From 6b31e3867e106ca7d01ffd5e034e0effa0f8569f Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Tue, 3 Sep 2024 20:02:12 +0000 Subject: [PATCH] build based on 3f507a3 --- dev/.documenter-siteinfo.json | 2 +- dev/assets/notebooks/SW10_Energy_cut.ipynb | 2 +- dev/assets/scripts/SW10_Energy_cut.jl | 2 +- dev/examples/01_LSWT_CoRh2O4-34ef3c72.png | Bin 109697 -> 0 bytes dev/examples/01_LSWT_CoRh2O4-51fbb4f9.png | Bin 0 -> 124357 bytes dev/examples/01_LSWT_CoRh2O4-8a494f8a.png | Bin 124334 -> 0 bytes dev/examples/01_LSWT_CoRh2O4-908eac7a.png | Bin 176056 -> 0 bytes dev/examples/01_LSWT_CoRh2O4-ca301888.png | Bin 0 -> 110455 bytes dev/examples/01_LSWT_CoRh2O4-d7cbc960.png | Bin 114206 -> 0 bytes dev/examples/01_LSWT_CoRh2O4-e59a338b.png | Bin 0 -> 107717 bytes dev/examples/01_LSWT_CoRh2O4-ea1db71e.png | Bin 0 -> 175490 bytes dev/examples/01_LSWT_CoRh2O4.html | 8 +- dev/examples/02_LLD_CoRh2O4-0e5115a2.png | Bin 81590 -> 0 bytes dev/examples/02_LLD_CoRh2O4-12081dec.png | Bin 0 -> 44373 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+{"documenter":{"julia_version":"1.10.5","generation_timestamp":"2024-09-03T20:02:04","documenter_version":"1.6.0"}} \ No newline at end of file diff --git a/dev/assets/notebooks/SW10_Energy_cut.ipynb b/dev/assets/notebooks/SW10_Energy_cut.ipynb index 9af1a9b1b..0a7446080 100644 --- a/dev/assets/notebooks/SW10_Energy_cut.ipynb +++ b/dev/assets/notebooks/SW10_Energy_cut.ipynb @@ -116,7 +116,7 @@ "outputs": [], "cell_type": "code", "source": [ - "res = intensities_static(swt, grid; bounds=(3.5, 4.0))\n", + "res = intensities_static(swt, grid; bounds=(3.5, 4.01))\n", "plot_intensities(res; units)" ], "metadata": {}, diff --git a/dev/assets/scripts/SW10_Energy_cut.jl b/dev/assets/scripts/SW10_Energy_cut.jl index dad918e4b..bd69ee673 100644 --- a/dev/assets/scripts/SW10_Energy_cut.jl +++ b/dev/assets/scripts/SW10_Energy_cut.jl @@ -16,5 +16,5 @@ swt = SpinWaveTheory(sys; measure=ssf_perp(sys)) res = intensities(swt, grid; energies=[3.75], kernel=gaussian(fwhm=0.2)) plot_intensities(res; units) -res = intensities_static(swt, grid; bounds=(3.5, 4.0)) +res = intensities_static(swt, grid; bounds=(3.5, 4.01)) plot_intensities(res; units) diff --git a/dev/examples/01_LSWT_CoRh2O4-34ef3c72.png b/dev/examples/01_LSWT_CoRh2O4-34ef3c72.png deleted file mode 100644 index 9ca80a05ea3d4fe3d8e03b9c9095c67af02a06e2..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 109697 zcmeEuWmuNk_V)vVpr9Zkpc0A*D5-QKf{2u~bc0BDhXqPX2+|4yN_VFSNTW!%NJ)3Y zyY2^P&YYP4b-upWb>_p2jt~1@d+)W@FV_9|^1h7tsgoB@qEM()5_d)AQK;kaODq&F zCj95aS?qJj?~HFr-$J2sg7EfqvEla^w({a4sLVF<8Tid5{k!tgD3l8=3gz_-h1!At zZuOVn)E zVscmpv5^f}wuch^Z?rkoqnRZJ@(WsRTJ@w9`$viNH|jY}I%z5z&HF!_Nt?M;oTxYU z)(sV-A&U5t?d$u3`ciJop6q0dYukK@W^QqH?e4m~V7x3u1OMn~-245DhyVKpO6Pxv z04edmL-^kz95sjkWy1e5;s1wBxD%i+ARt~^CPrg4*E^lfih)9rQG`L{{-+nI_#eET zJ)>X$Mmo8T&3J`l{?|+=K(!9>eZX-YJM*ETpmU@`Dso@>Sam|$w?s2XO4sRTW;Q2( zeh#EyfHl06u~4Y$VmIo=g^?63wZi$Xh=6VId$&v%%Gae)cW@Ifqn~@@qht&1(nIk 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output

To search for the ground state, call randomize_spins! and minimize_energy! in sequence. For this problem, optimization converges rapidly to the expected Néel order. See this with plot_spins, where spins are colored according to their global $z$-component.

randomize_spins!(sys)
 minimize_energy!(sys)
-plot_spins(sys; color=[S[3] for S in sys.dipoles])
Example block output

The diamond lattice is bipartite, allowing each spin to perfectly anti-align with its 4 nearest-neighbors. Each of these 4 bonds contribute $-J s^2$ to the total energy. Two sites participate in each bond, so the energy per site is $-2 J s^2$. Check this by calling energy_per_site.

@assert energy_per_site(sys) ≈ -2J*(3/2)^2

Reshaping the magnetic cell

The most compact magnetic cell for this Néel order is a primitive unit cell. Reduce the magnetic cell size using reshape_supercell, where columns of the shape matrix are primitive lattice vectors as multiples of the conventional cubic lattice vectors $(𝐚_1, 𝐚_2, 𝐚_3)$. One could alternatively use shape = cryst.latvecs \ cryst.prim_latvecs. Verify that the energy per site is unchanged after the reshaping the supercell.

shape = [0 1 1;
+plot_spins(sys; color=[S[3] for S in sys.dipoles])
Example block output

The diamond lattice is bipartite, allowing each spin to perfectly anti-align with its 4 nearest-neighbors. Each of these 4 bonds contribute $-J s^2$ to the total energy. Two sites participate in each bond, so the energy per site is $-2 J s^2$. Check this by calling energy_per_site.

@assert energy_per_site(sys) ≈ -2J*(3/2)^2

Reshaping the magnetic cell

The most compact magnetic cell for this Néel order is a primitive unit cell. Reduce the magnetic cell size using reshape_supercell, where columns of the shape matrix are primitive lattice vectors as multiples of the conventional cubic lattice vectors $(𝐚_1, 𝐚_2, 𝐚_3)$. One could alternatively use shape = cryst.latvecs \ cryst.prim_latvecs. Verify that the energy per site is unchanged after the reshaping the supercell.

shape = [0 1 1;
          1 0 1;
          1 1 0] / 2
 sys_prim = reshape_supercell(sys, shape)
-@assert energy_per_site(sys_prim) ≈ -2J*(3/2)^2

Plotting sys_prim shows the two spins within the primitive cell, as well as the larger conventional cubic cell for context.

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

Spin wave theory

With this primitive cell, we will perform a SpinWaveTheory calculation of the structure factor $\mathcal{S}(𝐪,ω)$. The measurement ssf_perp indicates projection of the spin structure factor $\mathcal{S}(𝐪,ω)$ perpendicular to the direction of momentum transfer, as appropriate for unpolarized neutron scattering. The isotropic FormFactor for Co²⁺ dampens intensities at large $𝐪$.

formfactors = [1 => FormFactor("Co2")]
+@assert energy_per_site(sys_prim) ≈ -2J*(3/2)^2

Plotting sys_prim shows the two spins within the primitive cell, as well as the larger conventional cubic cell for context.

