4 Numerical method

The effective field now contains an additional term due to the Lagrange constraint. For the three components of the Lagrange parameter the set of equations (2.24) is augmented by the equations . The stationary points found with this method are also stationary points of the Hamiltonian (2.1), since for these points . However, if the system has many metastable states, only part of these points, compatible with the behavior assumed by the direction of the net magnetization is determined. More precisely, in individual small particles where the exchange interaction is dominating, the deviations from the collinear state are small and thereby the individual spins adiabatically adjust to the net direction when the latter is rotated. In this case, it is possible to define a net magnetization and parameterize it, e.g., in the spherical system of coordinates as . The advantage of this technique is that it can produce highly non-collinear multi-dimensional stationary points [74,87,99]. Accordingly, in the present work and unlike the studies presented in Ref. [14], spin non-collinearities are taken into account. Moreover, in order to check the correct loci of the saddle points, we computed the eigenvalues and gradient of the Hessian matrix associated with the Hamiltonian (2.1).

Rocio Yanes