A spectral-neighbour representation for vector fields: machine-learning potentials including spin

TL;DR

This paper presents a novel spectral-neighbour representation for vector fields that is invariant under translation and rotation, enabling improved machine-learning models for systems with vector densities like magnetism.

Contribution

It introduces a new formalism based on the power spectrum of combined angular momentum for vector fields, applicable to various machine-learning schemes and physical systems.

Findings

Effective in modeling classical spin Hamiltonians

Compatible with linear and Gaussian ML methods

Captures spin-lattice coupling and energy fluctuations

Abstract

We introduce a translational and rotational invariant local representation for vector fields, which can be employed in the construction of machine-learning energy models of solids and molecules. This allows us to describe, on the same footing, the energy fluctuations due to the atomic motion, the longitudinal and transverse excitations of the vector field, and their mutual interplay. The formalism can then be applied to physical systems where the total energy is determined by a vector density, as in the case of magnetism. Our representation is constructed over the power spectrum of the combined angular momentum describing the local atomic positions and the vector field, and can be used in conjunction with different machine-learning schemes and data taken from accurate ab initio electronic structure theories. We demonstrate the descriptive power of our representation for a range of classical spin Hamiltonian and machine-learning algorithms. In particular, we construct energy models based on both linear Ridge regression, as in conventional spectral neighbour analysis potentials, and gaussian approximation. These are both built to represent a Heisenberg-type Hamiltonian including a longitudinal energy term and spin-lattice coupling.