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SW18 - Distorted kagome

This is a Sunny port of SpinW Tutorial 18, originally authored by Goran Nilsen and Sandor Toth. This tutorial illustrates spin wave calculations for KCu₃As₂O₇(OD)₃. The Cu ions are arranged in a distorted kagome lattice and exhibit an incommensurate helical magnetic order, as described in G. J. Nilsen, et al., Phys. Rev. B 89, 140412 (2014). The model follows Toth and Lake, J. Phys.: Condens. Matter 27, 166002 (2015).

using Sunny, GLMakie

Build the distorted kagome crystal, with spacegroup 12 ("C 1 2/m 1" setting).

units = Units(:meV, :angstrom)
latvecs = lattice_vectors(10.2, 5.94, 7.81, 90, 117.7, 90)
positions = [[0, 0, 0], [1/4, 1/4, 0]]
types = ["Cu1", "Cu2"]
cryst = Crystal(latvecs, positions, "C 1 2/m 1"; types)
view_crystal(cryst)

Define the interactions.

moments = [1 => Moment(s=1/2, g=2), 3 => Moment(s=1/2, g=2)]
sys = System(cryst, moments, :dipole)
J   = -2
Jp  = -1
Jab = 0.75
Ja  = -J/.66 - Jab
Jip = 0.01
set_exchange!(sys, J, Bond(1, 3, [0, 0, 0]))
set_exchange!(sys, Jp, Bond(3, 5, [0, 0, 0]))
set_exchange!(sys, Ja, Bond(3, 4, [0, 0, 0]))
set_exchange!(sys, Jab, Bond(1, 2, [0, 0, 0]))
set_exchange!(sys, Jip, Bond(3, 4, [0, 0, 1]))

Use minimize_spiral_energy! to optimize the generalized spiral order. This determines the propagation wavevector k and fits the spin values within the unit cell. One must provide a fixed axis perpendicular to the polarization plane. For this system, all interactions are rotationally invariant and the axis vector is arbitrary. In other cases, a good axis will frequently be determined from symmetry considerations.

axis = [0, 0, 1]
randomize_spins!(sys)
k = minimize_spiral_energy!(sys, axis; k_guess=randn(3))
plot_spins(sys; ndims=2)

If successful, the optimization process will find one two propagation wavevectors, ±k_ref, with opposite chiralities. In this system, the spiral_energy_per_site is independent of chirality.

k_ref = [0.785902495, 0.0, 0.107048756]
k_ref_alt = [1, 0, 1] - k_ref
@assert isapprox(k, k_ref; atol=1e-6) || isapprox(k, k_ref_alt; atol=1e-6)
@assert spiral_energy_per_site(sys; k, axis) ≈ -0.78338383838

Check the energy with a real-space calculation using a large magnetic cell. First, we must determine a lattice size for which k becomes approximately commensurate.

suggest_magnetic_supercell([k_ref]; tol=1e-3)

Possible magnetic supercell in multiples of lattice vectors:

    [1 0 -7; 0 1 0; 2 0 14]

for the rationalized wavevectors:

    [[11/14, 0, 3/28]]

Resize the system as suggested and perform a real-space calculation. Working with a commensurate wavevector increases the energy slightly. The precise value might vary from run-to-run due to trapping in a local energy minimum.

new_shape = [14 0 1; 0 1 0; 0 0 2]
sys2 = reshape_supercell(sys, new_shape)
randomize_spins!(sys2)
minimize_energy!(sys2)
energy_per_site(sys2)

-0.7833837597216421

Return to the original system (with a single chemical cell) and construct SpinWaveTheorySpiral for calculations on the incommensurate spiral phase.

measure = ssf_perp(sys; apply_g=false)
swt = SpinWaveTheorySpiral(sys; measure, k, axis)

