Definitions

All definitions are described in detail in G. P. Müller et al., Phys. Rev. B 99, 224414 (2019). Here we make brief summaries to give you an overview.

Heisenberg Hamiltonian

The Hamiltonian is defined as

where it is important to note that <ij> denotes the unique pairs of interacting spins i and j.

The quadruplet interaction is defined as

LLG dynamics

Spirit denotes the LLG equation as

γ is the electron gyromagnetic ratio, α is the damping parameter, β is a non-adiabaticity parameter, with

and

If temperature is used, a thermal component is added to the effective magnetic field:

Geodesic nudged elastic band method

The total force is

with the spring force

and the energy gradient force

The corresponding 3-component subvectors need to be orthogonalized with respect to the spins:

The spring forces need to be projected as well

Note the features of “climbing/falling images” and “path shortening”, which are described in detail in the paper.

Minimum mode following method

The mode following force is given by an inversion of the energy gradient force along the mode λ:

To calculate the energy eigenmodes, we calculate the Hessian matrix

Details on this method, the equations and their derivations can be found in G. P. Müller et al., Phys. Rev. Lett. 121, 197202 and [1].

Harmonic transition state theory

The transition rate reads

with

Details on these equations and their derivations can be found in [1].

Topological charge

The topological charge is defined as

On a discrete lattice, this corresponds to

with

Gaussian (test-) Hamiltonian

The Hamiltonian is defined as

[1]: G. P. Müller, Advanced methods for atomic scale spin simulations and application to localized magnetic states. PhD Thesis (2019) (availavle from Univ. of Iceland, RWTH Aachen and FZ Jülich)