Chapter 14 - Superoperator many-body theory of molecular currents: Non-equilibrium Green functions in real time

Abstract

This chapter discusses development of the nonequilibrium superoperator green function theory (NESGFT) and applied it to the computation of molecular current. The Liouville space-time ordering operator provides an elegant way for performing calculations in real time, thus avoiding the artificial backward and forward time evolution required in Hilbert space (Keldysh loop). Wick's theorem for superoperators is used to compute the self-energies perturbatively to the second order in phonon–electron coupling. Recently, Galperin et al. have used a fully self-consistent solution to study the influence of different interactions on molecular conductivity for a strong electron–phonon coupling. The main aim of the present work is to demonstrate that by doing calculations in Liouville space one can avoid the backward/forward time evolution (Keldysh loop) required in Hilbert space. This originates from the fact that in Liouville space both ket and bra evolve forward in time. Thus, one can couple the system with two independent, “left” and “right” fields. This property of Liouville space can be used to construct real (physical) time generating functionals for the nonperturbative calculation of the self-energies. The present model ignores electron-electron interactions. These may be treated using the GW technique formulated in terms of the superoperators and extended to nonequilibrium situations. All nonequilibrium observables can be obtained from a single generating functional in terms of “left” and “right” operators.