SpinWaveTheorySpiral(SpinWaveTheory(System([Dipole mode], Supercell (1×1×1)×6, Energy per site -0.3888), Sunny.SWTDataDipole(StaticArraysCore.SMatrix{3, 3, Float64, 9}[[0.36704042217671334 -0.5325964729196501 0.7626416757048414; -0.43276305627303785 0.6279637430991603 0.6468212075049112; -0.8234060150457729 -0.5674526714946723 -2.1630973088396975e-12], [-0.0051859508842808846 -0.030357100044005155 -0.9995256636977082; 0.16831223075204146 0.9852525287303838 -0.030796876621760064; 0.9857200915537317 -0.1683921052404258 4.9600936171553064e-12], [0.9312694458197368 0.06258711015391089 -0.3589151333187036; 0.3581073168096049 0.02406704328761839 0.9333701983001264; 0.06705496946555023 -0.9977492826707389 -1.9104212486199887e-11], [0.726035759682319 -0.6183917828897122 -0.3007718047241153; -0.22897344057313948 0.19502523433930366 -0.9536961368711728; 0.6484159461137713 0.7612862541944279 2.0343343846361475e-11], [0.9251401981043844 -0.1236751427301049 -0.3589151333143663; 0.3557503958652175 -0.04755763623026746 0.9333701983017944; -0.13250384784270328 -0.9911824909202533 -1.8523787888925855e-11], [0.5887079007018238 -0.7503061569329668 -0.3007718047286583; -0.185663683526612 0.23662771422439982 -0.9536961368697401; 0.7867350279926412 0.6172908518108934 2.0046692254181634e-11]], StaticArraysCore.SVector{3, Float64}[[0.36704042217671334, -0.5325964729196501, 0.7626416757048414] [-0.0051859508842808846, -0.030357100044005155, -0.9995256636977082] … [0.9251401981043844, -0.1236751427301049, -0.3589151333143663] [0.5887079007018238, -0.7503061569329668, -0.3007718047286583]; [-0.43276305627303785, 0.6279637430991603, 0.6468212075049112] [0.16831223075204146, 0.9852525287303838, -0.030796876621760064] … [0.3557503958652175, -0.04755763623026746, 0.9333701983017944] [-0.185663683526612, 0.23662771422439982, -0.9536961368697401]; [-0.8234060150457729, -0.5674526714946723, -2.1630973088396975e-12] [0.9857200915537317, -0.1683921052404258, 4.9600936171553064e-12] … [-0.13250384784270328, -0.9911824909202533, -1.8523787888925855e-11] [0.7867350279926412, 0.6172908518108934, 2.0046692254181634e-11]], Sunny.StevensExpansion[Sunny.StevensExpansion([0.0], [0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], 0), Sunny.StevensExpansion([0.0], [0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], 0), Sunny.StevensExpansion([0.0], [0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], 0), Sunny.StevensExpansion([0.0], [0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], 0), Sunny.StevensExpansion([0.0], [0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], 0), Sunny.StevensExpansion([0.0], [0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], 0)], [0.7071067811865476, 0.7071067811865476, 0.7071067811865476, 0.7071067811865476, 0.7071067811865476, 0.7071067811865476]), MeasureSpec, 1.0e-8), [0.785902494538363, -8.62725556726441e-13, 0.10704875274839978], 3, [0.0, 0.0, 1.0], Matrix{StaticArraysCore.SMatrix{3, 3, ComplexF64, 9}}[[[0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 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0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] … [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im]], [[0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] … [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im]; [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] … [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 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0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im]], [[0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] … [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im]; [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] … [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im]; … ; [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] … [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im]; [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] … [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im]], [[0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] … [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im]; [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] … [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im]; … ; [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] … [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im]; [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] … [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im] [0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im]]])

Plot intensities for a path through $𝐪$-space.

qs = [[0,0,0], [1,0,0]]
path = q_space_path(cryst, qs, 400)
res = intensities_bands(swt, path)
plot_intensities(res; units)

Plot the powder-averaged intensities

radii = range(0, 2, 100) # (1/Å)
energies = range(0, 5, 200)
kernel = gaussian(fwhm=0.05)
res = powder_average(cryst, radii, 400) do qs
    intensities(swt, qs; energies, kernel)
end
plot_intensities(res; units)