Classical ground states of symmetrical Heisenberg spin systems

TL;DR

This paper analyzes classical Heisenberg spin systems with symmetry, using group theory to classify ground states and establish energy bounds, thereby supporting the rotational band structure hypothesis.

Contribution

It provides a complete enumeration of ground states with symmetry and coplanarity, and derives quadratic energy bounds related to total spin S.

Findings

Energy of states depends quadratically on total spin S.

Coplanar S=0 ground states are key to quadratic energy dependence.

Bounds support the rotational band structure hypothesis.

Abstract

We investigate the ground states of classical Heisenberg spin systems which have point group symmetry. Examples are the regular polygons (spin rings) and the seven quasi-regular polyhedra including the five Platonic solids. For these examples, ground states with special properties, e.g. coplanarity or symmetry, can be completely enumerated using group-theoretical methods. For systems having coplanar (anti-) ground states with vanishing total spin we also calculate the smallest and largest energies of all states having a given total spin S. We find that these extremal energies depend quadratically on S and prove that, under certain assumptions, this happens only for systems with coplanar S=0 ground states. For general systems the corresponding parabolas represent lower and upper bounds for the energy values. This provides strong support and clarifies the conditions for the so-called rotational band structure hypothesis which has been numerically established for many quantum spin systems.