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

TL;DR

This paper develops a method combining DFT and topological concepts to derive effective spin Hamiltonians for magnetic nano-clusters, revealing the influence of Berry curvature and Chern numbers on their low-energy spin dynamics.

Contribution

It introduces a novel approach to construct topologically-informed spin Hamiltonians from ab-initio calculations for nanoclusters.

Findings

Chern numbers are quantized and relate to the Hilbert space dimension.

Berry curvature significantly influences spin dynamics.

Effective Hamiltonians accurately describe low-energy magnetic behavior.

Abstract

Combining field-theoretical methods and ab-initio calculations, we construct an effective Hamiltonian with a single giant-spin degree of freedom, capable of the describing the low-energy spin dynamics of ferromagnetic metal nanoclusters consisting of up to a few tens of atoms. In our procedure, the magnetic moment direction of the Kohn-Sham SDFT wave-function is constrained by means of a penalty functional, allowing 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, representing 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 article 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$_{2}$, Mn$_{x}$N$_{y}$, Co$_{3}$Fe$_{2}$.