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arXiv:cond-mat/0407413 · DoD High Performance Computing Modernization Program Users Group Conference (HPCMP-UGC)

Parallelizing the Keldysh formalism for strongly correlated electrons

James K. Freericks, Volodymyr Turkowski, V. Zlatić

Published Jan 1, 2004• 0 citations
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Abstract

Nonequilibrium quantum mechanics can be solved with the Keldysh formalism, which evolves the quantum mechanical states forward in time in the presence of a time-dependent field, and then evolves them backward in time, undoing the effect of the time-dependent field. The Feynman path integral over the Keldysh contour is employed to calculate the strongly correlated Green's function. We examine the accuracy of this procedure for the simplest problem that requires a nonequilibrium formulation: the electron spectral function of the spinless Falicov-Kimball model.

DOI
10.1109/dod_ugc.2004.32

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Nonequilibrium quantum mechanics can be solved with the Keldysh formalism, which evolves the quantum mechanical states forward in time in the presence of a time-dependent field, and then evolves them backward in time, undoing the effect of the time-dependent field. The Feynman path integral over the Keldysh contour is employed to calculate the strongly correlated Green's function. We examine the accuracy of this procedure for the simplest problem that requires a nonequilibrium formulation: the electron spectral function of the spinless Falicov-Kimball model.
We derive exact operator average expressions for the first two spectral moments of nonequilibrium Green's functions for the Falicov-Kimball model and the Hubbard model in the presence of a spatially uniform, time-dependent electric field. The moments are similar to the well-known moments in equilibrium, but we extend those results to systems in arbitrary time-dependent electric fields. Moment sum rules can be employed to estimate the accuracy of numerical calculations; we compare our theoretical results to numerical calculations for the nonequilibrium dynamical mean-field theory solution of the Falicov-Kimball model at half-filling.
This chapter examines the Schwinger-Keldysh technique, applying path integrals, and the functional method of 2PI effective actions—with the latter serving to construct powerful symmetry-preserving nonperturbative approximations for equilibrium as well as nonequilibrium descriptions of strongly interacting quantum many-body systems. It provides a rigorous mathematical basis for exploring the quantum mechanical basis of thermalization and decoherence. Mesoscopic systems with quantum processes, in particular, play a key role. The domain of nonequilibrium mesoscopic systems is flooded with open questions some of which are solved but still others have provisional or incomplete solutions. A correlation function encodes the correlation between the field operators at different times and is an inference to all thermodynamic quantities. The beauty of the Keldysh time path is the definition of the Green's function, which plays an analogous role in the nonequilibrium formalism similar to the role the causal Green's function plays in the equilibrium theory.
This paper gives an introduction to the Keldysh formalism, with emphasis on its usefulness in time-dependent density functional theory. In the first part we introduce the Keldysh contour and the one-particle Green function defined on this contour. We then discuss how to combine and manipulate functions with time-arguments on the contour. The effects of electron-electron interaction can be taken systematically into account, as we illustrate by propagating the Kadanoff-Baym equations for the second-order self-energy approximation. It is important to use conserving approximations such that the evolution of the electron density, momentum, and total energy agrees with the macroscopic conservation laws. One of the main topics in this paper is the non-interacting Green function, which is the relevant quantity for time-dependent density functional theory. We discuss this Green function in detail, and show how the Keldysh contour in a simple way allows us to derive the time-dependent Kohn-Sham potential from an action functional. The formalism in a similar way leads to response functions that obey the causality principle. To illustrate these points, we discuss the time-dependent optimized effective potential equations.
We present a review of our recent work in extending the successful dynamical mean-field theory from the equilibrium case to nonequilibrium cases. In particular, we focus on the problem of turning on a spatially uniform, but possibly time varying, electric field (neglecting all magnetic field effects). We show how to work with a manifestly gauge-invariant formalism, and compare numerical calculations from a transient-response formalism to different types of approximate treatments, including the semiclassical Boltzmann equation and perturbation theory in the interaction. In this review, we solve the nonequilibrium problem for the Falicov-Kimball model, which is the simplest many-body model and the easiest problem to illustrate the nonequilibrium behavior in both diffusive metals and Mott insulators. Due to space restrictions, we assume the reader already has some familiarity both with the Kadanoff-Baym-Keldysh nonequilibrium formalism and with equilibrium dynamical mean-field theory; we provide a guide to the literature where additional details can be found.
We have considered application the Keldysh formalism in time-dependent \ndensity functional theory. It is shown that effects of electron-electron interaction can \nbe taken systematically into account in the framework of the Kadanoff-Baym \nequations for Green functions determined on the Keldysh contour for the second-order \n47 \nself-energy approximation. With the help of these Green functions we have derived \nthe time-dependent Kohn-Sham potential from an action functional. The Keldysh \nformalism in a similar way leads to response functions that obey the causality \nprinciple that is illustrated on the case of the time-dependent optimized effective \npotential equations.
The fff-electron spectral function of the Falicov-Kimball model is calculated via a Keldysh-based many-body formalism originally developed by Brandt and Urbanek. We provide results for both the Bethe lattice and the hypercubic lattice at half filling. Since the numerical computations are quite sensitive to the discretization along the Kadanoff-Baym contour and to the maximum cutoff in time that is employed, we analyze the accuracy of the results using a variety of different moment sum rules and spectral formulas. We find that the fff-electron spectral function has interesting temperature dependence, becoming a narrow single-peaked function for small UUU and developing a gap, with two broader peaks for large UUU.