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

Spin wave theory

With this primitive cell, we will perform a SpinWaveTheory calculation of the structure factor $\mathcal{S}(𝐪,ω)$. The measurement ssf_perp indicates projection of the spin structure factor $\mathcal{S}(𝐪,ω)$ perpendicular to the direction of momentum transfer, as appropriate for unpolarized neutron scattering. The isotropic FormFactor for Co²⁺ dampens intensities at large $𝐪$.

formfactors = [1 => FormFactor("Co2")]
 measure = ssf_perp(sys_prim; formfactors)
 swt = SpinWaveTheory(sys_prim; measure)
SpinWaveTheory [Dipole mode]
   2 atoms
@@ -37,8 +37,8 @@
   [0, 0, 0] → [1/2, 0, 0] → [1/2, 1/2, 0] → [0, 0, 0]
 

Calculate single-crystal scattering intensities along this path, for energies between 0 and 6 meV. Use plot_intensities to visualize the result.

energies = range(0, 6, 300)
 res = intensities(swt, path; energies, kernel)
-plot_intensities(res; units, title="CoRh₂O₄ LSWT")
Example block output

Sometimes experimental data is only available as a powder average, i.e., as an average over all possible crystal orientations. Use powder_average to simulate these intensities. Each $𝐪$-magnitude defines a spherical shell in reciprocal space. Consider 200 radii from 0 to 3 inverse angstroms, and collect 2000 random samples per spherical shell. As configured, this calculation completes in about two seconds. Had we used the conventional cubic cell, the calculation would be an order of magnitude slower.

radii = range(0, 3, 200) # (1/Å)
+plot_intensities(res; units, title="CoRh₂O₄ LSWT")
Example block output

Sometimes experimental data is only available as a powder average, i.e., as an average over all possible crystal orientations. Use powder_average to simulate these intensities. Each $𝐪$-magnitude defines a spherical shell in reciprocal space. Consider 200 radii from 0 to 3 inverse angstroms, and collect 2000 random samples per spherical shell. As configured, this calculation completes in about two seconds. Had we used the conventional cubic cell, the calculation would be an order of magnitude slower.

radii = range(0, 3, 200) # (1/Å)
 res = powder_average(cryst, radii, 2000) do qs
     intensities(swt, qs; energies, kernel)
 end
-plot_intensities(res; units, saturation=1.0, title="CoRh₂O₄ Powder Average")
Example block output

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

What's next?

  • For more spin wave calculations of this type, browse the SpinW tutorials ported to Sunny.
  • Spin wave theory neglects thermal fluctuations of the magnetic order. The next CoRh₂O₄ tutorial demonstrates how to sample spins in thermal equilibrium, and measure dynamical correlations from the classical spin dynamics.
  • Sunny also offers features that go beyond the dipole approximation of a quantum spin via the theory of SU(N) coherent states. This can be especially useful for systems with strong single-ion anisotropy, as demonstrated in the FeI₂ tutorial.
+plot_intensities(res; units, saturation=1.0, title="CoRh₂O₄ Powder Average")Example block output

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

What's next?

  • For more spin wave calculations of this type, browse the SpinW tutorials ported to Sunny.
  • Spin wave theory neglects thermal fluctuations of the magnetic order. The next CoRh₂O₄ tutorial demonstrates how to sample spins in thermal equilibrium, and measure dynamical correlations from the classical spin dynamics.
  • Sunny also offers features that go beyond the dipole approximation of a quantum spin via the theory of SU(N) coherent states. This can be especially useful for systems with strong single-ion anisotropy, as demonstrated in the FeI₂ tutorial.
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Use repeat_periodically to extend the system to 10×10×10 chemical unit cells. The ground state Néel order is retained. Increasing the system size further would reduce finite-size artifacts and increase momentum-space resolution, but would also make the simulations slower.

sys = repeat_periodically(sys, (10, 10, 10))
-plot_spins(sys; color=[S[3] for S in sys.dipoles])
Example block output

Langevin dynamics for sampling

We will be using a Langevin spin dynamics to thermalize the system. This dynamics is a variant of the Landau-Lifshitz equation that incorporates noise and dissipation terms, which are linked by a fluctuation-dissipation theorem. The temperature 6 K ≈ 1.38 meV is slightly above ordering for this model. The dimensionless damping magnitude sets a timescale for coupling to the implicit thermal bath; 0.2 is usually a good choice.

langevin = Langevin(; damping=0.2, kT=16*units.K)
Langevin(<missing>; damping=0.2, kT=1.3787733219432283)
+plot_spins(sys; color=[S[3] for S in sys.dipoles])
Example block output

Use repeat_periodically to extend the system to 10×10×10 chemical unit cells. The ground state Néel order is retained. Increasing the system size further would reduce finite-size artifacts and increase momentum-space resolution, but would also make the simulations slower.

sys = repeat_periodically(sys, (10, 10, 10))
+plot_spins(sys; color=[S[3] for S in sys.dipoles])
Example block output

Langevin dynamics for sampling

We will be using a Langevin spin dynamics to thermalize the system. This dynamics is a variant of the Landau-Lifshitz equation that incorporates noise and dissipation terms, which are linked by a fluctuation-dissipation theorem. The temperature 6 K ≈ 1.38 meV is slightly above ordering for this model. The dimensionless damping magnitude sets a timescale for coupling to the implicit thermal bath; 0.2 is usually a good choice.

langevin = Langevin(; damping=0.2, kT=16*units.K)
Langevin(<missing>; damping=0.2, kT=1.3787733219432283)
 

Use suggest_timestep to select an integration timestep. A dimensionless error tolerance of 1e-2 is usually a good choice. The suggested timestep will vary according to the magnetic configuration. It is reasonable to start from an energy-minimized configuration.

suggest_timestep(sys, langevin; tol=1e-2)
 langevin.dt = 0.025;
Consider dt ≈ 0.02528 for this spin configuration at tol = 0.01.

Now run a Langevin trajectory to sample spin configurations. Keep track of the energy per site at each time step.

energies = [energy_per_site(sys)]
 for _ in 1:1000
     step!(sys, langevin)
     push!(energies, energy_per_site(sys))
 end

From the relaxed spin configuration, we can learn that dt was a little smaller than necessary; increasing it will make the remaining simulations faster.

suggest_timestep(sys, langevin; tol=1e-2)
-langevin.dt = 0.042;
Consider dt ≈ 0.04198 for this spin configuration at tol = 0.01. Current value is dt = 0.025.

Plot energy versus time using the Makie lines function. The plateau suggests that the system has reached thermal equilibrium.

lines(energies, color=:blue, figure=(size=(600,300),), axis=(xlabel="Timesteps", ylabel="Energy (meV)"))
Example block output

Plot the spins colored by their alignment with a reference spin at the origin. The field sys.dipoles is a 4D array storing the spin dipole data. The first three indices of label the chemical cell, while the fourth index labels an atom within the cell. Note that Julia arrays use 1-based indexing. Thermal fluctuations are apparent in the plot.

S0 = sys.dipoles[1,1,1,1]
-plot_spins(sys; color=[S'*S0 for S in sys.dipoles])
Example block output

Static structure factor

Use SampledCorrelationsStatic to estimate spatial correlations for configurations in classical thermal equilibrium. Each call to add_sample! will accumulate data for the current spin snapshot.

formfactors = [1 => FormFactor("Co2")]
+langevin.dt = 0.042;
Consider dt ≈ 0.04164 for this spin configuration at tol = 0.01. Current value is dt = 0.025.

