Quantum spin systems at positive temperature

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 S satisfy beta << root S. 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 S >> 1. 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.