  Efficient algorithms for the calculation of minimum energy paths of magnetic
transitions are implemented within the geodesic nudged elastic band (GNEB)
approach. While an objective function is not available for GNEB and a
traditional line search can, therefore, not be performed, the use of limited
memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) and conjugate gradient
algorithms in conjunction with orthogonal spin optimization (OSO) approach is
shown to greatly outperform the previously used velocity projection and
dissipative Landau-Lifschitz dynamics optimization methods. The implementation
makes use of energy weighted springs for the distribution of the discretization
points along the path and this is found to improve performance significantly.
The various methods are applied to several test problems using a
Heisenberg-type Hamiltonian, extended in some cases to include
Dzyaloshinskii-Moriya and exchange interactions beyond nearest neighbours.
Minimum energy paths are found for magnetization reversals in a nano-island,
collapse of skyrmions in two-dimensional layers and annihilation of a chiral
bobber near the surface of a three-dimensional magnet. The LBFGS-OSO method is
found to outperform the dynamics based approaches by up to a factor of 8 in
some cases.