Plot energy versus time using the Makie lines function. The plateau suggests that the system has reached thermal equilibrium.

lines(energies, color=:blue, figure=(size=(600,300),), axis=(xlabel="Timesteps", ylabel="Energy (meV)"))
Example block output

Plot the spins colored by their alignment with a reference spin at the origin. The field sys.dipoles is a 4D array storing the spin dipole data. The first three indices of label the chemical cell, while the fourth index labels an atom within the cell. Note that Julia arrays use 1-based indexing. Thermal fluctuations are apparent in the plot.

S0 = sys.dipoles[1,1,1,1]
+plot_spins(sys; color=[S'*S0 for S in sys.dipoles])
Example block output

Static structure factor

Use SampledCorrelationsStatic to estimate spatial correlations for configurations in classical thermal equilibrium. Each call to add_sample! will accumulate data for the current spin snapshot.

formfactors = [1 => FormFactor("Co2")]
 measure = ssf_perp(sys; formfactors)
 sc = SampledCorrelationsStatic(sys; measure)
 add_sample!(sc, sys)    # Accumulate the newly sampled structure factor into `sf`

Collect 20 additional samples. Perform 100 Langevin time-steps between measurements to approximately decorrelate the sample in thermal equilibrium.

for _ in 1:20
@@ -29,7 +29,7 @@
     end
     add_sample!(sc, sys)
 end

Use q_space_grid to define a slice of momentum space $[H, K, 0]$, where $H$ and $K$ each range from -10 to 10 in RLU. This command produces a 200×200 grid of sample points.

grid = q_space_grid(cryst, [1, 0, 0], range(-10, 10, 200), [0, 1, 0], (-10, 10))
Sunny.QGrid{2} (200×200 samples)

Calculate and plot the instantaneous structure factor on the slice by integrating over all energy values ω. We employ the appropriate FormFactor for Co2⁺. Selecting saturation = 1.0 sets the color saturation point to the maximum intensity value. This is reasonable because we are above the ordering temperature, and do not have sharp Bragg peaks.

res = intensities_static(sc, grid)
-plot_intensities(res; saturation=1.0, title="Static Intensities at T = 16 K")
Example block output

Dynamical structure factor

To collect statistics for the dynamical structure factor intensities $\mathcal{S}(𝐪,ω)$ at finite temperature, use SampledCorrelations. It requires a range of energies to resolve, which will be associated with frequencies of the classical spin dynamics. The integration timestep dt can be somewhat larger than that used by the Langevin dynamics.

dt = 2*langevin.dt
+plot_intensities(res; saturation=1.0, title="Static Intensities at T = 16 K")
Example block output

Dynamical structure factor

To collect statistics for the dynamical structure factor intensities $\mathcal{S}(𝐪,ω)$ at finite temperature, use SampledCorrelations. It requires a range of energies to resolve, which will be associated with frequencies of the classical spin dynamics. The integration timestep dt can be somewhat larger than that used by the Langevin dynamics.

dt = 2*langevin.dt
 energies = range(0, 6, 50)
 sc = SampledCorrelations(sys; dt, energies, measure=ssf_perp(sys))
SampledCorrelations (612.180 MiB)
 [S(q,ω) | nω = 50, Δω = 0.1224 | 0 sample]
@@ -50,8 +50,8 @@
 path = q_space_path(cryst, qs, 500)
QPath (500 samples)
   [3/4, 3/4, 0] → [0, 0, 0] → [0, 1/2, 1/2] → [1/2, 1, 0] → [0, 1, 0] → [1/4, 1, 1/4] → [0, 1, 0] → [0, -4, 0]
 

Calculate and plot the intensities along this path.

res = intensities(sc, path; energies, langevin.kT)
-plot_intensities(res; units, title="Intensities at 16 K")
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 350 $𝐪$-points

radii = range(0, 3.5, 200) # (1/Å)
+plot_intensities(res; units, title="Intensities at 16 K")
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 350 $𝐪$-points

radii = range(0, 3.5, 200) # (1/Å)
 res = powder_average(cryst, radii, 350) do qs
     intensities(sc, qs; energies, langevin.kT)
 end
-plot_intensities(res; units, title="Powder Average at 16 K")
Example block output +plot_intensities(res; units, title="Powder Average at 16 K")Example block output diff --git a/dev/examples/03_LSWT_SU3_FeI2.html b/dev/examples/03_LSWT_SU3_FeI2.html index a8abf193d..9f72db6fa 100644 --- a/dev/examples/03_LSWT_SU3_FeI2.html +++ b/dev/examples/03_LSWT_SU3_FeI2.html @@ -127,4 +127,4 @@ res = domain_average(cryst, path; rotations, weights) do path_rotated intensities(swt, path_rotated; energies, kernel) end -plot_intensities(res; units, colormap=:viridis, title="Domain Averaged Intensities")Example block output

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

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

To get this agreement, the theory of SU(3) coherent states is essential. The lower band has large quadrupolar character, and arises from the strong easy-axis anisotropy of FeI₂.

An interesting exercise is to repeat the same study, but using :dipole mode instead of :SUN. That alternative choice would constrain the coherent state dynamics to the space of dipoles only, and the flat band of single-ion bound states would be missing.

+plot_intensities(res; units, colormap=:viridis, title="Domain Averaged Intensities")Example block output

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

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

To get this agreement, the theory of SU(3) coherent states is essential. The lower band has large quadrupolar character, and arises from the strong easy-axis anisotropy of FeI₂.

An interesting exercise is to repeat the same study, but using :dipole mode instead of :SUN. That alternative choice would constrain the coherent state dynamics to the space of dipoles only, and the flat band of single-ion bound states would be missing.

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Use suggest_timestep to select an integration timestep for the error tolerance tol=1e-2. Initializing sys to some low-energy configuration usually works well.

randomize_spins!(sys)
 minimize_energy!(sys; maxiters=10)
 suggest_timestep(sys, langevin; tol=1e-2)
-langevin.dt = 0.03;
Consider dt ≈ 0.03215 for this spin configuration at tol = 0.01.

Run a Langevin trajectory for 10,000 time-steps and plot the spins. The magnetic order is present, but may be difficult to see.

for _ in 1:10_000
+langevin.dt = 0.03;
Consider dt ≈ 0.03062 for this spin configuration at tol = 0.01.

Run a Langevin trajectory for 10,000 time-steps and plot the spins. The magnetic order is present, but may be difficult to see.

for _ in 1:10_000
     step!(sys, langevin)
 end
-plot_spins(sys; color=[S[3] for S in sys.dipoles])
Example block output

Verify the expected two-up, two-down spiral magnetic order by calling print_wrapped_intensities. A single propagation wavevector $±𝐤$ dominates the static intensity in $\mathcal{S}(𝐪)$, indicating the expected 2 up, 2 down magnetic spiral order. A smaller amount of intensity is spread among many other wavevectors due to thermal fluctuations.

print_wrapped_intensities(sys)
Dominant wavevectors for spin sublattices:
+plot_spins(sys; color=[S[3] for S in sys.dipoles])
Example block output

Verify the expected two-up, two-down spiral magnetic order by calling print_wrapped_intensities. A single propagation wavevector $±𝐤$ dominates the static intensity in $\mathcal{S}(𝐪)$, indicating the expected 2 up, 2 down magnetic spiral order. A smaller amount of intensity is spread among many other wavevectors due to thermal fluctuations.

print_wrapped_intensities(sys)
Dominant wavevectors for spin sublattices:
 
-    [0, -1/4, 1/4]         38.46% weight
-    [0, 1/4, -1/4]         38.46%
-    [-1/4, 1/4, 1/4]        0.09%
-    [1/4, -1/4, -1/4]       0.09%
-    [3/8, 0.3125, 1/4]      0.08%
-    [-3/8, -0.3125, -1/4]    0.08%
-    [1/2, 0.4375, 1/4]      0.07%
-    [1/2, -0.4375, -1/4]    0.07%
-    [-1/8, 0.0625, 1/2]     0.07%
-    [1/8, -0.0625, 1/2]     0.07%
-    [-0.1875, 3/8, 1/2]     0.07%
-    ...                     ...

