Dynamics and equilibrium states of infinite systems of lattice bosons

TL;DR

This paper studies the dynamics of infinite lattice boson systems, establishing automorphism groups, propagation bounds, and the existence of equilibrium KMS states, advancing the mathematical understanding of quantum many-body systems.

Contribution

It introduces a new framework for analyzing infinite lattice boson dynamics using $C^*$-algebras and proves the existence of KMS equilibrium states.

Findings

Dynamics define a group of automorphisms on a $C^*$-algebra.

Propagation bounds of Lieb--Robinson type are derived.

Finite-volume Gibbs states have accumulation points satisfying the KMS condition.

Abstract

We consider the dynamics of systems of lattice bosons with infinitely many degrees of freedom. We show that their dynamics defines a group of automorphisms on a $C^*$--algebra introduced by Buchholz, which extends the resolvent algebra of local field operators. For states that admit uniform bounds on moments of the local particle number, we derive propagation bounds of Lieb--Robinson type. Using these bounds, we show that the dynamics of local observables gives rise to a strongly continuous unitary group in the GNS representation. Moreover, accumulation points of finite-volume Gibbs states satisfy the KMS condition with respect to this group. This, in particular, proves the existence of KMS states.