Phase transitions in classical spin systems are well understood,
however phase transitions in quantum spin systems are not ... at least
that is what most mathematicians would say. (If anything, they might
question how well we even understand classical spin systems.) Physicists,
on the other hand, say that if the classical model has a phase transition,
then the quantum model does as well. We will prove that, for some special
models. The main tools are reflection positivity, coherent states, and a
new result which generalizes the Berezin-Lieb inequality to the level of
matrix elements.