Thermalization has not substantially altered the suggested dt.

suggest_timestep(sys, langevin; tol=1e-2)
Consider dt ≈ 0.03174 for this spin configuration at tol = 0.01. Current value is dt = 0.03.

Structure factor in the paramagnetic phase

The remainder of this tutorial will focus on the paramagnetic phase. Re-thermalize the system to the temperature of 3.5 K ≈ 0.30 meV.

langevin.kT = 3.5 * units.K
+    [0, -1/4, 1/4]         29.12% weight
+    [0, 1/4, -1/4]         29.12%
+    [1/4, 0, 1/4]           0.90%
+    [-1/4, 0, -1/4]         0.90%
+    [-0.0625, -0.1875, 1/4]    0.66%
+    [0.0625, 0.1875, -1/4]    0.66%
+    [0.1875, 0.0625, 1/4]    0.60%
+    [-0.1875, -0.0625, -1/4]    0.60%
+    [0.3125, -0.0625, 1/4]    0.56%
+    [-0.3125, 0.0625, -1/4]    0.56%
+    [0.0625, -0.3125, 1/4]    0.51%
+    ...                     ...

Thermalization has not substantially altered the suggested dt.

suggest_timestep(sys, langevin; tol=1e-2)
Consider dt ≈ 0.03274 for this spin configuration at tol = 0.01. Current value is dt = 0.03.

Structure factor in the paramagnetic phase

The remainder of this tutorial will focus on the paramagnetic phase. Re-thermalize the system to the temperature of 3.5 K ≈ 0.30 meV.

langevin.kT = 3.5 * units.K
 for _ in 1:10_000
     step!(sys, langevin)
 end

The suggested timestep has increased slightly. Following this suggestion will make the simulations a bit faster.

suggest_timestep(sys, langevin; tol=1e-2)
-langevin.dt = 0.040;
Consider dt ≈ 0.0399 for this spin configuration at tol = 0.01. Current value is dt = 0.03.

Collect dynamical spin structure factor data using SampledCorrelations. This procedure involves sampling spin configurations from thermal equilibrium and using the spin dynamics of SU(N) coherent states to estimate dynamical correlations. With proper classical-to-quantum corrections, the associated structure factor intensities $S^{αβ}(q,ω)$ can be compared with finite-temperature inelastic neutron scattering data. Incorporate the FormFactor appropriate to Fe²⁺.

dt = 2*langevin.dt
+langevin.dt = 0.040;
Consider dt ≈ 0.04039 for this spin configuration at tol = 0.01. Current value is dt = 0.03.

Collect dynamical spin structure factor data using SampledCorrelations. This procedure involves sampling spin configurations from thermal equilibrium and using the spin dynamics of SU(N) coherent states to estimate dynamical correlations. With proper classical-to-quantum corrections, the associated structure factor intensities $S^{αβ}(q,ω)$ can be compared with finite-temperature inelastic neutron scattering data. Incorporate the FormFactor appropriate to Fe²⁺.

dt = 2*langevin.dt
 energies = range(0, 7.5, 120)
 formfactors = [1 => FormFactor("Fe2"; g_lande=3/2)]
 sc = SampledCorrelations(sys; dt, energies, measure=ssf_perp(sys; formfactors))
SampledCorrelations (37.824 MiB)
@@ -74,7 +74,7 @@
 fig = lines(res.energies, res.data[:, 1]; axis=(xlabel="meV", ylabel="Intensity"), label="(0,0,0)")
 lines!(res.energies, res.data[:, 2]; label="(π,π,π)")
 axislegend()
-fig
Example block output

Next, we will measure intensities along a q_space_path that connects high symmetry points. Because this is a real-space calculation, data is only available for discrete $𝐪$ modes, with resolution that scales inversely to linear system size. Intensities at $ω = 0$ dominate, so to enhance visibility, we restrict the color range empirically.

qs = [[0,   0, 0],  # List of wave vectors that define a path
+fig
Example block output

Next, we will measure intensities along a q_space_path that connects high symmetry points. Because this is a real-space calculation, data is only available for discrete $𝐪$ modes, with resolution that scales inversely to linear system size. Intensities at $ω = 0$ dominate, so to enhance visibility, we restrict the color range empirically.

qs = [[0,   0, 0],  # List of wave vectors that define a path
       [1,   0, 0],
       [0,   1, 0],
       [1/2, 0, 0],
@@ -82,6 +82,6 @@
       [0,   0, 0]]
 qpath = q_space_path(cryst, qs, 500)
 res = intensities(sc, qpath; energies, langevin.kT)
-plot_intensities(res; colorrange=(0.0, 1.0), title="Intensities at T = 2.3 K")
Example block output

One can also view the intensity along a q_space_grid for a fixed energy value. Alternatively, use intensities_static to integrate over all available energies.

grid = q_space_grid(cryst, [1, 0, 0], range(-1.5, 1.5, 300), [0, 1, 0], (-1.5, 1.5); orthogonalize=true)
+plot_intensities(res; colorrange=(0.0, 1.0), title="Intensities at T = 2.3 K")
Example block output

One can also view the intensity along a q_space_grid for a fixed energy value. Alternatively, use intensities_static to integrate over all available energies.

grid = q_space_grid(cryst, [1, 0, 0], range(-1.5, 1.5, 300), [0, 1, 0], (-1.5, 1.5); orthogonalize=true)
 res = intensities(sc, grid; energies=[3.5], langevin.kT)
-plot_intensities(res; title="Intensity slice at ω = 3.5 meV")
Example block output +plot_intensities(res; title="Intensity slice at ω = 3.5 meV")Example block output diff --git a/dev/examples/05_MC_Ising.html b/dev/examples/05_MC_Ising.html index 5bc6aa182..ac167dd18 100644 --- a/dev/examples/05_MC_Ising.html +++ b/dev/examples/05_MC_Ising.html @@ -13,4 +13,4 @@ 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[3] 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[3] for S in sys.dipoles], (L, L)))
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a/dev/examples/06_CP2_Skyrmions.html +++ b/dev/examples/06_CP2_Skyrmions.html @@ -31,7 +31,7 @@ set_onsite_coupling!(sys, S -> D*S[3]^2, 1)

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

damping = 0.05
 kT = 0;

The first step is to estimate a reasonable integration timestep dt. In this case, a random spin configuration gives a tighter bound than does an energy-minimized configuration. A relatively large error tolerance is OK for this phenomenological study.

randomize_spins!(sys)
 integrator = Langevin(; damping, kT)
-suggest_timestep(sys, integrator; tol=0.05)
Consider dt ≈ 0.01105 for this spin configuration at tol = 0.05.

Apply the suggested timestep.

integrator.dt = 0.01;

Now run the dynamical quench starting from a randomized configuration. The field frames stores the system spin configuration, as SU(3) coherent states, at three different times during the quenching process.

randomize_spins!(sys)
+suggest_timestep(sys, integrator; tol=0.05)
Consider dt ≈ 0.01107 for this spin configuration at tol = 0.05.

