Linear Spin Wave theory

Exercises

1. Compute magnon dispersion of a simple ferromagnet.

2. Compute magnon dispersion of a simple antiferromagnet.

All methods for linear spin-wave theory are grouped in the magnopy.LSWT class.

To create it we require two things

• Spin Hamiltonian

• Directions of spins in the ground state

import numpy as np
import magnopy

An example of the cubic ferromagnet with nearest-neighbor interactions and easy magnetic axis along \(\mathbf{z}\).

spinham = magnopy.examples.cubic_ferro_nn(S=1, J_21=(0, 0, -0.5))

# Get the LSWT object from it
lswt = magnopy.LSWT(spinham=spinham, spin_directions=[[0, 0, 1]])

Once created it can be used to compute parts of the magnon Hamiltonian

Correction to the classical energy

print(lswt.E_2())

-3.5

Coefficients of the one-operator terms

print(lswt.O())

[0.+0.j]

Magnon energies

print(lswt.omega(k=[0, 0, 0]))

[1.+0.j]

Parallelization

Typically one wants to compute magnon energies for a set of k-points. Often for a rather large set of k-points. Magnopy offers simple interface to parallelize the calculations over the k-points using multiprocessing. See magnopy.multiprocess_over_k() for details. For example, to compute magnon energies for the set of k-points (that are given as absolute coordinates in reciprocal space) using two processes use

kpoints = np.linspace([0, 0, 0], [1, 0, 0], 1000)

results = magnopy.multiprocess_over_k(
    kpoints=kpoints,
    relative=False,
    function=lswt.omega,
    number_processors=2,
)

Note that results[i] is equivalent to lswt.omega(k=kpoints[i], relative=False).

print(results[42])

print(lswt.omega(k=kpoints[42], relative=False))

[1.00176727+0.j]
[1.00176727+0.j]