A quantum Mermin-Wagner theorem for quantum rotators on two-dimensional graphs

Kelbert, Mark; Suhov, Yurii
March 2013
Journal of Mathematical Physics;Mar2013, Vol. 54 Issue 3, p033301
Academic Journal
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, 'Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models,' Phys. Rev. Lett. 17, 1133-1136 (1966)]. 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 (Γ,E) satisfying a special bi-dimensionality property. The kinetic energy part of the Hamiltonian is minus a half of the Laplace operator -Δ/2 on M. We assume that the interaction potential is C2-smooth and invariant under the action of a connected Lie group G (i.e., a Euclidean space Rd′ or a torus M′ of dimension d′ <= 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 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, 'Absence of breakdown of continuous symmetry in two-dimensional models of statistical physics,' Commun. Math. Phys. 42, 31-40 (1975); C.-E. Pfister, 'On the symmetry of the Gibbs states in two-dimensional lattice systems,' ibid. 79, 181-188 (1981); J. Fröhlich and C. Pfister, 'On the absence of spontaneous symmetry breaking and of crystalline ordering in two-dimensional systems,' ibid. 81, 277-298 (1981); B. Simon and A. Sokal, 'Rigorous entropy-energy arguments,' J. Stat. Phys. 25, 679-694 (1981); D. Ioffe, S. Shlosman and Y. Velenik, '2D models of statistical physics with continuous symmetry: The case of singular interactions,' Commun. Math. Phys. 226, 433-454 (2002)] in combination with the Feynman-Kac representation, to prove that any state lying in the class G (defined in the text) is 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 ∈ Γ to one of its neighbors (a generalized Hubbard model).


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