Uniqueness of Gibbs states of a quantum system on graphs

TL;DR

This paper investigates the uniqueness of Gibbs states in infinite quantum systems on graphs, establishing conditions for uniqueness in both deterministic and random potential scenarios.

Contribution

It introduces new criteria for the uniqueness of Gibbs states on graphs with bounded and unbounded degrees, including cases with random potentials.

Findings

Uniqueness of Gibbs states proven for locally finite graphs with small bounded interactions.

Almost sure uniqueness established for graphs with random interaction potentials.

Conditions on interaction potentials and probability distributions are identified for uniqueness.

Abstract

Gibbs states of an infinite system of interacting quantum particles are considered. Each particle moves on a compact Riemannian manifold and is attached to a vertex of a graph (one particle per vertex). Two kinds of graphs are studied: (a) a general graph with locally finite degree; (b) a graph with globally bounded degree. In case (a), the uniqueness of Gibbs states is shown under the condition that the interaction potentials are uniformly bounded by a sufficiently small constant. In case (b), the interaction potentials are random. In this case, under a certain condition imposed on the probability distribution of these potentials the almost sure uniqueness of Gibbs states has been shown.