Phase transitions and quantum effects in anharmonic crystals

TL;DR

This paper reviews recent advances in understanding phase transitions and quantum effects in anharmonic crystals, providing conditions for phase transitions and discussing their thermodynamic and probabilistic descriptions.

Contribution

It offers a unified framework for describing phase transitions and quantum effects in anharmonic crystals using probability measures on path spaces, with explicit conditions involving interaction and oscillator parameters.

Findings

Conditions for phase transitions are expressed as simple inequalities.

Phase transitions are characterized by multiple thermodynamic phases at the same parameters.

The theory links phase transitions to non-differentiability of free energy and ordering phenomena.

Abstract

The most important recent results in the theory of phase transitions and quantum effects in quantum anharmonic crystals are presented and discussed. In particular, necessary and sufficient conditions for a phase transition to occur at some temperature are given in the form of simple inequalities involving the interaction strength and the parameters describing a single oscillator. The main characteristic feature of the theory is that both mentioned phenomena are described in one and the same setting, in which thermodynamic phases of the model appear as probability measures on path spaces. Then the possibility of a phase transition to occur is related to the existence of multiple phases at the same values of the relevant parameters. Other definitions of phase transitions, based on the non-differentiability of the free energy density and on the appearance of ordering, are also discussed.