Reformulation of steady state nonequilibrium quantum statistical mechanics

Abstract

Starting from the usual formulation of nonequilibrium quantum statistical mechanics, the expectation value of an operator A in a steady state nonequilibrium quantum system is shown to have the form {A) ==: Tr{e ~f iiH~Y) A}/Tv{e~p (H~Y) }, where H is the Hamiltonian, p is the inverse of the temperature, and Y is an operator which depends on how the system is driven out of equilibrium. Because {A) is not expressed as a sum of correlation functions integrated over real time, one can now consider performing nonperturbative calculations in interacting nonequilibrium quantum problems.PACS numbers: 72.10. Bg, 05.60,+w Nonequilibrium problems have come under increasing study in condensed matter physics. On the one hand, there exists a growing number of classical systems which undergo a phase transition as one drives them out of equilibrium. On the other hand, with technological advances allowing one to make measurements on smaller samples, it becomes easier to drive systems out of equilibrium. This is particularly true in making resistance measurements on very small (mesoscopic) devices at low temperatures [1], which is an inherently quantum problem. Furthermore, while linear response measurements do probe some equilibrium correlation functions, much more information can be obtained from the nonlinear response, which probes the full nonequilibrium problem. For example, nonlinear current-voltage characteristics in metal-insulator-superconductor tunnel junctions have long been used to determine the superconducting gap and the phonon density of states. In mesoscopic systems similar kinds of information about the density of states can be obtained from nonlinear transport [2][3][4][5].In spite of the experimental importance of studying nonlinear response there are far fewer theoretical techniques available for studying nonequilibrium quantum systems than for studying equilibrium ones. Again using the example of mesoscopic systems, the noninteracting quantum problem may be solved exactly using the scattering states for electrons coming from reservoirs which are at different chemical potentials. This is the essence of the Landauer formula [6] and its subsequent generalizations [7,8]. For an interacting system the nonlinear current-voltage characteristic can be computed by doing perturbation theory in the part of the Hamiltonian which drives the system out of equilibrium, e.g., an electric field or the tunneling between two leads at different chemical potentials. This perturbation theory, which we will call nonequilibrium quantum statistical mechanics, involves summing a set of real time correlation functions for the linear, quadratic, cubic, etc., response to this interaction [9,10].While there are many problems for which a perturbation theory is perfectly satisfactory, there is a large class of problems of current interest for which a nonperturbative approach would be quite useful and perhaps even essential. In the case of mesoscopic systems it is now possible to tunnel through a one-dimensional wire [11], a small quantum...

New Method for Studying Steady States in Quantum Impurity Problems: The Interacting Resonant Level Model

“…In quantum impurity systems, steady states are also time-independent quantum states. They are defined by prescribing the baths to be, at x < 0, in thermal equilibrium at temperature T with a chemical potential V associated to a local conserved charge Q of H 0 , and by asking for time-independence with respect to the dynamics given by the full hamiltonian H. That is, quantum averages · · · ≡ Tr (ρ · · ·) /Tr (ρ) are described by the density matrix [5] …”

“…The latter describes directly the expected end result just from "how the state looks" asymptotically far from the impurity. Hershfield's Y operator [5] (see also the studies [3,6,7]) gives a "steady-state density matrix" that encodes these scattering states. This is interesting, since a non-equilibrium steady state is not described by the usual density matrix, but it is still hard to apply to interacting systems.…”

Imaginary-Time Formulation of Steady-State Nonequilibrium: Application to Strongly Correlated Transport

“…Unfortunately they often suffer from long-time behaviors associated with low energy strongly correlated states and finite size effects. Direct construction of nonequilibrium ensembles through the scattering state formalism [2,8,9, 10] and field theoretic approach [11] have provided new perspectives to the problem.The main goal of this work is to provide a critical step toward the time-independent description of equilibrium and steady-state nonequilibrium quantum statistics. In addition to the resolution of this fundamental problem, we provide a strong application.…”

“…Unfortunately they often suffer from long-time behaviors associated with low energy strongly correlated states and finite size effects. Direct construction of nonequilibrium ensembles through the scattering state formalism [2,8,9, 10] and field theoretic approach [11] have provided new perspectives to the problem.…”

Impurity model for non-equilibrium steady states

“…Building on Hershfield's expression for the steady-state density matrix [14], Han and Heary gave an effective Matsubara description of the steady state. This allows the use of standard many-body equilibrium tools to tackle the strong interaction at the cost of introducing some imaginary chemical potentials:…”