2 Quantum Lattice Models at Intermediate

We analyze the free energy and construct the Gibbs-KMS states for a class of quantum lattice systems, at low temperature and when the interactions are almost diagonal in a suitable basis. We study systems with continuous symmetry, but our results are valid for discrete symmetry breaking only. Such phase transitions occur at intermediate temperatures where the continuous symmetry is not broken, while at very low temperature continuous symmetry breaking may occur. 1. Introduction In this paper we study the low temperature phase diagram for a class of quantum lattice systems. Starting with PS, Sin], Pirogov-Sinai theory has evolved KP, Zah, BKL, BS, BI, BK] into a very powerful tool to study the pure phases, their coexistence and the rst-order phase transitions in classical spin systems at low temperature. In recent years large part of the Pirogov-Sinai theory has been extended to quantum systems Pir, BKU, DFF, DFFR, KU], including quantum spin systems as well as fermionic and bosonic lattice gases and applied to a variety of models FR, DFF2, GKU] to describe insulating phases associated with discrete symmetry breaking. Here we formulate the Pirogov-Sinai theory focusing on linear functionals that are tangent to the free energy. This allows both to avoid the diiculties associated with boundary conditions, and to make the link with the notion of KMS states, using well-known results Ara, Isr, Sim]. We reformulate results of BKU, DFF, DFFR, KU] in this framework, and extend the theory to a class of models where discrete symmetries are broken at intermediate temperatures. This applies in particular to some systems with continuous symmetries. For this, we consider the restricted ensembles introduced in BKL] that are very useful to analyze phases which are associated to a family of conngurations rather than to a single connguration. The models that we consider have Hamiltonians, for nite volumes , of the form