A class of quantum lattice models is considered, with Hamiltonians consisti
ng of a classical (diagonal) part and a small off-diagonal part (e.g. hoppi
ng terms), In some cases when the classical part has an infinite degeneracy
of ground states, the quantum perturbation may stabilize some of them. The
mechanism of this stabilization stems from effective potential created by
the quantum perturbation.
Conditions are found when this strategy can be rigorously controlled and th
e low temperature phase diagram of the full quantum model can be proven to
be a small deformation of the zero temperature phase diagram of the classic
al part with the effective potential added. As illustrations we discuss the
asymmetric Hubbard model and the Bose-Hubbard model.