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{"documenter":{"julia_version":"1.10.5","generation_timestamp":"2024-09-19T01:02:01","documenter_version":"1.7.0"}} | ||
{"documenter":{"julia_version":"1.10.5","generation_timestamp":"2024-09-20T16:59:54","documenter_version":"1.7.0"}} |
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{ | ||
"cells": [ | ||
{ | ||
"cell_type": "markdown", | ||
"source": [ | ||
"# 9. Disordered system with KPM\n", | ||
"\n", | ||
"This example uses the kernel polynomial method (KPM) to efficiently calculate\n", | ||
"the neutron scattering spectrum of a disordered triangular antiferromagnet.\n", | ||
"The model is inspired by YbMgGaO4, as studied in [Paddison et al, Nature\n", | ||
"Phys., **13**, 117–122 (2017)](https://doi.org/10.1038/nphys3971) and [Zhu et\n", | ||
"al, Phys. Rev. Lett. **119**, 157201\n", | ||
"(2017)](https://doi.org/10.1103/PhysRevLett.119.157201). Disordered occupancy\n", | ||
"of non-magnetic Mg/Ga sites can be modeled as a stochastic distribution of\n", | ||
"exchange constants and $g$-factors. Including this disorder introduces\n", | ||
"broadening of the spin wave spectrum." | ||
], | ||
"metadata": {} | ||
}, | ||
{ | ||
"outputs": [], | ||
"cell_type": "code", | ||
"source": [ | ||
"using Sunny, GLMakie" | ||
], | ||
"metadata": {}, | ||
"execution_count": null | ||
}, | ||
{ | ||
"cell_type": "markdown", | ||
"source": [ | ||
"Set up minimal triangular lattice system. Include antiferromagnetic exchange\n", | ||
"interactions between nearest neighbor bonds. Energy minimization yields the\n", | ||
"magnetic ground state with 120° angles between spins in triangular plaquettes." | ||
], | ||
"metadata": {} | ||
}, | ||
{ | ||
"outputs": [], | ||
"cell_type": "code", | ||
"source": [ | ||
"latvecs = lattice_vectors(1, 1, 10, 90, 90, 120)\n", | ||
"cryst = Crystal(latvecs, [[0, 0, 0]])\n", | ||
"sys = System(cryst, [1 => Moment(s=1/2, g=1)], :dipole; dims=(3, 3, 1))\n", | ||
"set_exchange!(sys, +1.0, Bond(1, 1, [1,0,0]))\n", | ||
"\n", | ||
"randomize_spins!(sys)\n", | ||
"minimize_energy!(sys)\n", | ||
"plot_spins(sys; color=[S[3] for S in sys.dipoles], ndims=2)" | ||
], | ||
"metadata": {}, | ||
"execution_count": null | ||
}, | ||
{ | ||
"cell_type": "markdown", | ||
"source": [ | ||
"Select a $𝐪$-space path for the spin wave calculations." | ||
], | ||
"metadata": {} | ||
}, | ||
{ | ||
"outputs": [], | ||
"cell_type": "code", | ||
"source": [ | ||
"qs = [[0, 0, 0], [1/3, 1/3, 0], [1/2, 0, 0], [0, 0, 0]]\n", | ||
"labels = [\"Γ\", \"K\", \"M\", \"Γ\"]\n", | ||
"path = q_space_path(cryst, qs, 150; labels)\n", | ||
"kernel = lorentzian(fwhm=0.4);" | ||
], | ||
"metadata": {}, | ||
"execution_count": null | ||
}, | ||
{ | ||
"cell_type": "markdown", | ||
"source": [ | ||
"Perform a traditional spin wave calculation. The spectrum shows sharp modes\n", | ||
"associated with coherent excitations about the K-point ordering wavevector,\n", | ||
"$𝐪 = [1/3, 1/3, 0]$." | ||
], | ||
"metadata": {} | ||
}, | ||
{ | ||
"outputs": [], | ||
"cell_type": "code", | ||
"source": [ | ||
"energies = range(0.0, 3.0, 150)\n", | ||
"swt = SpinWaveTheory(sys; measure=ssf_perp(sys))\n", | ||
"res = intensities(swt, path; energies, kernel)\n", | ||
"plot_intensities(res)" | ||
], | ||
"metadata": {}, | ||
"execution_count": null | ||
}, | ||
{ | ||
"cell_type": "markdown", | ||
"source": [ | ||
"Use `repeat_periodically` to enlarge the system by a factor of 10 in\n", | ||
"each dimension. Use `to_inhomogeneous` to disable symmetry\n", | ||
"constraints, and allow for the addition of disordered interactions." | ||
], | ||
"metadata": {} | ||
}, | ||
{ | ||
"outputs": [], | ||
"cell_type": "code", | ||
"source": [ | ||
"sys_inhom = to_inhomogeneous(repeat_periodically(sys, (10, 10, 1)))" | ||
