- 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.
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
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.