Thermal phase transitions on spin-orbit coupled magnets

We study thermal phase transitions of models of XY spins with generic direction dependent interactions, which preserve as symmetries only a combination of lattice and spin rotations in both two and three dimensions. Our models may be thought of as generalizations of the well studied classical versions of the compass models. Using both renormalization group calculations and Monte Carlo simulations we present a theory for the continuous transitions found in our models. Apart from their applications to magnets with spin-orbit coupling we argue that such models are effective theories of the thermal and quantum melting of valence bond solid order in quantum magnets.