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

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)]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 \documentclass[12pt]{minimal}\begin{document}$(\Gamma ,\mathcal {E})$\end{document}(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 \documentclass[12pt]{minimal}\begin{document}${\tt G}$\end{document}G (i.e., a Euclidean space \documentclass[12pt]{minimal}\begin{document}${\mathbb {R}}^{d^{\prime }}$\end{document}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 \documentclass[12pt]{minimal}\begin{document}$\mathfrak {G}$\end{document}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)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 \documentclass[12pt]{minimal}\begin{document}$\mathfrak {G}$\end{document}G (defined in the text) is \documentclass[12pt]{minimal}\begin{document}${\tt G}$\end{document}G-invariant. An example is given where the interaction potential is singular and there exists a Gibbs state which is not \documentclass[12pt]{minimal}\begin{document}${\tt G}$\end{document}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).