Nonequilibrium density matrix for thermal transport in quantum field theory

Abstract

In these notes I explain how to describe one-dimensional quantum systems that are simultaneously near to, but not exactly at, a critical point, and in a far-from-equilibrium steady state. This description uses a density matrix on scattering states (of the type of Hershfield's density matrix), or equivalently a Gibbs-like ensemble of scattering states. The steady state I am considering is one where there is a steady flow of energy along the chain, coming from the steady draining / filling of two far-away reservoirs put at different temperatures. The context I am using is that of massive relativistic quantum field theory, which is the framework for describing the region near quantum critical points in any universality class with translation invariance and with dynamical exponent z equal to 1. I show, in this completely general setup, that a particular steady-state density matrix occurs naturally from the physically motivated Keldysh formulation of the steady state. In this formulation the steady state occurs as a result of a large-time evolution from an initial state where two halves of the system are separately thermalized. I also show how this suggests a particular dependence (a "factorization") of the average current on the left and right temperatures. The idea of this density matrix was proposed already in a recent publication with my collaborator Denis Bernard, where we studied it in the context of conformal field theory.