Euclidean Gibbs states of interacting quantum anharmonic oscillators

TL;DR

This paper rigorously characterizes the equilibrium states of an infinite system of interacting quantum anharmonic oscillators using Euclidean Gibbs measures, revealing conditions for uniqueness and phase transitions.

Contribution

It provides a complete mathematical description of Euclidean Gibbs states for quantum anharmonic oscillators, including existence, uniqueness, and phase transition conditions.

Findings

The set of Euclidean Gibbs measures is non-empty and compact.

At high temperatures, the Gibbs measure is unique.

Multiple Gibbs measures appear at low temperatures under attractive interactions.

Abstract

A rigorous description of the equilibrium thermodynamic properties of an infinite system of interacting $\nu$-dimensional quantum anharmonic oscillators is given. The oscillators are indexed by the elements of a countable set $\mathbb{L}\subset \mathbb{R}^d$, possibly irregular; the anharmonic potentials vary from site to site. The description is based on the representation of the Gibbs states in terms of path measures -- the so called Euclidean Gibbs measures. It is proven that: (a) the set of such measures $\mathcal{G}^{\rm t}$ is non-void and compact; (b) every $\mu \in \mathcal{G}^{\rm t}$ obeys an exponential integrability estimate, the same for the whole set $\mathcal{G}^{\rm t}$; (c) every $\mu \in \mathcal{G}^{\rm t}$ has a Lebowitz-Presutti type support; (d) $\mathcal{G}^{\rm t}$ is a singleton at high temperatures. In the case of attractive interaction and $\nu=1$ we prove that $|\mathcal{G}^{\rm t}|>1$ at low temperatures. The uniqueness of Gibbs measures due to quantum effects and at a nonzero external field are also proven in this case. Thereby, a qualitative theory of phase transitions and quantum effects, which interprets most important experimental data known for the corresponding physical objects, is developed. The mathematical result of the paper is a complete description of the set $\mathcal{G}^{\rm t}$, which refines and extends the results known for models of this type.