Quantum spin systems at positive temperature

TL;DR

This paper introduces a new method linking classical and quantum spin models to demonstrate phase transitions in quantum systems at low temperatures, especially for large spin magnitudes.

Contribution

It establishes a framework connecting classical phase transition proofs to quantum models under specific conditions, extending Berezin-Lieb inequalities to matrix elements.

Findings

Proves phase transitions in quantum orbital-compass and 120-degree models.

Shows symmetry breaking at low temperatures despite classical ground state degeneracy.

Applies the theory to models with large quantum spins rom 1 to ew.

Abstract

We develop a novel approach to phase transitions in quantum spin models based on a relation to their classical counterparts. Explicitly, we show that whenever chessboard estimates can be used to prove a phase transition in the classical model, the corresponding quantum model will have a similar phase transition, provided the inverse temperature $\beta$ and the magnitude of the quantum spins $\CalS$ satisfy $\beta\ll\sqrt\CalS$. From the quantum system we require that it is reflection positive and that it has a meaningful classical limit; the core technical estimate may be described as an extension of the Berezin-Lieb inequalities down to the level of matrix elements. The general theory is applied to prove phase transitions in various quantum spin systems with $\CalS\gg1$. The most notable examples are the quantum orbital-compass model on $\Z^2$ and the quantum 120-degree model on $\Z^3$ which are shown to exhibit symmetry breaking at low-temperatures despite the infinite degeneracy of their (classical) ground state.