AMP Advances in Mathematical Physics 1687-9139 1687-9120 Hindawi Publishing Corporation 637375 10.1155/2013/637375 637375 Research Article A Quantum Mermin-Wagner Theorem for a Generalized Hubbard Model Kelbert Mark mark.kelbert@gmail.com 1,2 Suhov Yurii i.m.soukhov@statslab.cam.ac.uk 2,3,4 Maes Christian 1 Swansea University, Singleton Park, Swansea SA2 8PP UK swansea.ac.uk 2 Instituto de Mathematica e Estatistica, USP, Rua de Matão, 1010, Cidada Universitária 05508-090 São Paulo, SP Brazil usp.br 3 Statistical Laboratory University of Cambridge Wilberforce Road, Cambridge CB3 0WB UK cam.ac.uk 4 IITP RAS Bolshoy Karetny per. 18 Moscow 127994 Russia ras.ru 2013 15 9 2013 2013 20 03 2013 14 05 2013 2013 Copyright © 2013 Mark Kelbert and Yurii Suhov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 1 = L 2 ( M ), where M is a d -dimensional unit torus M = d / d with a flat metric. The phase space of k spins is k = L 2 s y m ( M k ) , the subspace of L 2 ( M k ) formed by functions symmetric under the permutations of the arguments. The Fock space H = k = 0,1 , k 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 , ) 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.

http://dx.doi.org/10.13039/501100001807 São Paulo Research Foundation 2011/20133-0 http://dx.doi.org/10.13039/501100001809 National Natural Science Foundation of China 2012/04372-7 Universidade de São Paulo 2011.5.764.45.0