96-226 Datta N., Fern\'andez R., Froehlich J., Rey-Bellet L.

Low-temperature phase diagrams of quantum lattice systems. II. Convergent perturbation expansions and stability in systems with infinite degeneracy. (208K, LaTeX) May 22, 96

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Abstract. 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 $$ H\,=\,H_0\,+\,tV, $$ where $H_0$ 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 $H_0$ has infinitely many groundstates. We construct a unitary conjugation transforming $H$ to a form that enables us to find its low-energy spectrum (to some finite order $>1$ in $t$) and to understand how the perturbation $tV$ lifts the degeneracy of the groundstate energy of $H_0$. 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.

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