Apply the suggested timestep.

integrator.dt = 0.01;

Now run the dynamical quench starting from a randomized configuration. The field frames stores the system spin configuration, as SU(3) coherent states, at three different times during the quenching process.

randomize_spins!(sys)
 τs = [4, 16, 256]   # Times to record snapshots
 frames = []         # Empty array to store snapshots
 for i in eachindex(τs)
@@ -53,4 +53,4 @@
 
 plot_triangular_plaquettes(sun_berry_curvature, frames; size=(600, 200),
     offset_spacing=10, texts=["\tt = "*string(τ) for τ in τs], text_offset=(0, 6)
-)
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. At late times, there are well-formed skyrmions of positive (red) and negative (blue) charge, and various 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. At late times, there are well-formed skyrmions of positive (red) and negative (blue) charge, and various metastable spin configurations. A full-sized version of this figure is available in Dahlbom et al..

diff --git a/dev/examples/07_Dipole_Dipole.html b/dev/examples/07_Dipole_Dipole.html index 69c2d256b..de6d64833 100644 --- a/dev/examples/07_Dipole_Dipole.html +++ b/dev/examples/07_Dipole_Dipole.html @@ -35,4 +35,4 @@ for c in eachrow(res3.disp) lines!(ax, eachindex(c), c; linestyle=:dash, color=:black) end -figExample block output +figExample block output diff --git a/dev/examples/08_Momentum_Conventions-2c8ec3cd.png b/dev/examples/08_Momentum_Conventions-2c8ec3cd.png deleted file mode 100644 index a52dfee78c5b801057adbdd54226b73f93116bdb..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 110197 zcmeFZX&_Z^_cy-HlBrS-$xsp{5eb=3nGh10WjG;9N~z3K6GbUWWFA5?mLwrV zIAw^Eqzunz>wDk#Z}`9Xzj|Ihy{K@`KG(jkwbo}?YrAmpz#a}Zel~(2IJEcfJWLSG z_%FQ?D?R?%Bv_b-e;DmGbTtSfCu-@e6%)Rub3eRClgMro_<}ElZT244B?vz;f}n;H z#BY3y`k5d+r3qr#k|31g2!hu&sqm08UP@^1)G+X&8?9JkaAfeK+Slfi=|Dld>&EKp z%QV~wj$PFfZi4!oHtTWLKdsPTllS(`)(q!2ueR>rw0X6pZ1^il6684!On)x? 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References

[1] - H. Zhang, C. D. Batista, "Classical spin dynamics based on SU(N) coherent states," PRB (2021)

[2] - T. Huberman, D. A. Tennant, R. A. Cowley, R. Coldea and C. D. Frost, "A study of the quantum classical crossover in the spin dynamics of the 2D S = 5/2 antiferromagnet Rb2MnF4: neutron scattering, computer simulations and analytic theories" (2008)

[3] - D. Dahlbom, D. Brooks, M. S. Wilson, S. Chi, A. I. Kolesnikov, M. B. Stone, H. Cao, Y.-W. Li, K. Barros, M. Mourigal, C. D. Batista, X. Bai, "Quantum to classical crossover in generalized spin systems," arXiv:2310.19905 (2023)

+end

Finally, we evaluate the sum.

total_spectral_weight(sc; kT) / prod(sys.dims)
5.142433967821603

The result is something slightly greater than 5, substantially closer to the expected quantum sum rule. We can now adjust $\kappa$ and iterate until we reach a value sufficiently close to 16/3. In general, this should be done while collecting substantially more statistics.

Note that $\kappa (T)$ needs to be determined empirically for each model. A detailed example, demonstrating the calculations used in [3], is available here.

References

[1] - H. Zhang, C. D. Batista, "Classical spin dynamics based on SU(N) coherent states," PRB (2021)

[2] - T. Huberman, D. A. Tennant, R. A. Cowley, R. Coldea and C. D. Frost, "A study of the quantum classical crossover in the spin dynamics of the 2D S = 5/2 antiferromagnet Rb2MnF4: neutron scattering, computer simulations and analytic theories" (2008)

[3] - D. Dahlbom, D. Brooks, M. S. Wilson, S. Chi, A. I. Kolesnikov, M. B. Stone, H. Cao, Y.-W. Li, K. Barros, M. Mourigal, C. D. Batista, X. Bai, "Quantum to classical crossover in generalized spin systems," arXiv:2310.19905 (2023)

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Set a nearest neighbor exchange interaction of $J = -1$ meV between neighboring atoms. That is, the total energy along each bond is $J S_i S_{i+1}$. The exchange interaction will be propagated to all symmetry equivalent bonds in the system.

J = -1
 set_exchange!(sys, J, Bond(1, 1, [1, 0, 0]))

Find the energy minimum, which is ferromagnetic. The energy per site is $-J$ for this unfrustrated FM order.

randomize_spins!(sys)
 minimize_energy!(sys)
-energy_per_site(sys)
-1.0000000000000002

Because the interaction is Heisenberg (isotropic), the minimization procedure selects an arbitrary direction in spin-space.

plot_spins(sys; ndims=2, ghost_radius=8)
Example block output

Build a SpinWaveTheory object to measure the dynamical spin-spin structure factor (SSF). Select ssf_perp to project intensities onto the space perpendicular to the momentum transfer $𝐪$, which is appropriate for an unpolarized neutron beam.

swt = SpinWaveTheory(sys; measure=ssf_perp(sys))
SpinWaveTheory [Dipole mode]
+energy_per_site(sys)
-0.9999999999999998

Because the interaction is Heisenberg (isotropic), the minimization procedure selects an arbitrary direction in spin-space.

plot_spins(sys; ndims=2, ghost_radius=8)
Example block output

Build a SpinWaveTheory object to measure the dynamical spin-spin structure factor (SSF). Select ssf_perp to project intensities onto the space perpendicular to the momentum transfer $𝐪$, which is appropriate for an unpolarized neutron beam.

swt = SpinWaveTheory(sys; measure=ssf_perp(sys))
SpinWaveTheory [Dipole mode]
   1 atoms
 

Define a path from $[0,0,0]$ to $[1,0,0]$ in reciprocal lattice units (RLU) containing 400 sampled $𝐪$-points.

qs = [[0,0,0], [1,0,0]]
 path = q_space_path(cryst, qs, 400)
QPath (400 samples)
   [0, 0, 0] → [1, 0, 0]
 

Calculate and plot the intensities along this path.

res = intensities_bands(swt, path)
-plot_intensities(res; units)
Example block output

Perform a powder average over the intensities for 200 radii between 0 and 2.5 inverse Å. Each radial distance defines a spherical shell in recripocal space, which will be sampled approximately uniformly, involving 1000 sample points. Measure intensities for 200 energy values between 0 and 5 meV. Gaussian line-broadening is applied with a full-width at half-maximum (FWHM) of 0.1 meV. With the above parameters, this calculation takes about a second on a modern laptop. To decrease stochastic error, one can increase the number of sample points on each spherical shell.

radii = range(0, 2.5, 200) # 1/Å
+plot_intensities(res; units)
Example block output

Perform a powder average over the intensities for 200 radii between 0 and 2.5 inverse Å. Each radial distance defines a spherical shell in recripocal space, which will be sampled approximately uniformly, involving 1000 sample points. Measure intensities for 200 energy values between 0 and 5 meV. Gaussian line-broadening is applied with a full-width at half-maximum (FWHM) of 0.1 meV. With the above parameters, this calculation takes about a second on a modern laptop. To decrease stochastic error, one can increase the number of sample points on each spherical shell.

