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

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. In the model considered here (quantum rotators) the phase space of a single spin is a $d-$dimensional torus, 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. We assume that the interaction potential is C$^2$-smooth and invariant under the action of a connected Lie group ${\ttG}$. 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 $\fG$). This class contains the so-called limit Gibbs states, with or without boundary conditions. We use ideas and techniques originated from various past papers, in combination with the Feynman--Kac representation, to prove that any state lying in the class $\fG$ (defined in the text) is ${\ttG}$-invariant. An example is given where the interaction potential is singular and there exists a Gibbs state which is not ${\ttG}$-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 of the graph to one of its neighbors (a generalized Hubbard model).