]
    3.7312072301840273

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

    params = unit_resolution_binning_parameters(sc)
     params.binstart[1:2] .= -1 # Expand plot range slightly
     
    @@ -124,7 +124,7 @@
     hm = heatmap!(ax,bcs[1],bcs[2],is[:,:,1,1] ./ counts[:,:,1,1])
     add_lines!(ax,params)
     Colorbar(fig[1,2], hm);
    -fig
    Example block output

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



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

    latvecs = sys.crystal.latvecs
    +fig
    Example block output

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



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

    latvecs = sys.crystal.latvecs
     metric = latvecs' * I(3) * latvecs
    3×3 Matrix{Float64}:
      16.4035   -8.20174   0.0
      -8.20174  16.4035    0.0
    @@ -161,7 +161,7 @@
     is = intensities_interpolated(sc, qs, new_formula; interpolation=:linear)
     
     add_lines!(ax_left,params)
    -fig
    Example block output

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

    metric = (latvecs * inv(A'))' * I(3) * (latvecs * inv(A'))
    3×3 Matrix{Float64}:
    +fig
    Example block output

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

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

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

    is_static = instant_intensities_interpolated(sc, qs, new_formula; interpolation = :linear)
    @@ -175,4 +175,4 @@
         )
     )
     Colorbar(hm.figure[1,2], hm.plot)
    -hm
    Example block output
    +hm
    Example block output
    diff --git a/dev/examples/05_MC_Ising.html b/dev/examples/05_MC_Ising.html index 3c5cf840f..f0b270ae8 100644 --- a/dev/examples/05_MC_Ising.html +++ b/dev/examples/05_MC_Ising.html @@ -1,5 +1,5 @@ -5. Monte Carlo sampling of the Ising model · Sunny documentation

    Download this example as Jupyter notebook or Julia script.

    5. Monte Carlo sampling of the Ising model

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

    using Sunny, GLMakie

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

    a = 1
    +5. Monte Carlo sampling of the Ising model · Sunny documentation

    Download this example as Jupyter notebook or Julia script.

    5. Monte Carlo sampling of the Ising model

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

    using Sunny, GLMakie

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

    a = 1
     latvecs = lattice_vectors(a,a,10a,90,90,90)
     crystal = Crystal(latvecs, [[0,0,0]])
    Crystal
     Spacegroup 'P 4/m m m' (123)
    @@ -14,4 +14,4 @@
     sampler = LocalSampler(kT=Tc, propose=propose_flip)
     for i in 1:nsweeps
         step!(sys, sampler)
    -end

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

    heatmap(reshape([s.z for s in sys.dipoles], (L,L)))
    Example block output
    +end

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

    heatmap(reshape([s.z for s in sys.dipoles], (L,L)))
    Example block output
    diff --git a/dev/examples/06_CP2_Skyrmions.html b/dev/examples/06_CP2_Skyrmions.html index 670ac2a10..80bc6ff7d 100644 --- a/dev/examples/06_CP2_Skyrmions.html +++ b/dev/examples/06_CP2_Skyrmions.html @@ -1,5 +1,5 @@ -6. Dynamical quench into CP² skyrmion liquid · Sunny documentation

    Download this example as Jupyter notebook or Julia script.

    6. Dynamical quench into CP² skyrmion liquid

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

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

    using Sunny, GLMakie

    The Hamiltonian we will implement,

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

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

    lat_vecs = lattice_vectors(1.0, 1.0, 2.0, 90, 90, 120)
    +6. Dynamical quench into CP² skyrmion liquid · Sunny documentation

    Download this example as Jupyter notebook or Julia script.

    6. Dynamical quench into CP² skyrmion liquid

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

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

    using Sunny, GLMakie

    The Hamiltonian we will implement,

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

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

    lat_vecs = lattice_vectors(1.0, 1.0, 2.0, 90, 90, 120)
     basis_vecs = [[0,0,0]]
     cryst = Crystal(lat_vecs, basis_vecs)
    Crystal
     Spacegroup 'P 6/m m m' (191)
    @@ -52,4 +52,4 @@
     
     plot_triangular_plaquettes(sun_berry_curvature, frames; resolution=(1800,600),
         offset_spacing=10, texts = ["\tt = "*string(τ) for τ in τs], text_offset = (0.0, 6.0)
    -)
    Example block output

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

    +)
    Example block output

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

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