Theory of ground states for classical Heisenberg spin systems I

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.