Conformal field theory out of equilibrium: a review

TL;DR

This review explores non-equilibrium conformal field theory, focusing on quantum transport, steady states, and large deviations in critical systems, connecting theoretical frameworks with physical models and highlighting open questions.

Contribution

It provides a comprehensive pedagogical overview of non-equilibrium CFT, including transport phenomena, fluctuation relations, and extensions to higher dimensions and non-integrable models.

Findings

Description of energy transport and large deviations in 1D critical systems

Extension of non-equilibrium concepts to higher-dimensional and non-integrable theories

Connection between free-particle models and generalized Gibbs ensembles

Abstract

We provide a pedagogical review of the main ideas and results in non-equilibrium conformal field theory and connected subjects. These concern the understanding of quantum transport and its statistics at and near critical points. Starting with phenomenological considerations, we explain the general framework, illustrated by the example of the Heisenberg quantum chain. We then introduce the main concepts underlying conformal field theory (CFT), the emergence of critical ballistic transport, and the CFT scattering construction of non-equilibrium steady states. Using this we review the theory for energy transport in homogeneous one-dimensional critical systems, including the complete description of its large deviations and the resulting (extended) fluctuation relations. We generalize some of these ideas to one-dimensional critical charge transport and to the presence of defects, as well as beyond one-dimensional criticality. We describe non-equilibrium transport in free-particle models, where connections are made with generalized Gibbs ensembles, and in higher-dimensional and non-integrable quantum field theories, where the use of the powerful hydrodynamic ideas for non-equilibrium steady states is explained. We finish with a list of open questions. The review does not assume any advanced prior knowledge of conformal field theory, large-deviation theory or hydrodynamics.