], | ||
"metadata": {}, | ||
"execution_count": null | ||
}, | ||
{ | ||
"cell_type": "markdown", | ||
"source": [ | ||
"Use `symmetry_equivalent_bonds` to iterate over all nearest neighbor\n", | ||
"bonds of the inhomogeneous system. Modify each AFM exchange with a noise term\n", | ||
"that has variance of 1/3. The newly minimized energy configuration allows for\n", | ||
"long wavelength modulations on top of the original 120° order." | ||
], | ||
"metadata": {} | ||
}, | ||
{ | ||
"outputs": [], | ||
"cell_type": "code", | ||
"source": [ | ||
"for (site1, site2, offset) in symmetry_equivalent_bonds(sys_inhom, Bond(1,1,[1,0,0]))\n", | ||
" noise = randn()/3\n", | ||
" set_exchange_at!(sys_inhom, 1.0 + noise, site1, site2; offset)\n", | ||
"end\n", | ||
"\n", | ||
"minimize_energy!(sys_inhom, maxiters=5_000)\n", | ||
"plot_spins(sys_inhom; color=[S[3] for S in sys_inhom.dipoles], ndims=2)" | ||
], | ||
"metadata": {}, | ||
"execution_count": null | ||
}, | ||
{ | ||
"cell_type": "markdown", | ||
"source": [ | ||
"Traditional spin wave theory calculations become impractical for large system\n", | ||
"sizes. Significant acceleration is possible with the [kernel polynomial\n", | ||
"method](https://arxiv.org/abs/2312.08349). Enable it by selecting\n", | ||
"`SpinWaveTheoryKPM` in place of the traditional\n", | ||
"`SpinWaveTheory`. Using KPM, the cost of an `intensities`\n", | ||
"calculation becomes linear in system size and scales inversely with the width\n", | ||
"of the line broadening `kernel`. Error tolerance is controlled through the\n", | ||
"dimensionless `tol` parameter. A relatively small value, `tol = 0.01`, helps\n", | ||
"to resolve the large intensities near the ordering wavevector. The alternative\n", | ||
"choice `tol = 0.1` would be twice faster, but would introduce significant\n", | ||
"numerical artifacts.\n", | ||
"\n", | ||
"Observe from the KPM calculation that disorder in the nearest-neighbor\n", | ||
"exchange serves to broaden the discrete excitation bands into a continuum." | ||
], | ||
"metadata": {} | ||
}, | ||
{ | ||
"outputs": [], | ||
"cell_type": "code", | ||
"source": [ | ||
"swt = SpinWaveTheoryKPM(sys_inhom; measure=ssf_perp(sys_inhom), tol=0.01)\n", | ||
"res = intensities(swt, path; energies, kernel)\n", | ||
"plot_intensities(res)" | ||
], | ||
"metadata": {}, | ||
"execution_count": null | ||
}, | ||
{ | ||
"cell_type": "markdown", | ||
"source": [ | ||
"Now apply a magnetic field of magnitude 7.5 (energy units) along the global\n", | ||
"$ẑ$ axis. This field fully polarizes the spins. Because gap opens, a larger\n", | ||
"tolerance of `tol = 0.1` can be used to accelerate the KPM calculation without\n", | ||
"sacrificing much accuracy. The resulting spin wave spectrum shows a sharp mode\n", | ||
"at the Γ-point (zone center) that broadens into a continuum along the K and M\n", | ||
"points (zone boundary)." | ||
], | ||
"metadata": {} | ||
}, | ||
{ | ||
"outputs": [], | ||
"cell_type": "code", | ||
"source": [ | ||
"set_field!(sys_inhom, [0, 0, 7.5])\n", | ||
"randomize_spins!(sys_inhom)\n", | ||
"minimize_energy!(sys_inhom)\n", | ||
"\n", | ||
"energies = range(0.0, 9.0, 150)\n", | ||
"swt = SpinWaveTheoryKPM(sys_inhom; measure=ssf_perp(sys_inhom), tol=0.1)\n", | ||
"res = intensities(swt, path; energies, kernel)\n", | ||
"plot_intensities(res)" | ||
], | ||
"metadata": {}, | ||
"execution_count": null | ||
}, | ||
{ | ||
"cell_type": "markdown", | ||
"source": [ | ||
"Add disorder to the $z$-component of each magnetic moment $g$-tensor. This\n", | ||
"further broadens intensities, now across the entire path. Some intensity\n", | ||
"modulation within the continuum is also apparent. This modulation is a\n", | ||
"finite-size effect, and would be mitigated by enlarging the system beyond\n", | ||
