A Ferromagnetic Lieb-Mattis Theorem

TL;DR

This paper proves a theorem demonstrating ferromagnetic ordering of energy levels in specific quantum spin chains, extending the Lieb-Mattis theorem to ferromagnetic systems and providing new insights into their spectral properties.

Contribution

It establishes a ferromagnetic analogue of the Lieb-Mattis theorem for XXX and XXZ spin chains, showing energy level ordering based on total spin.

Findings

Energy levels decrease monotonically with increasing total spin.

The result applies to all spins in XXX chains and spin-1/2 in XXZ chains.

Provides a new theoretical framework for ferromagnetic systems.

Abstract

We prove ferromagnetic ordering of energy levels for XXX Heisenberg chains of any spin and XXZ spin chains with all spins equal to 1/2. Ferromagnetic ordering means that the minimum energies in the invariant subspaces of fixed total spin are monotone decreasing as a function of the total spin. This result provides a ferromagnetic analogue of the well-known theorem by Lieb and Mattis about ordering of energy levels in antiferromagnetic and ferrimagnetic systems on bipartite graphs.