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

Author(s)

Kelbert, Mark ^{[1, 2]} ; Suhov, Yurii ^{[1, 3, 4]}

Affiliation

^[1] Univ Sao Paulo, IME, BR-05508 Sao Paulo - Brazil

^[2] Swansea Univ, Dept Math, Swansea SA2 8PP, West Glamorgan - Wales

^[3] Univ Cambridge, DPMMS, StatsLab, Cambridge CB2 1TN - England

^[4] RAS, IITP, Moscow 117901 - Russia

• Document type: Journal article
• Source: Journal of Mathematical Physics; v. 54, n. 3 MAR 2013.
• Web of Science Citations: 3

Abstract

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 (Gamma, epsilon) 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 R-d' 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 B). 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. Frohlich 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 B (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 is an element of Gamma to one of its neighbors (a generalized Hubbard model). (C) 2013 American Institute of Physics. {[}http://dx.doi.org/10.1063/1.4790885] (AU)

• FAPESP's process: 12/04372-7 - Probabilistic aspects of causal dynamical triangulations
• Grantee: Anatoli Iambartsev
• Support Opportunities: Regular Research Grants
• FAPESP's process: 11/20133-0 - Absence of continuous symmetry-breaking in 2-dimensional quantum systems
• Grantee: Anatoli Iambartsev
• Support Opportunities: Research Grants - Visiting Researcher Grant - International