A Quantum Mermin-Wagner Theorem for a Generalized Hubbard Model

This paper is the second in a series of papers considering symmetry properties of bosonic quantum systems over 2D graphs, with continuous spins, in the spirit of the Mermin-Wagner theorem. In the model considered here the phase space of a single spin is where is a -dimensional unit torus with a flat metric. The phase space of spins is , the subspace of formed by functions symmetric under the permutations of the arguments. The Fock space yields the phase space of a system of a varying (but finite) number of particles. We associate a space with each vertex of a graph satisfying a special bidimensionality property. (Physically, vertex represents a heavy “atom” or “ion” that does not move but attracts a number of “light” particles.) The kinetic energy part of the Hamiltonian includes (i) , the minus a half of the Laplace operator on , responsible for the motion of a particle while “trapped” by a given atom, and (ii) an integral term describing possible “jumps” where a particle may join another atom. The potential part is an operator of multiplication by a function (the potential energy of a classical configuration) which is a sum of (a) one-body potentials , , describing a field generated by a heavy atom, (b) two-body potentials , , showing the interaction between pairs of particles belonging to the same atom, and (c) two-body potentials , , scaled along the graph distance between vertices , which gives the interaction between particles belonging to different atoms. The system under consideration can be considered as a generalized (bosonic) Hubbard model. We assume that a connected Lie group acts on , represented by a Euclidean space or torus of dimension , preserving the metric and the volume in . Furthermore, we suppose that the potentials , , and are -invariant. The result of the paper is that any (appropriately defined) Gibbs states generated by the above Hamiltonian is -invariant, provided that the thermodynamic variables (the fugacity and the inverse temperature ) satisfy a certain restriction. The definition of a Gibbs state (and its analysis) is based on the Feynman-Kac representation for the density matrices. 1. Introduction 1.1. Basic Facts on Bi-Dimensional Graphs As in [1], we suppose that the graph has been given, with the set of vertices and the set of edges . The graph has the property that whenever edge , the reversed edge belongs to as well. Furthermore, graph is without multiple edges and has a bounded degree; that is, the number of edges with a fixed initial or terminal vertex is uniformly bounded: The bi-dimensionality property is