A Quantum Mermin-Wagner Theorem for a Generalized Hubbard Model (original) (raw)

2013, Advances in Mathematical Physics

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...

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