A quantum Mermin--Wagner theorem for a generalized Hubbard model on a 2D graph

This paper is the second in a series of papers considering symmetry properties of a bosonic quantum system over an 2D graph, with continuous spins, in the spirit of the Mermin–Wagner theorem. Here we consider bosonic systems on bi-dimensional graphs where particles can jump from a vertex to another (a generalized Hubbard model). The Feynman–Kac representation is used for proving that if the local Hamiltonians are invariant under a continuous group of transformations $\mathtt{G}$ (a Euclidean space or a torus of dimension $d^{\mathrm{\prime }}$ acting on a torus of dimension $d\ge d^{\mathrm{\prime }}$) then any infinite-volume Gibbs state from a certain class (introduced in the paper) is also $\mathtt{G}$-invariant.