radii = range(0, 2.5, 200) # 1/Å
 energies = range(0, 5, 200) # meV
 kernel = gaussian(fwhm=0.1)
 res = powder_average(cryst, radii, 1000) do qs
     intensities(swt, qs; energies, kernel)
 end
-plot_intensities(res; units)
Example block output +plot_intensities(res; units)Example block output diff --git a/dev/examples/spinw/SW02_AFM_Heisenberg_chain-00d6baa4.png b/dev/examples/spinw/SW02_AFM_Heisenberg_chain-00d6baa4.png new file mode 100644 index 0000000000000000000000000000000000000000..a83848963a91d37ab9a99191b66077d40bc2b7d7 GIT binary patch literal 57532 zcmdqJhdKulc;5*Y)z@shq^w)0a=9P^hy~l8+QoD8f7x>bN)Y zF?dDzoNhKe5gLoiilR{YK_vV7C*WrSTSW;mR8Bk14E%7-KvGc_g>qp;p}f3Ns9ku; zYZ`^J=SHDs^-w7OPbd_nRZO{pAY6))dL*jsL@@u2D5iVk6&}B}B@oXdP#7?5M3QHB z>5=Hu^LbwqpGlgVa7KN4@Z5irJ^wG~t!C=buaPmAlhX9=U%1=o)cED=ErR^>?2^GZ zq%{W>Fk#nNfCB=~^udB_($n&q8)c^c4{r~m9%y%YRt4y(if|8O1HeMT?g2AX( z_+J&$AbNM6RNkN|FkD$#*${_8jzy!c)t8-{8@WFAj&AnLtz+el_S)Dra~y_U2Icjp z(V9j|I9?w4ZcVVyo};w@ttsRMAM*I|;}9L!h2lSdp4(@prly+kXgqyNM*8jqijR+` zDLp$oTv@QVk6a{3id8 zX%5lI|IgsOe%@P)ZOfU%U@+1x!(n;oJze>$%qUd(PcTYD%(;xHcrC3I?bIMPycz+@ z3PnYL^7?s6hHuyZSshq2(QEkf!@T2>bo_bQWI|N!D?Kpz+n?r~^Sy627tpK6O*N1Q 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zp2~|jAetL}_-U@s+sXWog_d*qklZ4&X;bsc#N=lU!NT7MUrGMLk@6q6D;LjH&X#8G zdju3c_c9Ygnwr(O+KWH4HVQJ*~ zPP*&kKe}g-;P}rzA931S4<4WV|MBwt_cpYB{O=ZExample block output

swt = SpinWaveTheory(sys; measure=ssf_perp(sys))
+plot_spins(sys; ndims=2, ghost_radius=8)
Example block output

Perform a SpinWaveTheory calculation for a path between $[0,0,0]$ and $[1,0,0]$ in RLU.

swt = SpinWaveTheory(sys; measure=ssf_perp(sys))
 qs = [[0,0,0], [1,0,0]]
 path = q_space_path(cryst, qs, 400)
 res = intensities_bands(swt, path)
 plot_intensities(res; units)
Example block output

This system includes two bands that are fully degenerate in their dispersion.

isapprox(res.disp[1, :], res.disp[2, :])
true

Plot the intensities summed over the two degenerate bands using the Makie lines function.

xs = [q[1] for q in path.qs]
 ys = log10.(res.data[1, :] + res.data[2, :])
-lines(xs, ys; axis=(; xlabel="[H, 0, 0]", ylabel="Log intensity (dimensionless)"))
Example block output +lines(xs, ys; axis=(; xlabel="[H, 0, 0]", ylabel="Log intensity (dimensionless)"))Example block output diff --git a/dev/examples/spinw/SW03_Frustrated_chain-45de5a7a.png b/dev/examples/spinw/SW03_Frustrated_chain-45de5a7a.png new file mode 100644 index 0000000000000000000000000000000000000000..9f8ce37520e3979123f6a6bc9dc668c841feb3d8 GIT binary patch literal 42610 zcmeEuc{r4R_y0Xe^C`u&q6L%EB1=RfltB_omdKK1$zJw-Dlv&fWnYR=7>b7MTaj!@ zM#z>uG4`>~@7&Y#Jm23R-@kv?^}DX;)8`VCx#xYK_v@VVITO1^#-yeZx-lC#$m>XA$IG*ydHU4e&F@SzGlSlHIg-41VA@Kd-HU zAYO+MgcyJz%kUC$6hU0Y5ajoD1UVgzAiJI7i!RH-uOe!MvpOD_Kb4LRz3#@M>mwtx zzQ*lsG)i{YRp}S|RHE~O9&B-Rvg-`}C$O(j;Gtc}tb`!jbKVYIih9ZIuiRRKY$vqN zJMK+6B&aaaJdl(Z{rdh}gY;9M?7!aS*y^8NT4Z8D?7Di(+gQK6oxZX}iJJSmZ*b5~ zx@e)hcHnZg7#q4eH(mIOSR10ZLQMPb5C65ne{t|%U-)l4_-{=7Z(jIsI{Yss`2UR- zHF-1o%hw9hsnn{x?m}Ph-ALbt{0|=%)1*s}AR`5vDm?$t$`(K6>^rj@`uw@-V?ItR zPqjO}yUV^F?1kTbFn3>J@qW)^_XpA)v2NA*Y*`;QM<-w;)cV-)<8KVmm7{mnI 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Assuming a spiral order, optimize the propagation wavevector $𝐤$ starting from a random initial guess. Because all interactions are isotropic in spin space, the polarization axis is arbitrary.

axis = [0, 0, 1]
 randomize_spins!(sys)
 k = minimize_spiral_energy!(sys, axis; k_guess=randn(3))
3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
- 0.23005345616037434
- 0.0612694338713399
- 0.373620977745638

The first component of the order wavevector $𝐤$ has a unique value up to reflection symmetry, $𝐤 → -𝐤$. The second and third components of $𝐤$ are arbitrary for this 1D chain system. In all cases, the minimized energy has a precise value of -33/16 in units of $|J₁|$.

@assert k[1] ≈ 0.2300534561 || k[1] ≈ 1 - 0.2300534561
+ 0.769946543837592
+ 0.297299303151331
+ 0.9159665936112013

The first component of the order wavevector $𝐤$ has a unique value up to reflection symmetry, $𝐤 → -𝐤$. The second and third components of $𝐤$ are arbitrary for this 1D chain system. In all cases, the minimized energy has a precise value of -33/16 in units of $|J₁|$.

@assert k[1] ≈ 0.2300534561 || k[1] ≈ 1 - 0.2300534561
 @assert spiral_energy_per_site(sys; k, axis) ≈ -33/16 * abs(J1)
sys_enlarged = repeat_periodically_as_spiral(sys, (8, 1, 1); k, axis)
-plot_spins(sys_enlarged; ndims=2)
Example block output

Use SpinWaveTheorySpiral on the original sys to calculate the dispersion and intensities for the incommensurate ordering wavevector.

swt = SpinWaveTheorySpiral(sys; measure=ssf_perp(sys), k, axis)
+plot_spins(sys_enlarged; ndims=2)
Example block output

Use SpinWaveTheorySpiral on the original sys to calculate the dispersion and intensities for the incommensurate ordering wavevector.

