Equilibrium States, Phase Transitions and Dynamics in Quantum Anharmonic Crystals

TL;DR

This paper reviews the mathematical framework for understanding equilibrium states, phase transitions, and dynamics in quantum anharmonic crystals, highlighting conditions for phase transitions and quantum stabilization.

Contribution

It introduces a rigorous approach to characterize local and global equilibrium states, and provides conditions for phase transitions and their suppression in quantum crystals.

Findings

A sufficient condition for phase transitions at certain temperatures.

A criterion for quantum stabilization preventing phase transitions.

Insights into how phase transitions influence local equilibrium dynamics.

Abstract

The basic elements of the mathematical theory of states of thermal equilibrium of infinite systems of quantum anharmonic oscillators (quantum crystals) are outlined. The main concept of this theory is to describe the states of finite portions of the whole system (local states) in terms of stochastically positive KMS systems and path measures. The global states are constructed as Gibbs path measures satisfying the corresponding DLR equation. The multiplicity of such measures is then treated as the existence of phase transitions. This effect can be established by analyzing the properties of the Matsubara functions corresponding to the global states. The equilibrium dynamics of finite subsystems can also be described by means of these functions. Then three basic results of this theory are presented and discussed: (a) a sufficient condition for a phase transition to occur at some temperature; (b) a sufficient condition for the suppression of phase transitions at all temperatures (quantum stabilization); (c) a statement showing how the phase transition can affect the local equilibrium dynamics.