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摘要
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 H-1 = L-2 (M), where M is a d-dimensional unit torus M = R-d/Z(d) with a flat metric. The phase space of k spins is H-k = L-2(sym) (M-k), the subspace of L-2 (M-k) formed by functions symmetric under the permutations of the arguments. The Fock space H = circle plus(k=0,1, ...) H-k yields the phase space of a system of a varying (but finite) number of particles. We associate a space H similar or equal to H(i) with each vertex i epsilon Gamma of a graph (Gamma, E) satisfying a special bidimensionality property. (Physically, vertex.. represents a heavy %26quot;atom%26quot; or %26quot;ion%26quot; that does not move but attracts a number of %26quot;light%26quot; particles.) The kinetic energy part of the Hamiltonian includes (i) -Delta/2, the minus a half of the Laplace operator on M, responsible for the motion of a particle while %26quot;trapped%26quot; by a given atom, and (ii) an integral term describing possible %26quot;jumps%26quot; where a particle may join another atom. The potential part is an operator of multiplication by a the potential energy of a classical configuration) which is a sum of (a) one-body potentials U-(1) (x), x epsilon M, describing a field generated by a heavy atom, (b) two-body potentials U-(2) (x,y), x, y epsilon M, showing the interaction between pairs of particles belonging to the same atom, and (c) two-body potentials V(x,y), x, y epsilon M, scaled along the graph distance d(i,j) between vertices i, j epsilon Gamma, 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.., represented by a Euclidean space or torus of dimension d(1) %26lt;= 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 beta) satisfy a certain restriction. The definition of a Gibbs state (and its analysis) is based on the Feynman-Kac representation for the density matrices.
- 出版日期2013
