Low-Temperature Phase Diagrams of Quantum Lattice Systems. II. Convergent Perturbation Expansions and Stability in Systems with Infinite Degeneracy

We study groundstates and low-temperature phases of quantum lattice systems in statistical mechanics: quantum spin systems and fermionic or bosonic lattice gases. The Hamiltonians of such systems have the form H = H0 + tV, where H0 is a classical Hamiltonian, V is a quantum perturbation, and t is the perturbation parameter. Conventional methods to study such systems cannot be used when H0 has infinitely many groundstates. We construct a unitary conjugation transforming H to a form that enables us to find its lowenergy spectrum (to some finite order > 1 in t) and to understand how the perturbation tV lifts the degeneracy of the groundstate energy of H0. The purpose of the unitary conjugation is to cast H in a form that enables us to determine the low-temperature phase diagram of the system. Our main tools are a generalization of a form of Rayleigh-Ritz analytic perturbation theory analogous to Nekhoroshev’s form of classical perturbation theory and an extension of Pirogov-Sinai theory.