  A method for finding minimum energy paths of transitions in magnetic systems
is presented and used to determine the mechanism and estimate the activation
energy of skyrmion and antivortex annihiliation in nano-systems. The path is
optimized with respect to orientation of the magnetic vectors while their
magnitudes are fixed or obtained from separate calculations. The curvature of
the configuration space is taken into account by: (1) using geodesics to
evaluate distances and displacements of the system during the optimization, and
(2) projecting the path tangent and the magnetic force on the tangent space of
the manifold defined by all possible orientations of the magnetic vectors. The
method, named geodesic nudged elastic band (GNEB), and its implementation are
illustrated with calculations of complex transitions involving annihilation and
creation of skyrmion and antivortex states. The lifetime of the latter was
determined within harmonic transition state theory using a noncollinear
extension of the Alexander-Anderson model.