swt = SpinWaveTheorySpiral(sys; measure=ssf_perp(sys), k, axis)
 qs = [[0,0,0], [1,0,0]]
 path = q_space_path(cryst, qs, 400)
 res = intensities_bands(swt, path)
-plot_intensities(res; units)
Example block output +plot_intensities(res; units)Example block output diff --git a/dev/examples/spinw/SW04_Frustrated_square-83ba5d19.png b/dev/examples/spinw/SW04_Frustrated_square-83ba5d19.png new file mode 100644 index 0000000000000000000000000000000000000000..46c31938a6bd8fb83eacddee657cace33914c4d4 GIT binary patch literal 76249 zcmeFZc|4SB_y_Eia7s}rDqA9wkYrz4ERiLO>>^~}vagkswa9)V6yZn+AzvatW@2K2Z<(3cHo=!ac5m5({B-TK{%Izr%A{?}c3a@*U#=T#pJ6KR<{gJ01h1Ym z)@Nc0In2a#<2NRzCHSWs-T<-}d8-&)Lv*kw4*Y}%%k`}%l(Pn4ibL+6xTwIb?L^3w*Bwg#g46q>Yq zRwv2w{k=|!0o=%E8r8^%>sMGF{7E^#@y#y%|Gr`R&jT!!At7FUBO(;6xnQ{nfLSaTUy^SGcgU-KZIAoM?%Q~&1+NR<7LCcb^NL! zB-&blBn{I%bCk6FRbG90g|ue-mAW>Se%T7|A-lh!e2g0W3lq~hF%dY>U*GRzpUTM{ zR8or|_Y}~ghWwPa%V(#DP1x!cH@|b`3mps`qc#dPFT_NO3F&GW7E12fYAbuE!uo~x z)oWwtGbIVVPvB#OZ|?$;d+ombepFyU0C{0yoWk@ zaVuqB5f|w-w!4$PCMK2#Y?n3T{8^$xi;L?LlSm=6-&O<_atvC#oJ0gHW1sht`@9!k zdoMKe^Np?!aYz&f#J5THk~fyJ&C)VG%m!J6J!2^{1%!!*y&*dp&ybf+&xgW)ztT+81C(da&$Yd>Fus}<^l6mG%Z)fK9$d|77Q928R-Hy!D812 z!Ukz8KZ2SF?jyFAk|OLP?9FqZW7f=8;&pPb~jl26x#jOZmc1A#bp z(7s0LdjRFZ19&4E=P9s*WpdP)S2I~fOZV|(7f-py%Vp@Q8vm=I^im=*^Vs=9XMb(Ibl*NrP%F8Bt|Z_al>bcr!yAuJLC1GYqjE!CgrZL*frEK>mx4HU%m>g7846Y_LF}5+v z*hZM}u6>$4&Xq4ncz888p~4AOm&6&skYl_?&pBP+ z%Z!*LIBXe3rE<_$sEc7dng@c}^&bn`Uz53fG(D4VOg((E##}P6xAwYj=ZT}G9{}c3 z?j9@jie^b6*g=#jd2)EzQdAU9m$lXITK$7~jhW=T9Ti&1Z^zzG>ehwQ1_IZdxI9LE zr9QaY4t!cl^6ZypFSsD)oCTRSllnuJPx_+bQK$$$K@Z=jmMAHu6!>%Z%Z6peN%W|_eikh zo+^TrZNu#7`OMYmFbCP}W5pM-)K8NEfRlhmJD%>b>v3gxa>T^v));xsh{^d$S8@O2 zne=?S<;#()So!_it?|^wCb;&pa~iy=mrS72v)D93jkcSZoQ&)6mZJ|XFZcEIOyY3Bp*=wp75M9m zA9Y`m+?U5^r~H;8m!~M=J5|#gti+ba9X>7l9w~! zDF+LPAkhqqa92<^Z-yN;fx7GFY1VjDqXnt~weqDcA*51l;ke1n;(^9S`G7!?y#|ND zcI)pPB+q5xT%OrTja)sNJm`cwkhJz*VEJaO!+Hep9+0&~`HlBZPT~Uts|imL1Ka=) znGW0bSL9GrMU4e_b-uE#;GPNEx4vqtx~FSjuq*Xi1xBU4RfMOp%UfECwWRhoqnp>} zR>I?g3wlD6_y?4f8o~qS!?DWufxo{try2HJRVWwqO!j<>bd#%&T!?8luK?SN_BqQw z>2*Dvx)ZLGfb05##fPlH6O@M$#|yw!1Jwvaezb_MVC4 zTK_hwW%%#h(-S=X%Bm=TfKGung|(&A?+HLB%>Cf|1QcyBR44^A>RMw_(Bo}C7lN`V z6E6gFs~Ezm-*}uYv6r>1Z?d2Oe2Fq2!l8Z03ze&BBWY#1#Z?txLLRG&FIWplU2lN zeDn8)im8dF=cw==X=ZF}m>T7^hVX`{@!}VVtms`5WEbdiY86W{6PCG~Z|hQ5IFQi@ zv4JGUzLG*bL)%Dd)SsCx%c&}?KD44zA@k0c8+cN$DB`^7GxH<%tr^tS8K~dlvQE@V+G5@z65FZF;@0gv2@+6qF;CExh$rY_ErXxHMLIKD-y!2M~cGE>|WGv%Xi7D z^-w;JlUD(#oV{kwe16}y|M-%Ew6%*qlQ1}HfU`Zg%OuE6u0J0b2I;03JXGch}R zNkdj2PQ1wFQ|)9!IbzGb37Yyi1aWf+;)>sgRD(DTG%)bw6KD>PShZ`8RLkQia6Xh5 zJF;7Ir*dk&MM;wbF>-7Ue+(m@iGdy55bY43`?IePftb~n9|cTPP0$?Jf?Kuw1RGv4PG z03I*7Y;cy$nASnple^pmJ5h%-#=(~}kSAA~dv_cYhGD%fE zQ#nvPJ>Jngari|!(OK5d8aqY>icu3quzdE31_L`s?J85qj=r>=^v$Tc7sP zwPve5D=Vu|N~Z&GD#|91x7~jr1ZBhEX@DH|4szX+?}mpdd3hv;<$ST!tkk$>2ok&H zM|bO%06kXV`Pvt|dcW5ky=>F$E0W!{by}~}L1r#wMSU?L(#SrcEuTaQY#cBx?fsc2 zl$pwl)LISBs?D^p3dEiR%bAw`({f!7GKkq#5wja!tFhS55@l0CGepw|LUY#-&PvVa zqS~XhCzr%C92M8i_0C^46BF0beB{&_z`54M(j174JZ!7&r>^wd16|si;#`_7KHTLpW9T#wAxkR-P#)lJb9w42@{bi zN&RW+A_!DRLr|W?T54I!n$*kdhcYpWm88X)%tJ`7Xiy#82+2giyj}B}C2m|W zz7MWT4HgZc465$z9HTarBk}AXJU=~r0d~E*_7qRMlv6iLvwu^71B1Vr2?LzVZ(5gt1ip8o=(AHuI z6Y99lOXQqpNqI6 zR&`83r_i|2xKmc2SZj^1@gPdyv9B~+n6s^mZ#&=ny}X=a*@t$qIe7;I;nGaPNoJEu z#bJzNpp4=;O@O)OCPg4X}+}7K-?0DAM6f^3}(Bsp^$O=KjI?3E$K$8;vmH0?B(NrB^^0N%)45(ZH zs*@V1zrK9e>-QpaAfM(R2ZrTIcSd$6MYn3Vem-#R+OG3%S-Ds*6YVvCiy;aw!an|8eRqpxD(s^}S1;Y_*Xo|qX>LZH4?azzeJl#%Yu@rUvm8eL;d5)BOxxU?yiC#5A3>0$#A+DH#^vPX#Kji5wAuF{Hp}_lC9BQ; zI3t{4RRLQnnfYQ_dWX>zi2MO>krWBuxXj7e`Oi2)nh01c^9nz&f&P(0S+N#rrcL=bw!2T9i!%1HA^hwCl%rs^+T=Yvvsh;&= zns}Ptb&@Q+XyaQKn~WtQhV$00?_4}UX`)s{jAz=o$%c`<+Ldw=DRkVpY`oCUxO`_` zWktoEaDKXlDK1$CZ)-hpvf%zjE}NKMQnewHU}MDMi*PQS%sB?deAMtr+5H}J2ZA4* zd31#*+Jk8Dw(6;G{R7>?5mw_|QT2O$vWN2IXBN51{&AW@siRfO{CVswwD7f;G>-7K z)0_{sv#k0l*;a_`jk~iqp~v1woeh26Pq@-b$zxAK=9Avi=*nwZ^rGM{S*gpVbM}?L zcd||U{VtJ->Kt*9enDBcAjhpCKk4bisf!`U3^UFbcb)lcUlY%8^3YTZHUms zuic&Wh&$Hj7OA9qVEwQnzXfyLdPtxI0@UnxcD8>d&lP;0?twA~;r=TK-RB@X}a;o)Ce1&>@>HR8$Th*?<-t08pkb_U4Q z8cBHKUGW>oj+d4+6?b}cPrrDy_?~Xzlgfbs>z(Em_}0(ws7(P+apS+)DP(sE*Am=W zDhmn@?bIp8!N#j3kd2p<8A%$W`X$`qF1M}q8F8}qD$2%7fwvs*7W}|UvX~sQA*_u{ z#0p1Fr0N$B_!5p~Le81;Q7N8<8tf*kw4?PbNqu?+UbeROIe*CO>N66q>~$K}FCO$; z@+>|CzWAW)Q{nr7vVbn|=IdD5!0p5Kh4x%q?}Pfu``}ysYZ0#c^Ryi&FS2J6ZN;gX zv@c0JK~vZ;t?#*o8YKg>!34;A{RWO@meyNY;6|#`42tJI*0|v{93cF}wAxhQ6a5

