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A quantum Mermin-Wagner theorem for quantum rotators on two-dimensional graphs. (English) Zbl 1282.82024

Summary: This is the first of a series of papers considering symmetry properties of quantum systems over 2D graphs or manifolds, with continuous spins, in the spirit of the Mermin-Wagner theorem [N. D. Mermin and H. Wagner, Phys. Rev. Lett.17, No. 22, 1133–1136 (1966; doi:10.1103/PhysRevLett.17.1133)]. In the model considered here (quantum rotators), the phase space of a single spin is a \(d\)-dimensional torus \(M\), and spins (or particles) are attached to sites of a graph \((\Gamma ,\mathcal{E})\) satisfying a special bi-dimensionality property. The kinetic energy part of the Hamiltonian is minus a half of the Laplace operator {\(-\Delta\)}/2 on \(M\). We assume that the interaction potential is \(C^2\)-smooth and invariant under the action of a connected Lie group \(G\) (i.e., a Euclidean space \({\mathbb{R}}^{d^{\prime }}\) or a torus \(M'\) of dimension \(d'\leq d\)) on M preserving the flat Riemannian metric. A part of our approach is to give a definition (and a construction) of a class of infinite-volume Gibbs states for the systems under consideration (the class \(\mathfrak{G}\)). This class contains the so-called limit Gibbs states, with or without boundary conditions. We use ideas and techniques originated from papers [R. L. Dobrushin and S. B. Shlosman, Commun. Math. Phys. 42, 31–40 (1975; doi:10.1007/BF01609432); C.-E. Pfister, Commun. Math. Phys. 79, 181–188 (1981; doi:10.1007/BF01942060); J. Fröhlich and C. Pfister, Commun. Math. Phys.81, 277–298 (1981; doi:10.1007/BF01208901); B. Simon and A. Sokal, J. Stat. Phys.25, 679–694 (1981; doi:10.1007/BF01022362); D. Ioffe, S. Shlosman and Y. Velenik, Commun. Math. Phys. 226, No. 2, 433–454 (2002; Zbl 0990.82004)] in combination with the Feynman-Kac representation, to prove that any state lying in the class \(\mathfrak{G}\) (defined in the text) is \({\mathtt G}\)-invariant. An example is given where the interaction potential is singular and there exists a Gibbs state which is not \(G\)-invariant. In the next paper, under the same title we establish a similar result for a bosonic model where particles can jump from a vertex \(i \in\Gamma\) to one of its neighbors (a generalized Hubbard model).{
©2013 American Institute of Physics}

MSC:

82B30 Statistical thermodynamics
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices

Citations:

Zbl 0990.82004
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References:

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