TY - JOUR
T1 - A quantum Mermin-Wagner theorem for quantum rotators on two-dimensional graphs
AU - Kelbert, Mark
AU - Suhov, Yurii
N1 - Funding Information:
This work has been conducted under Grant No. 2011/20133-0 provided by the FAPESP, Grant No. 2011.5.764.45.0 provided by The Reitoria of the Universidade de São Paulo and Grant No. 2012/04372-7 provided by the FAPESP. The authors express their gratitude to NUMEC and IME, Universidade de São Paulo, Brazil, for the warm hospitality. The authors thank the referees for remarks and suggestions.
PY - 2013/3/6
Y1 - 2013/3/6
N2 - 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)]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 (Γ,ε) 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 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)10.1007/BF01609432; C.-E. Pfister, "On the symmetry of the Gibbs states in two-dimensional lattice systems," Commun. Math. Phys.79, 181-188 (1981)10.1007/BF01942060; J. Fröhlich and C. Pfister, "On the absence of spontaneous symmetry breaking and of crystalline ordering in two-dimensional systems," Commun. Math. Phys.81, 277-298 (1981)10.1007/BF01208901; B. Simon and A. Sokal, "Rigorous entropy-energy arguments," J. Stat. Phys.25, 679-694 (1981)10.1007/BF01022362; 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)]10.1007/s002200200627 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 ∈ Γ to one of its neighbors (a generalized Hubbard model).
AB - 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)]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 (Γ,ε) 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 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)10.1007/BF01609432; C.-E. Pfister, "On the symmetry of the Gibbs states in two-dimensional lattice systems," Commun. Math. Phys.79, 181-188 (1981)10.1007/BF01942060; J. Fröhlich and C. Pfister, "On the absence of spontaneous symmetry breaking and of crystalline ordering in two-dimensional systems," Commun. Math. Phys.81, 277-298 (1981)10.1007/BF01208901; B. Simon and A. Sokal, "Rigorous entropy-energy arguments," J. Stat. Phys.25, 679-694 (1981)10.1007/BF01022362; 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)]10.1007/s002200200627 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 ∈ Γ to one of its neighbors (a generalized Hubbard model).
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U2 - 10.1063/1.4790885
DO - 10.1063/1.4790885
M3 - Article
AN - SCOPUS:84875874235
VL - 54
JO - Journal of Mathematical Physics
JF - Journal of Mathematical Physics
SN - 0022-2488
IS - 3
M1 - 033301
ER -