  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.
