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 (Year: 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 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 (i.e., a Euclidean space 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 ). 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 (Year: 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 (Year: 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 (Year: 1981)10.1007/BF01208901; B. Simon and A. Sokal, “Rigorous entropy-energy arguments,” J. Stat. Phys.25, 679–694 (Year: 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 (Year: 2002)]10.1007/s002200200627 in combination with the Feynman–Kac representation, to prove that any state lying in the class (defined in the text) is -invariant. An example is given where the interaction potential is singular and there exists a Gibbs state which is not -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).
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.
I. INTRODUCTION: EXISTENCE AND INVARIANCE OF A LIMITING GIBBS STATE
A. Bi-dimensional graphs
B. The phase space and the group action
C. The Hamiltonian of the model and assumptions about the potential
D. Properties of limiting Gibbs states
II. THE FEYNMAN–KAC (FK) FORMULA AND DOBRUSHIN–LANFORD–RUELLE (DLR) EQUATIONS
A. The Feynman–Kac representation for the partition function
B. The FK representation for the RDMK in a finite volume
C. The FK-DLR equations in a finite volume
III. THE CLASS OF GIBBS STATES IN THE INFINITE VOLUME
A. Definition of the class
B. Properties of class
IV. PROOF OF THE MAIN RESULTS
A. Proof of Theorem 3.1
B. Proof of Theorem 3.2
C. Proof of Theorem 3.4 and Lemma 1.1
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