A Quantum Mermin‐Wagner Theorem for a Generalized Hubbard Model

Mark Kelbert and Yurii Suhov

Advances in Mathematical Physics, 2013, vol. 2013, issue 1

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 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.

Date: 2013
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