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7. Long-range dipole interactions

This example shows how long-range dipole-dipole interactions can affect a spin wave calculation. These interactions can be included two ways: Ewald summation or in real-space with a distance cutoff. The study follows Del Maestro and Gingras, J. Phys.: Cond. Matter, 16, 3339 (2004).

using Sunny, GLMakie

Create a pyrochlore crystal from spacegroup 227.

units = Units(:K, :angstrom)
latvecs = lattice_vectors(10.19, 10.19, 10.19, 90, 90, 90)
positions = [[1/8, 1/8, 1/8]]
cryst = Crystal(latvecs, positions, 227, setting="1")
view_crystal(cryst)
Example block output

Create a system and reshape to the primitive cell, which contains four atoms. Add antiferromagnetic nearest neighbor exchange interactions.

primitive_cell = [1/2 1/2 0; 0 1/2 1/2; 1/2 0 1/2]
sys = System(cryst, [1 => Moment(s=7/2, g=2)], :dipole; seed=0)
sys = reshape_supercell(sys, primitive_cell)
J1 = 0.304 # (K)
set_exchange!(sys, J1, Bond(1, 2, [0,0,0]))

Create a copy of the system and enable long-range dipole-dipole interactions using Ewald summation.

sys_lr = clone_system(sys)
enable_dipole_dipole!(sys_lr, units.vacuum_permeability)

Create a copy of the system and add long-range dipole-dipole interactions up to a 5 Å cutoff distance.

sys_lr_cut = clone_system(sys)
modify_exchange_with_truncated_dipole_dipole!(sys_lr_cut, 5.0, units.vacuum_permeability)

Find an energy minimizing spin configuration accounting for the long-range dipole-dipole interactions. This will arbitrarily select from a discrete set of possible ground states based on the system seed.

randomize_spins!(sys_lr)
minimize_energy!(sys_lr)
plot_spins(sys_lr; ghost_radius=8, color=[:red, :blue, :yellow, :purple])
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Copy this configuration to the other two systems. Note that the original sys has a continuum of degenerate ground states.

sys.dipoles .= sys_lr.dipoles
sys_lr_cut.dipoles .= sys_lr.dipoles;

Calculate dispersions for the three systems. The high-symmetry $𝐪$-points are specified in reciprocal lattice units with respect to the conventional cubic cell.

qs = [[0,0,0], [0,1,0], [1,1/2,0], [1/2,1/2,1/2], [3/4,3/4,0], [0,0,0]]
labels = ["Γ", "X", "W", "L", "K", "Γ"]
path = q_space_path(cryst, qs, 500; labels)

measure = ssf_trace(sys)
swt = SpinWaveTheory(sys; measure)
res1 = intensities_bands(swt, path)

swt = SpinWaveTheory(sys_lr; measure)
res2 = intensities_bands(swt, path)

swt = SpinWaveTheory(sys_lr_cut; measure)
res3 = intensities_bands(swt, path);

Create a panel corresponding to Fig. 2 of Del Maestro and Gingras. Dashed lines show the effect of truncating dipole-dipole interactions at 5 Å. The Del Maestro and Gingras paper underreported the energy scale by a factor of two, and requires slight corrections to its third dispersion band.

fig = Figure(size=(768, 300))
plot_intensities!(fig[1, 1], res1; units, title="Without long-range dipole")
ax = plot_intensities!(fig[1, 2], res2; units, title="With long-range dipole")
for c in eachrow(res3.disp)
    lines!(ax, eachindex(c), c; linestyle=:dash, color=:black)
end
fig
Example block output