Theory of ground states for classical Heisenberg spin systems I

TL;DR

This paper develops a rigorous theoretical framework for identifying ground states in classical finite Heisenberg spin systems, linking them to eigenvectors of a specific symmetric matrix derived from system parameters.

Contribution

It introduces a method to construct all ground states from eigenvectors of a matrix related to coupling constants and Lagrange multipliers, advancing the understanding of spin system ground states.

Findings

Ground states correspond to eigenvectors of a symmetric matrix.

All ground states can be constructed from these eigenvectors.

Rare cases with unphysical dimensions may require extending the theory.

Abstract

We formulate part I of a rigorous theory of ground states for classical, finite, Heisenberg spin systems. The main result is that all ground states can be constructed from the eigenvectors of a real, symmetric matrix with entries comprising the coupling constants of the spin system as well as certain Lagrange parameters. The eigenvectors correspond to the unique maximum of the minimal eigenvalue considered as a function of the Lagrange parameters. However, there are rare cases where all ground states obtained in this way have unphysical dimensions $M>3$ and the theory would have to be extended. Further results concern the degree of additional degeneracy, additional to the trivial degeneracy of ground states due to rotations or reflections. The theory is illustrated by a couple of elementary examples.