Non-equilibrium steady states in the Klein–Gordon theory

Abstract

We construct non-equilibrium steady states in the Klein-Gordon theory in\narbitrary space dimension $d$ following a local quench. We consider the\napproach where two independently thermalized semi-infinite systems, with\ntemperatures $T_{\\rm L}$ and $T_{\\rm R}$, are connected along a\n$d-1$-dimensional hypersurface. A current-carrying steady state, described by\nthermally distributed modes with temperatures $T_{\\rm L}$ and $T_{\\rm R}$ for\nleft and right-moving modes, respectively, emerges at late times. The\nnon-equilibrium density matrix is the exponential of a non-local conserved\ncharge. We obtain exact results for the average energy current and the complete\ndistribution of energy current fluctuations. The latter shows that the\nlong-time energy transfer can be described by a continuum of independent\nPoisson processes, for which we provide the exact weights. We further describe\nthe full time evolution of local observables following the quench. Averages of\ngeneric local observables, including the stress-energy tensor, approach the\nsteady state with a power-law in time, where the exponent depends on the\ninitial conditions at the connection hypersurface. We describe boundary\nconditions and special operators for which the steady state is reached\ninstantaneously on the connection hypersurface. A semiclassical analysis of\nfreely propagating modes yields the average energy current at large distances\nand late times. We conclude by comparing and contrasting our findings with\nresults for interacting theories and provide an estimate for the timescale\ngoverning the crossover to hydrodynamics. As a modification of our Klein-Gordon\nanalysis we also include exact results for free Dirac fermions.\n