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Optimizing spins

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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zZT9S>AYcrhGW*aW=4YzsgRtr(;E}nPHWZFochR3CMCh^#R&zOSS9{;^N1H`E-qZa% zROz&6uUv~pjxyPm&%{re!*-XlIPECmK5#QD-%#S-H zeFCq)Oi|PF&l0r$HGJ3f_C2k_Yf{_xiI{5hDBQ~4m-MDjRC5TH!>d-UvQ8Cc8~O2| zLF|d}-wquRmm!2(ogmaNhb)xLHWstdc16B%$+6A(vY8vhH}sq_ zdv2n$ zsNL>L5}TC2Zn$agEO6d7y!wHjKw#|NCQI|E<0PVOg*%d_S*% z0+XzT%hrj?Lx1=)O+5dKs~_GDdT^$D^z`=S;=-AV3S?qMMD>{e_KTa+YUA92FTyNc zr+GFNY zahzWe0AEW*`gCCC3NQ?|J;-GJ?h|~DAX_nZV1cqUBK-Mrx3?PhIu1bRQUB2M%qncG zLjvy#EkZrD!R~)QXp;3ae}YlLR&kCI0*a$9z{^CkjlrHoD#gz_&((iu+TkN-expRV z4|vaXemZmJy0Example 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)
    +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 ≈ 0.04201 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 ≈ 0.04192 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 (612.974 MiB)
     [S(q,ω) | nω = 49, 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/previews/PR324/examples/03_LSWT_SU3_FeI2.html b/previews/PR324/examples/03_LSWT_SU3_FeI2.html index 9e7d9399e..8c13534b7 100644 --- a/previews/PR324/examples/03_LSWT_SU3_FeI2.html +++ b/previews/PR324/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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    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 ≈ 0.01096 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 ≈ 0.01102 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/previews/PR324/examples/07_Dipole_Dipole.html b/previews/PR324/examples/07_Dipole_Dipole.html index ec48924ab..173998258 100644 --- a/previews/PR324/examples/07_Dipole_Dipole.html +++ b/previews/PR324/examples/07_Dipole_Dipole.html @@ -30,4 +30,4 @@ 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/previews/PR324/examples/08_Momentum_Conventions-337d8958.png b/previews/PR324/examples/08_Momentum_Conventions-337d8958.png deleted file mode 100644 index ddd2cea1002c367f87aad355fe00ade354856eac..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 110129 zcmeFZX&_eZ*EhV)lBrUiNrsXrWJ<^^N|8#0%=47USeZhUNJ*I`6{R9cMCKyNl%zt2 z$|*8LGGusvTmA3*e(vY}{C;|`4_DD??_(dwTI)BgwOu%PU=JG$Hw!@!Y}$KwA0`M! z{FT;-nHGO+<1H@0AM_4tx@rWG7qfiMiUD8Kcpl!PPUJN5e8(63HhT~262xgCf(Q;H zh*^9qcz__hB?w~Bk{}ck2x6^!O7S5@e3a1Mt#-tZ=11kSBZl8TtBf?2P6hJX#~G`t 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    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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z?g{L~rXddFlKV~BD%d*g60n)5gAae6?1Zdxed1Om}K_&R#JZy(iRbw-RF4=wl zgK|92^&ei}4}e2U0K@{~viFZuUtb@{v&JB414)Osey40}%c_BFefGY2a;a(e-D9z> z+#lf88K_ccr{v}1*c(Bnp_UGTz~^y10LB3 zP8>kqsB{k$1dz7w2S2#U1xfJgNb=TMTU(o44%BHJu*D(m|MEQ}6AMoX9|tS?|E6MpOKF9FmHUl{2!CI Bf^z@> literal 0 HcmV?d00001 diff --git a/previews/PR324/examples/04_GSD_FeI2.html b/previews/PR324/examples/04_GSD_FeI2.html index 7be9b11a0..21885bbd0 100644 --- a/previews/PR324/examples/04_GSD_FeI2.html +++ b/previews/PR324/examples/04_GSD_FeI2.html @@ -39,27 +39,27 @@
    randomize_spins!(sys)
     minimize_energy!(sys; maxiters=10)
     suggest_timestep(sys, langevin; tol=1e-2)
    -langevin.dt = 0.03;
    Consider dt ≈ 0.03059 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 ≈ 0.0318 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:
     
