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arXiv:cond-mat/0506130

Introduction to the Keldysh formalism and applications to time-dependent density-functional theory

Robert van Leeuwen, Nils Erik Dahlen, Gianluca Stefanucci, Carl-Olof Almbladh, Ulf von Barth

Published Jun 6, 2005• 18 citations
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Abstract

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.

DOI
10.48550/arxiv.cond-mat/0506130

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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 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 Runge-Gross [E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984)] action functional of time-dependent density-functional theory leads to a well-known causality paradox; that is, a perturbation of the electronic density in the future affects the response of the system in the present. This paradox is known to be caused by an inconsistent application of the Dirac-Frenkel variational principle. In view of the recent solutions to this problem, the action functional employed by Runge and Gross in their formulation of time-dependent density-functional theory is analyzed in the context of the Keldysh contour technique. The time-dependent electronic density and the concept of causality are extended to the contour. We derive a variational equation that obeys causality and relates the exchange-correlation potential with its kernel and the functional derivative of the exchange-correlation action functional with respect to the density. It is shown that the adiabatic local-density approximation is a consistent solution of this equation and that the time-dependent optimized potential method can also be derived from it. The formalism presented here can be used to find new approximation methods for the exchange-correlation potential and to avoid the causality dilemma.
Most present applications of time-dependent density functional theory use adiabatic functionals, i.e., the effective potential at time t is determined solely by the density at the same time. This paper discusses a method that aims to go beyond this approximation, by incorporating "memory" effects: the derived exchange-correlation potential will depend not only on present densities but also on the past. In order to ensure the potentials are causal, we formulate the action on the Keldysh contour for electrons in electromagnetic fields, from which we derive suitable Kohn-Sham equations. The exchange-correlation action is now a functional of the electron density and velocity field. A specific action functional is constructed which is Galilean invariant and yields a causal exchange-correlation vector potential for the Kohn-Sham equations incorporating memory effects. We show explicitly that the net exchange-correlation Lorentz force is zero. The potential is consistent with known dynamical properties of the homogeneous electron gas (in the linear response limit).
We resolve an existing paradox regarding the causality and symmetry properties of response functions within time-dependent density-functional theory. We do this by defining a new action functional within the Keldysh formalism. By functional differentiation the new functional leads to response functions which are symmetric in the Keldysh time contour parameter, but which become causal when a transition to physical time is made. The new functional is further used to derive the equations of the time-dependent optimized potential method.
Within the Landauer approach to conduction discussed in the previous chapter, electron interactions have been included only at the mean-field level. This is quite a strong approximation, especially in nanojunctions, where large current densities are common. I therefore want to go beyond this level of description. I can follow two different routes. (1) I can develop a functional theory of quantum correlations via time-dependent effective single-particle equations, as I will do in Chapter 7 where I will use dynamical density-functional theories within the micro-canonical approach to conduction. (2) I can try to solve the time-dependent Schrödinger equation 1.16 – or its mixed-state version, the Liouville-von Neumann equation 1.60 – directly. Written this way, this last proposition seems hopeless. In reality, we can employ a many-body technique, known as the non-equilibrium Green's function formalism (NEGF), also referred to as the Keldysh formalism (Keldysh, 1964; Kadanoff and Baym, 1962), which allows us, at least in principle, to do just that: solve the time-dependent Schrödinger equation for an interacting many-body system exactly, from which one can, in principle, calculate the time-dependent current. This is done by solving equations of motion for specific time-dependent single-particle Green's functions, from which the physical properties of interest, such as the charge and current densities, can be obtained. I have stressed the term “time-dependent” several times, because, as I will show in a moment, the NEGF is “exact” only when one solves the time-dependent Schrödinger equation for a closed quantum system, subject to deterministic perturbations: the system is closed but not necessarily isolated. These perturbations may drive the system far away from its initial state of thermodynamic equilibrium (Keldysh, 1964).