With the help of time-dependent scattering theory on the observable algebra of infinitely extended quasifree fermionic chains, we introduce a general class of so-called right mover/left mover states which are inspired by the nonequilibrium steady states for the prototypical nonequilibrium configuration of a finite sample coupled to two thermal reservoirs at different temperatures. Under the assumption of spatial translation invariance, we relate the 2-point operator of such a right mover/left mover state to the asymptotic velocity of the system and prove that the system is thermodynamically nontrivial in the sense that its entropy production rate is strictly positive. Our study of these not necessarily gauge-invariant systems covers and substantially generalizes well-known quasifree fermionic chains and opens the way for a more systematic analysis of the heat flux in such systems.
With the help of time-dependent scattering theory on the observable algebra of infinitely extended quasifree fermionic chains, we introduce a general class of so-called right mover/left mover states which are inspired by the nonequilibrium steady states for the prototypical nonequilibrium configuration of a finite sample coupled to two thermal reservoirs at different temperatures. Under the assumption of spatial translation invariance, we relate the 2-point operator of such a right mover/left mover state to the asymptotic velocity of the system and prove that the system is thermodynamically nontrivial in the sense that its entropy production rate is strictly positive. Our study of these not necessarily gauge-invariant systems covers and substantially generalizes well-known quasifree fermionic chains and opens the way for a more systematic analysis of the heat flux in such systems.
Abstract Within the rigorous axiomatic framework for the description of quantum mechanical systems with a large number of degrees of freedom, we construct the so-called nonequilibrium steady state for the quasifree fermionic system corresponding to the isotropic XY chain in which a finite sample, subject to a local gauge breaking anisotropy perturbation, is coupled to two thermal reservoirs at different temperatures. Using time dependent and stationary scattering theory, we rigorously prove, from first principles, that the nonequilibrium system under consideration is thermodynamically nontrivial, i. e., that its entropy production rate is strictly positive.
For the general class of quasifree fermionic right mover/left mover systems over the infinitely extended two-sided discrete line introduced in Aschbacher [Rev. Math. Phys. 35, 2330001 (2023)] within the algebraic framework of quantum statistical mechanics, we study the von Neumann entropy of a contiguous subsystem of finite length in interaction with its environment. In particular, under the assumption of spatial translation invariance, we analyze the asymptotic behavior of the von Neumann entropy for large subsystem lengths and prove that its leading order density is, in general, nonvanishing and displays the signature of a mixture of the independent thermal species underlying the right mover/left mover system. As special cases, the formalism covers so-called nonequilibrium steady states, thermal equilibrium states, and ground states. Moreover, for general Fermi functions, we derive a necessary and sufficient criterion for the von Neumann entropy density to vanish.
We study the time evolution of a one-dimensional interacting fermion system described by the Luttinger model starting from a nonequilibrium state defined by a smooth temperature profile T(x). As a specific example we consider the case when T(x) is equal to TL(TR) far to the left (right). Using a series expansion in $\ensuremath{\epsilon}=2({T}_{R}\ensuremath{-}{T}_{L})/({T}_{L}+{T}_{R})$, we compute the energy density, the heat current density, and the fermion two-point correlation function for all times $t\ensuremath{\ge}0$. For local (delta-function) interactions, the first two are computed to all orders, giving simple exact expressions involving the Schwarzian derivative of the integral of T(x). For nonlocal interactions, breaking scale invariance, we compute the nonequilibrium steady state (NESS) to all orders and the evolution to first order in $\ensuremath{\epsilon}$. The heat current in the NESS is universal even when conformal invariance is broken by the interactions, and its dependence on TL,R agrees with numerical results for the XXZ spin chain. Moreover, our analytical formulas predict peaks at short times in the transition region between different temperatures and show dispersion effects that, even if nonuniversal, are qualitatively similar to ones observed in numerical simulations for related models, such as spin chains and interacting lattice fermions.
We suggest a generalization of the expression of the nonequilibrium (NE) density matrix obtained by Hershfield's method for the cases where both heat and charge steady-state currents are present in a quantum open system. The finite-size quantum system, connected to two temperature and particle reservoirs, is driven out of equilibrium by the presence of both a temperature gradient and a chemical potential gradient between the two reservoirs. We show that the NE density matrix is given by a generalized Gibbs-like ensemble and is in full agreement with the general results of the McLennan-Zubarev nonequilibrium ensembles. The extra nonequilibrium terms are related to the entropy production in the system and characterize the fluxes of heat and particle. An explicit example, for the lowest-order expansion, is provide for a model system of noninteracting fermions.
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.
We consider two semi-infinite quantum Ising chains initially at thermal equilibrium at two different temperatures and subsequently joined by an interaction between their end points. Transport properties such as the heat current are determined by the dynamics of the left- and right-moving fermionic quasiparticles which characterize the ensuing unitary dynamics. Within the so-called semiclassical space-time scaling limit we extend known results by determining the full space and time dependence of the density and current of energy and of fermionic quasiparticles. Upon approaching the edge of the propagating front, these quantities as well as the two-point correlation function display qualitatively different behaviors depending on the transverse field of the chain being critical or not. While in the latter case corrections to the leading behavior are described, as expected, by the Airy kernel, in the former a novel scaling form emerges with universal features.