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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)

-0.7805426292365657

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}[[0.17838906803448953 0.33500780883548886 -0.9251740962786577; 0.4348412254823169 0.816615097243535 0.3795430035911203; 0.8826610045916967 -0.4700101604999373 -1.4930798945178516e-15], [0.023259307945917695 -0.27856092096915985 0.9601368745666893; -0.07989159017435711 0.9568072698940487 0.2795302883361829; -0.9965321562364606 -0.08320854275079248 -1.3011201525207262e-12], [0.9370227118949787 -0.34522775471175493 0.0529739064980133; -0.04970754804921152 0.018313777234518915 0.9985958968623593; -0.34571317166101034 -0.9383402383677707 4.0734695096896564e-12], [0.7945533827940469 0.17735092045431725 0.5807164307168716; 0.5667692627617544 0.12650761120596846 -0.8141059065597404; -0.21784748031371412 0.9759828253206948 -3.115668663968082e-12], [0.8497584756051684 -0.5245038592492948 0.0529739064991153; -0.04507832064604846 0.027824086283453252 0.9985958968623009; -0.5252413522799905 -0.8509533018063254 -1.995675676726224e-12], [0.813949403470543 0.015962320841620158 0.5807164307157981; 0.5806047942381378 0.011386211441301614 -0.8141059065605062; -0.019607179747144963 0.9998077607732214 2.6248505071587526e-12]], StaticArraysCore.SVector{3, Float64}[[0.17838906803448953, 0.33500780883548886, -0.9251740962786577] [0.023259307945917695, -0.27856092096915985, 0.9601368745666893] … [0.8497584756051684, -0.5245038592492948, 0.0529739064991153] [0.813949403470543, 0.015962320841620158, 0.5807164307157981]; [0.4348412254823169, 0.816615097243535, 0.3795430035911203] [-0.07989159017435711, 0.9568072698940487, 0.2795302883361829] … [-0.04507832064604846, 0.027824086283453252, 0.9985958968623009] [0.5806047942381378, 0.011386211441301614, -0.8141059065605062]; [0.8826610045916967, -0.4700101604999373, -1.4930798945178516e-15] [-0.9965321562364606, -0.08320854275079248, -1.3011201525207262e-12] … [-0.5252413522799905, -0.8509533018063254, -1.995675676726224e-12] [-0.019607179747144963, 0.9998077607732214, 2.6248505071587526e-12]], 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), [0.21409750546099118, -1.713634939040664e-14, 0.8929512472513177], 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)