EFFECTIVE INTERACTIONS DUE TO QUANTUM FLUCTUATIONSRoman

A class of quantum lattice models is considered, with Hamiltonians consisting of a classical (diagonal) part and a small oo-diagonal part (e.g. hopping terms). In some cases when the classical part has an innnite degeneracy of ground states, the quantum perturbation may stabilize some of them. The mechanism of this stabilization stems from eeective potential created by the quantum perturbation. Conditions are found when this strategy can be rigorously controlled and the 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 classical part with the eeective potential added. As illustrations we discuss the asymmetric Hubbard model and the hard-core Bose-Hubbard model. Contents 1. Introduction 1 2. Assumptions and statements 4 2.1. Classical Hamiltonian with quantum perturbation 4 2.2. The eeective potential 6 2.3. Quantum uctuations with less than four transitions 7 2.4. Stability of the dominant states 8 2.5. Characterization of stable phases 10 2.6. Phase diagram 11 3. Examples 13 3.1. The asymmetric Hubbard model 13 3.2. The hard-core Bose-Hubbard model 14 4. Contour representation of a quantum model 16 5. Exponential decay of the weight of the contours 25 6. Expectation values of local observables and construction of pure states 30 References 34