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 $\text{ttG}$. 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 $\text{fG}$). 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 $\text{fG}$ (defined in the text) is $\text{ttG}$-invariant. An example is given where the interaction potential is singular and there exists a Gibbs state which is not $\text{ttG}$-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).