Calculation of Chern number spin Hamiltonians for magnetic nano-clusters by DFT methods

By combining field-theoretical methods and ab initio calculations, we construct an effective Hamiltonian with a single giant-spin degree of freedom, which is capable of describing the low-energy spin dynamics of ferromagnetic metal nano-clusters consisting of up to a few tens of atoms. In our procedure, the magnetic moment direction of the Kohn-Sham spin density functional wave function is constrained by means of a penalty functional, which allows us to explore the entire parameter space of directions, and to extract the magnetic anisotropy energy and Berry curvature functionals. The average of the Berry curvature over all magnetization directions is a Chern number-a topological invariant that can only take on values equal to multiples of one-half, which represents the dimension of the Hilbert space of the effective spin system. The spin Hamiltonian is obtained by quantizing the classical anisotropy energy functional, after performing a change of variables to a constant Berry curvature space. The purpose of this paper is to examine the impact of the topological effect from the Berry curvature on the low-energy total-spin-system dynamics. To this end, we study small transition-metal clusters: Co-n (n=2,...,5), Rh-2, Ni-2, Pd-2, MnxNy, and Co3Fe2.