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.

Inspired by recent results on the non-equilibrium dynamics of many-body quantum systems, we study the classical hard rod problem in one dimension with initial domain wall condition. Hard rods are an integrable system, in the sense that for each velocity the density of particles is locally conserved. It was proven by Boldrighini et al (1983 J. Stat. Phys. 31 577) that on the hydrodynamic space-time scale, the fluid of hard rods satisfies Euler-type equations which comprise all conservation laws. We provide the general solution to these equations on the line, with an initial condition where the left and right halves are, asymptotically, in different states. The solution is interpreted as being composed of a continuum of contact discontinuities, one for each velocity. This is a classical counterpart of the transport problem solved recently in quantum integrable systems. We also discuss the Navier–Stokes (viscous) corrections, and study its effect on the broadening of the contact discontinuity and on entropy production.

We consider two quantum Ising chains initially prepared at thermal equilibrium but with different temperatures and coupled at a given time through one of their end points. In the long-time limit the system reaches a nonequilibrium steady state. We discuss properties of this nonequilibrium steady state and characterize the convergence to the steady regime. We compute the mean energy flux through the chain and show that the heat transport is ballistic. We derive also the large-deviation function for the quantum and thermal fluctuations of this energy transfer.

We review the recent advances on exact results for dynamical correlation functions at large scales and related transport coefficients in interacting integrable models. We discuss Drude weights, conductivity and diffusion constants, as well as linear and nonlinear response on top of equilibrium and non-equilibrium states. We consider the problems from the complementary perspectives of the general hydrodynamic theory of many-body systems, including hydrodynamic projections, and form-factor expansions in integrable models, and show how they provide a comprehensive and consistent set of exact methods to extract large scale behaviours. Finally, we overview various applications in integrable spin chains and field theories.

Recently, it was understood that modified concepts of locality played an important role in the study of extended quantum systems out of equilibrium, in particular in so-called generalized Gibbs ensembles. In this paper, we rigorously study pseudolocal charges and their involvement in time evolutions and in the thermalization process of arbitrary states with strong enough clustering properties. We show that the densities of pseudolocal charges form a Hilbert space, with inner product determined by thermodynamic susceptibilities. Using this, we define the family of pseudolocal states, which are determined by pseudolocal charges. This family includes thermal Gibbs states at high enough temperatures, as well as (a precise definition of) generalized Gibbs ensembles. We prove that the family of pseudolocal states is preserved by finite time evolution, and that, under certain conditions, the stationary state emerging at infinite time is a generalized Gibbs ensemble with respect to the evolution dynamics. If the evolution dynamics does not admit any conserved pseudolocal charges other than the evolution Hamiltonian, we show that any stationary pseudolocal state with respect to these dynamics is a thermal Gibbs state, and that Gibbs thermalization occurs. The framework is that of translation-invariant states on hypercubic quantum lattices of any dimensionality (including quantum chains) and finite-range Hamiltonians, and does not involve integrability.