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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}[[0.21110511457808598 0.3599329917941948 -0.9087809813244627; 0.45976640844090383 0.7838990506987529 0.41727344509679337; 0.8625830280432867 -0.5059155262804995 -1.193206474527161e-12], [0.010108140570072291 -0.23959104486696509 0.9708212794916331; -0.04092171407215044 0.9699584372699529 0.23980417695742584; -0.9991112244448127 -0.042151645144492045 -1.2413901537958928e-12], [0.9192390230982288 -0.3823133969412315 0.09400045176843888; -0.08679319027802845 0.03609746603340117 0.9955721546263383; -0.3840137504501319 -0.9233273739390726 6.305461878396071e-12], [0.8240024245768808 0.14863927847965328 0.5467415926966508; 0.5380576207864004 0.09705856942150236 -0.8373013978344535; -0.17752183247448894 0.984116862468528 -4.888028700479682e-12], [0.8249614968619478 -0.5573171841624016 0.09400045176799222; -0.07789164555557104 0.052621065033234785 0.9955721546263804; -0.5597958737325063 -0.8286305447858292 5.48056617109958e-12], [0.8371073023733155 -0.018027621272290618 0.5467415926967085; 0.5466148521204988 -0.011771687466920467 -0.8373013978344157; 0.021530623645310325 0.9997681892546112 -3.978089909496904e-12]], StaticArraysCore.SVector{3, Float64}[[0.21110511457808598, 0.3599329917941948, -0.9087809813244627] [0.010108140570072291, -0.23959104486696509, 0.9708212794916331] … [0.8249614968619478, -0.5573171841624016, 0.09400045176799222] [0.8371073023733155, -0.018027621272290618, 0.5467415926967085]; [0.45976640844090383, 0.7838990506987529, 0.41727344509679337] [-0.04092171407215044, 0.9699584372699529, 0.23980417695742584] … [-0.07789164555557104, 0.052621065033234785, 0.9955721546263804] [0.5466148521204988, -0.011771687466920467, -0.8373013978344157]; [0.8625830280432867, -0.5059155262804995, -1.193206474527161e-12] [-0.9991112244448127, -0.042151645144492045, -1.2413901537958928e-12] … [-0.5597958737325063, -0.8286305447858292, 5.48056617109958e-12] [0.021530623645310325, 0.9997681892546112, -3.978089909496904e-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), [0.21409750546129427, -4.674164034085471e-13, 0.892951247257677], 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)