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

TL;DR

This paper extends the Mermin--Wagner theorem to a generalized bosonic Hubbard model on 2D graphs, showing that continuous symmetries are preserved in the infinite-volume Gibbs states.

Contribution

It proves a quantum Mermin--Wagner theorem for a generalized Hubbard model on 2D graphs, demonstrating symmetry invariance of Gibbs states under continuous transformations.

Findings

Symmetry invariance of Gibbs states under group ${ t G}$

Extension of Mermin--Wagner theorem to bosonic systems on 2D graphs

Use of Feynman--Kac representation for proof

Abstract

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 ${\tt G}$ (a Euclidean space or a torus of dimension $d'$ acting on a torus of dimension $d\geq d'$) then any infinite-volume Gibbs state from a certain class (introduced in the paper) is also ${\tt G}$-invariant.