"30×30 chemical cells." | ||
], | ||
"metadata": {} | ||
}, | ||
{ | ||
"outputs": [], | ||
"cell_type": "code", | ||
"source": [ | ||
"for site in eachsite(sys_inhom)\n", | ||
" noise = randn()/6\n", | ||
" sys_inhom.gs[site] = [1 0 0; 0 1 0; 0 0 1+noise]\n", | ||
"end\n", | ||
"\n", | ||
"swt = SpinWaveTheoryKPM(sys_inhom; measure=ssf_perp(sys_inhom), tol=0.1)\n", | ||
"res = intensities(swt, path; energies, kernel)\n", | ||
"plot_intensities(res)" | ||
], | ||
"metadata": {}, | ||
"execution_count": null | ||
}, | ||
{ | ||
"cell_type": "markdown", | ||
"source": [ | ||
"For reference, the equivalent non-disordered system shows a single coherent\n", | ||
"mode." | ||
], | ||
"metadata": {} | ||
}, | ||
{ | ||
"outputs": [], | ||
"cell_type": "code", | ||
"source": [ | ||
"set_field!(sys, [0, 0, 7.5])\n", | ||
"randomize_spins!(sys)\n", | ||
"minimize_energy!(sys)\n", | ||
"swt = SpinWaveTheory(sys; measure=ssf_perp(sys))\n", | ||
"res = intensities(swt, path; energies, kernel)\n", | ||
"plot_intensities(res)" | ||
], | ||
"metadata": {}, | ||
"execution_count": null | ||
} | ||
], | ||
"nbformat_minor": 3, | ||
"metadata": { | ||
"language_info": { | ||
"file_extension": ".jl", | ||
"mimetype": "application/julia", | ||
"name": "julia", | ||
"version": "1.10.5" | ||
}, | ||
"kernelspec": { | ||
"name": "julia-1.10", | ||
"display_name": "Julia 1.10.5", | ||
"language": "julia" | ||
} | ||
}, | ||
"nbformat": 4 | ||
} |
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Original file line number | Diff line number | Diff line change |
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using Sunny, GLMakie | ||
|
||
latvecs = lattice_vectors(1, 1, 10, 90, 90, 120) | ||
cryst = Crystal(latvecs, [[0, 0, 0]]) | ||
sys = System(cryst, [1 => Moment(s=1/2, g=1)], :dipole; dims=(3, 3, 1)) | ||
set_exchange!(sys, +1.0, Bond(1, 1, [1,0,0])) | ||
|
||
randomize_spins!(sys) | ||
minimize_energy!(sys) | ||
plot_spins(sys; color=[S[3] for S in sys.dipoles], ndims=2) | ||
|
||
qs = [[0, 0, 0], [1/3, 1/3, 0], [1/2, 0, 0], [0, 0, 0]] | ||
labels = ["Γ", "K", "M", "Γ"] | ||
path = q_space_path(cryst, qs, 150; labels) | ||
kernel = lorentzian(fwhm=0.4); | ||
|
||
energies = range(0.0, 3.0, 150) | ||
swt = SpinWaveTheory(sys; measure=ssf_perp(sys)) | ||
res = intensities(swt, path; energies, kernel) | ||
plot_intensities(res) | ||
|
||
sys_inhom = to_inhomogeneous(repeat_periodically(sys, (10, 10, 1))) | ||
|
||
for (site1, site2, offset) in symmetry_equivalent_bonds(sys_inhom, Bond(1,1,[1,0,0])) | ||
noise = randn()/3 | ||
set_exchange_at!(sys_inhom, 1.0 + noise, site1, site2; offset) | ||
end | ||
|
||
minimize_energy!(sys_inhom, maxiters=5_000) | ||
plot_spins(sys_inhom; color=[S[3] for S in sys_inhom.dipoles], ndims=2) | ||
|
||
swt = SpinWaveTheoryKPM(sys_inhom; measure=ssf_perp(sys_inhom), tol=0.01) | ||
res = intensities(swt, path; energies, kernel) | ||
plot_intensities(res) | ||
|
||
set_field!(sys_inhom, [0, 0, 7.5]) | ||
randomize_spins!(sys_inhom) | ||
minimize_energy!(sys_inhom) | ||
|
||
energies = range(0.0, 9.0, 150) | ||
swt = SpinWaveTheoryKPM(sys_inhom; measure=ssf_perp(sys_inhom), tol=0.1) | ||
res = intensities(swt, path; energies, kernel) | ||
plot_intensities(res) | ||
|
||
for site in eachsite(sys_inhom) | ||
noise = randn()/6 | ||
sys_inhom.gs[site] = [1 0 0; 0 1 0; 0 0 1+noise] | ||
end | ||
|
||
swt = SpinWaveTheoryKPM(sys_inhom; measure=ssf_perp(sys_inhom), tol=0.1) | ||
res = intensities(swt, path; energies, kernel) | ||
plot_intensities(res) | ||
|
||
set_field!(sys, [0, 0, 7.5]) | ||
randomize_spins!(sys) | ||
minimize_energy!(sys) | ||
swt = SpinWaveTheory(sys; measure=ssf_perp(sys)) | ||
res = intensities(swt, path; energies, kernel) | ||
plot_intensities(res) |
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