An efficient and scalable implementation of a method for locating first-order saddle points on the energy surface of a magnetic system is presented, along with several applications in which the mechanisms of various magnetic transitions are identified. The starting point for the iterative search algorithm can be anywhere, even close to a local energy minimum representing an initial state of the system, and the final state need not be specified. Convergence on a saddle point is obtained by inverting the component of the gradient along the minimum mode, thereby effectively transforming the neighborhood of the saddle point to that of a local minimum. The method requires only the lowest two eigenvalues and corresponding eigenvectors of the Hessian of the system's energy and they are found using a quasi-Newton limited-memory Broyden-Fletcher-Goldfarb-Shanno solver for the minimization of the Rayleigh quotient without explicit evaluation of the Hessian. The method is applicable to large systems, as it does not introduce additional scaling overhead to the computational complexity determined by the interactions present in the system. Applications are presented to transitions in systems that reveal significant complexity of coexisting magnetic states, such as skyrmions, skyrmion bags, skyrmion tubes, chiral bobbers, and globules. The identification of new metastable three-dimensional (3D) textures, such as magnetic bobbers with extended equilibrium distance between the base and the terminating Bloch point, and magnetic globules appearing as isolated states in 3D due to magnetostatic interactions, demonstrates the usefulness of the method for the characterization of complex energy surfaces of magnetic systems. When combined with rate theory within the harmonic approximation, the method can be used for simulations of the long timescale dynamics of complex magnetic systems characterized by multiple metastable states.
Schrautzer et al. (Wed,) studied this question.