A Quantum Mermin-Wagner Theorem for a Generalized Hubbard Model

Abstract

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 isℋ1=L2(M),whereMis ad-dimensional unit torusM=ℝd/ℤdwith a flat metric. The phase space ofkspins isℋk=L2sym(Mk), the subspace ofL2(Mk)formed by functions symmetric under the permutations of the arguments. The Fock spaceH=⊕k=0,1,…ℋkyields the phase space of a system of a varying (but finite) number of particles. We associate a spaceH≃H(i)with each vertexi∈Γof a graph(Γ,ℰ)satisfying a special bidimensionality property. (Physically, vertexirepresents 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)-Δ/2, the minus a half of the Laplace operator onM, 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 potentialsU(1)(x),x∈M, describing a field generated by a heavy atom, (b) two-body potentialsU(2)(x,y),x,y∈M, showing the interaction between pairs of particles belonging to the same atom, and (c) two-body potentialsV(x,y),x,y∈M, scaled along the graph distanced(i,j)between verticesi,j∈Γ, 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 groupGacts onM, represented by a Euclidean space or torus of dimensiond'≤d, preserving the metric and the volume inM. Furthermore, we suppose that the potentialsU(1),U(2), andVareG-invariant. The result of the paper is that any (appropriately defined) Gibbs states generated by the above Hamiltonian isG-invariant, provided that the thermodynamic variables (the fugacityzand the inverse temperatureβ) satisfy a certain restriction. The definition of a Gibbs state (and its analysis) is based on the Feynman-Kac representation for the density matrices.