    -    [1/4, 0, 1/4]          39.39% weight
    -    [-1/4, 0, -1/4]        39.39%
    -    [1/8, 0, 1/4]           0.08%
    -    [-1/8, 0, -1/4]         0.08%
    -    [0.1875, 0.0625, 1/4]    0.07%
    -    [-0.1875, -0.0625, -1/4]    0.07%
    -    [-1/4, 0.3125, 0]       0.07%
    -    [1/4, -0.3125, 0]       0.07%
    -    [-0.1875, 0.3125, 1/4]    0.06%
    -    [0.1875, -0.3125, -1/4]    0.06%
    -    [-1/4, 0.4375, 0]       0.06%
    -    ...                     ...

    Thermalization has not substantially altered the suggested dt.

    suggest_timestep(sys, langevin; tol=1e-2)
    Consider dt ≈ 0.0314 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]         39.99% weight
    +    [0, 1/4, -1/4]         39.99%
    +    [1/8, -0.1875, 1/4]     0.11%
    +    [-1/8, 0.1875, -1/4]    0.11%
    +    [-0.1875, 0.1875, 1/4]    0.09%
    +    [0.1875, -0.1875, -1/4]    0.09%
    +    [0, 0.1875, 1/2]        0.08%
    +    [0, -0.1875, 1/2]       0.08%
    +    [0.1875, 1/8, 0]        0.07%
    +    [-0.1875, -1/8, 0]      0.07%
    +    [-0.3125, 1/8, 0]       0.07%
    +    ...                     ...

    Thermalization has not substantially altered the suggested dt.

    suggest_timestep(sys, langevin; tol=1e-2)
    Consider dt ≈ 0.03129 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 ≈ 0.04031 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 ≈ 0.04075 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 (37.950 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/previews/PR324/examples/05_MC_Ising.html b/previews/PR324/examples/05_MC_Ising.html index 1ae34be77..d07756d9a 100644 --- a/previews/PR324/examples/05_MC_Ising.html +++ b/previews/PR324/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)))
    Example block output diff --git a/previews/PR324/examples/06_CP2_Skyrmions-ab57592c.png b/previews/PR324/examples/06_CP2_Skyrmions-ab57592c.png deleted file mode 100644 index 949155524cd578189f8443a449af0b6f25d7e79e..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 274955 zcmeFY^-~-{+xI)TOK_JYxVt+9g1fse7918|Ja~e;1`n3tF2P}eV8Puz1h=#I{XFk` z{)F?(sp_iPncnKD>7M>xpGziMT~!_noeUiS0IZJ+vYG&ZSO)-bUr^y;SGX3i(O?He zJ1G?@0H{yIc(Oo(ox^!(%1Z+^P>Ms?35}J4rV0S~(E|V^8~`3*mmmiK;Kd05N9F(^ zoB;sDt~qTQqOiXLA7!Pq!Eh(NXgRqyfdWtKjVrf{3ybN>O(WbNhvCsgzW(V%wL*!sF-VWDQ82sA$$W5Cw*hX{n3{DaeOiTG3T+X z=2+z%{M$o;{_b#J>v8L;+sr-=jy4=u8kIKu%Yc07e}CXg4;Q_cf?fH~nYBg|&Hwb{ z{$Tw0zx}8NDewQcKb%z&?|=KHiRjS&Gthtj6q|(je}FfMrHl4U6@FA14oD`L}FN8fy=te5LyEaoH2NKsbEQDG@2 zBqJf-fPb}DB;AFZN-biy)~0=@XL2xS!pjMX3QZ#b?J}8M@sc@>u=TnTEoiB(%JCo= zv4p+SI3Fu?5n&C3B_rA3ARhE2{Tw~lDOx_$xD9Md=s5#AZD;np;L&jBE)P-a6!;!V zZbTX<4Co(FDYsFL^P$>{HMquDPvE%H_*}=4%83iHbR}lE}qOr&AANo;ZeW++7R5vXCIMu|;bdg`o 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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.

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