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)