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Calculate and plot intensities for a path through $𝐪$-space.

swt = SpinWaveTheory(sys; measure=ssf_perp(sys))
+plot_spins(sys)
Example block output

Calculate and plot intensities for a path through $𝐪$-space.

swt = SpinWaveTheory(sys; measure=ssf_perp(sys))
 qs = [[0, 0, 0], [1/2, 0, 0], [1/2, 1/2, 0], [0, 0, 0]]
 path = q_space_path(cryst, qs, 400);
 res = intensities_bands(swt, path)
-plot_intensities(res; units)
Example block output
+plot_intensities(res; units)Example block output diff --git a/dev/examples/spinw/SW05_Simple_kagome_FM-0cbea8a8.png b/dev/examples/spinw/SW05_Simple_kagome_FM-0cbea8a8.png deleted file mode 100644 index 09f9d52a2c9ff99ce4c7e7a0872db6048da0ea36..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 49994 zcmeFZi9eKW`#(OUa4SNEHnLXAlCA8OFoau$5R#7{($Fxxk96;ExO_CVKetj&u87islFooCsWr^=#^`PHzEGFJox`#5yg*APvu$6tq#+9 zx;OeRCTv4Vn=zvrnmcU$ZnbBl6j?-O%NK*klQ(9ae0sjI3(%vajhIop?tT7r$0A<# zBnsciHd{6uTE4&+rxmm{f+~*ecB`eNO%P#XpZ!~L0-MXWnTO#cRa^r`SG#Kl}qi|9oPBF>`O+=Nc|g^ zv3-UjDrUvb3)#leD_tTeRn8dHuM>>Kq8>>FqU#EI4NY;G>E+yaqSI!BxJ}|2o9+we z`QDNOZ*T7?ytxl?(1&OsuK?@mGNYcltg~xeT&tDaccra5r~T#Sx8&buvwu~o${v)R zu4X*6)F_%at^M|-nRk0(M zw@u3HHw@SLY%ac#AUu)DmzTDJlb7H85 zIASMdQ@MG6TT$1|Xw=S;C4q?Yp(%^m%uJ0QqMqK%d-KOKz1Dk*HDWlNrAoD^?`AZPS84szL`%>2y|}f82%Dd$y?;Uvp`%Z)>O#b|=+;$| zM=f5;iGSa|IG*A;z~xw#gb{iSy(PGcA@D!o6<_NwokNa+{ujv1EbC1bTl1m|7(W(>OHn2 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To view part of the incommensurate spiral spin structure, one can construct an enlarged system with repeat_periodically_as_spiral.

Perform a SpinWaveTheory calculation for a path between $[0,0,0]$ and $[1,0,0]$ in RLU.

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Consider dt ≈ 0.0995 for this spin configuration at tol = 0.01. Current value is dt = 0.1.

The Zeeman coupling polarizes the magnetic moments in the $𝐁 ∝ ẑ$ direction. The spin dipoles, however, are anti-aligned with the magnetic moments, and therefore point towards $-ẑ$. This is shown below.

plot_spins(sys)
Example block output

Estimate the dynamical structure factor using classical dynamics.

sc = SampledCorrelations(sys; dt, energies=range(0, 15D, 100), measure=ssf_trace(sys))
+end
Consider dt ≈ 0.0995 for this spin configuration at tol = 0.01. Current value is dt = 0.1.

The Zeeman coupling polarizes the magnetic moments in the $𝐁 ∝ ẑ$ direction. The spin dipoles, however, are anti-aligned with the magnetic moments, and therefore point towards $-ẑ$. This is shown below.

plot_spins(sys)
Example block output

Estimate the dynamical structure factor using classical dynamics.

sc = SampledCorrelations(sys; dt, energies=range(0, 15D, 100), measure=ssf_trace(sys))
 add_sample!(sc, sys)
 nsamples = 100
 for _ in 1:nsamples
@@ -35,4 +35,4 @@
 res2 = intensities_bands(swt, path)
Sunny.BandIntensities{Float64, Sunny.QPath, 2} (1×400 elements)

This model system has a single magnon band with dispersion $ϵ(𝐪) = 1 - D/B \sin(2πq₃)$ and uniform intensity. Both calculation methods reproduce this analytical solution. Observe that $𝐪$ and $-𝐪$ are inequivalent. The structure factor calculated from classical dynamics additionally shows an elastic peak at $𝐪 = [0,0,0]$, reflecting the ferromagnetic ground state.

fig = Figure(size=(768, 300))
 plot_intensities!(fig[1, 1], res1; title="Classical dynamics")
 plot_intensities!(fig[1, 2], res2; title="Spin wave theory")
-fig
Example block output
+figExample block output diff --git a/dev/examples/contributed/MgCr2O4-tutorial.html b/dev/examples/contributed/MgCr2O4-tutorial.html index f0a8ad432..271fcf22a 100644 --- a/dev/examples/contributed/MgCr2O4-tutorial.html +++ b/dev/examples/contributed/MgCr2O4-tutorial.html @@ -360,4 +360,4 @@ Sq_mgcro = intensities_static(sc_mgcro, qpts_mgcro; kT);

We can plot the results below. It is useful to compare these to the plot above generated with an SampledCorrelationsStatic.

fig = Figure(; size=(1200,500))
 ax = plot_intensities!(fig[1,1], Sq_pyro; title="Pyrochlore")
 ax = plot_intensities!(fig[1,2], Sq_mgcro; title="MgCr₂O₄")
-fig

+fig

diff --git a/dev/examples/contributed/kappa_tutorial.html b/dev/examples/contributed/kappa_tutorial.html index 79d965a4c..655e52142 100644 --- a/dev/examples/contributed/kappa_tutorial.html +++ b/dev/examples/contributed/kappa_tutorial.html @@ -98,4 +98,4 @@ # Turn off κ renormalization before generating a new equilibrium sample. set_spin_rescaling!(sys, 1.0) -end

Finally, we evaluate the sum.

total_spectral_weight(sc; kT) / prod(sys.dims)
5.142433967821603

The result is something slightly greater than 5, substantially closer to the expected quantum sum rule. We can now adjust $\kappa$ and iterate until we reach a value sufficiently close to 16/3. In general, this should be done while collecting substantially more statistics.

Note that $\kappa (T)$ needs to be determined empirically for each model. A detailed example, demonstrating the calculations used in [3], is available here.

We will study a temperature quench process using a generalized Langevin spin dynamics. In this SU(3) treatment of quantum spin-1, the dynamics include coupled dipoles and quadrupoles. Select a relatively small damping magnitude to overcome local minima, and disable thermal fluctuations.

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Optimizing spins