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 H1=L2(M), where M is a d-dimensional unit torus M=Rd/Zd with a flat metric. The phase space of k spins is Hk=L2sym(Mk), the subspace of L2(Mk) formed by functions symmetric under the permutations of the arguments. The Fock space H=⊕k=0,1,.Hk yields the phase space of a system of a varying (but finite) number of particles. We associate a space H≃H(i) with each vertex i∈Γ of a graph (Γ,ℰ) satisfying a special bidimensionality property. (Physically, vertex i 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) -Δ/2, the minus a half of the Laplace operator on M, 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 U(1)(x), x∈M, describing a field generated by a heavy atom, (b) two-body potentials U(2)(x,y), x,y∈M, showing the interaction between pairs of particles belonging to the same atom, and (c) two-body potentials V(x,y), x,y∈M, scaled along the graph distance d(i,j) between vertices i,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 group G acts on M, represented by a Euclidean space or torus of dimension d'≤d, preserving the metric and the volume in M. Furthermore, we suppose that the potentials U(1), U(2), and V are G-invariant. The result of the paper is that any (appropriately defined) Gibbs states generated by the above Hamiltonian is G-invariant, provided that the thermodynamic variables (the fugacity z and the inverse temperature β) satisfy a certain restriction. The definition of a Gibbs state (and its analysis) is based on the Feynman-Kac representation for the density matrices.