Quantum Spin Systems at Finite Temperature

We develop a novel approach to phase transitions in quantum spin models based on a relation to the corresponding classical spin systems. 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 β and the magnitude of the quantum spins S satisfy β ≪ √ 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 further 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 and the quantum 120degree model on Z which are shown to exhibit symmetry breaking at low-temperatures despite the infinite degeneracy of their (classical) ground state.