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Effective Hamiltonians and Phase Diagrams for Tight-Binding Models

We present rigorous results for several variants of the Hubbard model in the strong-coupling regime. We establish a mathematically controlled perturbation expansion which shows how previously proposed effective interactions are, in fact, leading-order terms of well defined (volume-independent) unitarily equivalent interactions. In addition, in the very asymmetric (Falicov-Kimball) regime, we are able to apply recently developed phase-diagram technology (quantum Pirogov-Sinai theory) to conclude that the zero-temperature phase diagrams obtained for the leading classical part remain valid, except for thin excluded regions and small deformations, for the full-fledged quantum interaction at zero or small temperature. Moreover, the phase diagram is stable upon the addition of arbitrary, but sufficiently small, further quantum terms that do not break the ground-states symmetries. This generalizes and unifies a number of previous results on the subject; in particular published results on the zero-temperature phase diagram of the Falikov-Kimball model (with and without magnetic flux) are extended to small temperatures and/or small ionic hopping. We give explicit expressions for the first few orders, in the hopping amplitude, of these equivalent interactions, and we describe the resulting phase diagram. Our approach, however, yields algorithms to compute arbitrary high orders.