Geometry of quantum phase transitions

TL;DR

This paper reviews geometric methods, including the Uhlmann phase, for analyzing quantum and non-equilibrium phase transitions, highlighting their ability to characterize critical phenomena and the quantum nature of steady states.

Contribution

It introduces a geometric framework based on mixed-state generalizations of Berry phases to study non-equilibrium quantum phase transitions, extending traditional approaches.

Findings

Geometric phase curvature signals criticality in lattice fermion systems.

The approach distinguishes quantum from classical phase transitions.

Relations between geometric phases, correlation length divergence, and gap behavior are established.

Abstract

In this article we provide a review of geometrical methods employed in the analysis of quantum phase transitions and non-equilibrium dissipative phase transitions. After a pedagogical introduction to geometric phases and geometric information in the characterisation of quantum phase transitions, we describe recent developments of geometrical approaches based on mixed-state generalisation of the Berry-phase, i.e. the Uhlmann geometric phase, for the investigation of non-equilibrium steady-state quantum phase transitions (NESS-QPTs ). Equilibrium phase transitions fall invariably into two markedly non-overlapping categories: classical phase transitions and quantum phase transitions, whereas in NESS-QPTs this distinction may fade off. The approach described in this review, among other things, can quantitatively assess the quantum character of such critical phenomena. This framework is applied to a paradigmatic class of lattice Fermion systems with local reservoirs, characterised by Gaussian non-equilibrium steady states. The relations between the behaviour of the geometric phase curvature, the divergence of the correlation length, the character of the criticality and the gap - either Hamiltonian or dissipative - are reviewed.