July 8, 2014 13:58 WSPC/Guidelines-IJMPB S0217979214300138 International Journal of Modern Physics B Vol. 28, No. 23 (2014) 1430013 (103 pages) c World Scientific Publishing Company DOI: 10.1142/S0217979214300138 Electron systems out of equilibrium: Nonequilibrium Green’s function approach∗ V´aclav Spiˇckaˇ †,§, Bedˇrich Velick´y†,‡ and Andˇela Kalvov´a† †Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 182 21 Praha 8, Czech Republic ‡Charles University, Faculty of Mathematics and Physics, DCMP, Ke Karlovu 5, 121 16 Praha 2, Czech Republic §[email protected] Received 3 May 2014 Accepted 28 May 2014 Published 7 July 2014 This review deals with the state of the art and perspectives of description of nonequi- librium many-body systems using the nonequilibrium Green’s function (NGF) method. The basic aim is to describe time evolution of the many-body system from its initial state over its transient dynamics to its long time asymptotic evolution. First, we discuss basic aims of transport theories to motivate the introduction of the NGF techniques. Sec- ond, this article summarizes the present view on construction of the electron transport equations formulated within the NGF approach to nonequilibrium. We discuss incorpo- ration of complex initial conditions to the NGF formalism, and the NGF reconstruction theorem, which serves as a tool to derive simplified kinetic equations. Three stages of evolution of the nonequilibrium, the first described by the full NGF description, the second by a non-Markovian generalized master equation and the third by a Markovian master equation will be related to each other. Int. J. Mod. Phys. B 2014.28. Downloaded from www.worldscientific.com Keywords: Nonequilibrium statistical physics; transients; quantum transport theory. PACS numbers: 05.30.-d, 72.10.Bg, 73.23.-b 1. Introduction by INSTITUTE OF PHYSICS THE CZECH ACADEMY SCIENCES on 08/25/14. For personal use only. The aim of this article is to show, how to describe time evolution of one par- ticle observables of many-body electron systems out of equilibrium within the nonequilibrium Green’s function (NGF) approach. This article will be orientated on the nonequilibrium quantum field theory on the real time Schwinger–Keldysh contour. We will demonstrate that NGF provides useful tools, how to deal with ∗Lecture given at the “Advanced School on Quantum Foundations and Open Quantum Systems” held in Jo˜ao Pessoa, Brazil, 16–28 July 2012. §Corresponding author. 1430013-1 July 8, 2014 13:58 WSPC/Guidelines-IJMPB S0217979214300138 V. Spiˇcka,ˇ B. Velick´y& A. Kalvov´a several open questions of nonequilibrium statistical physics and enables to formu- late the consistent quantum field theory of description of nonequilibrium quantum systems. Recently developed experimental techniques enable us to observe details of time evolution of various electron systems far from equilibrium and to perform many interesting measurements on various natural or artificially prepared structures in- cluding mesoscopic (nanoscopic) systems,1–3 where quantum processes, like quan- tum coherences, play decisive role. The growing area of nonequilibrium mesoscopic systems is naturally pervaded by open questions. Some of the problem open up newly during the research, some others have been resolved already, but in a pro- visional or an incomplete fashion. In general, the possibilities of the description of nonequilibrium many-body systems, not only mesoscopic ones, is far from being satisfactory due to the complexity of problems involved. To understand complex behavior of many-body systems out of equilibrium, and to interpret results of various recent experiments on mesoscopic systems, it is nec- essary to combine and to further improve methods of quantum field theory,4–7 many-body physics,8–14 statistical physics,15–40 and quantum transport theory.38–58 Before going to details of the NGF approach, which uses knowledge of all these fields, we will mention problems, which every candidate on a successful theory of nonequilibrium systems has to tackle. There are several key problems of nonequilibrium statistical physics to be un- derstood on the way to adequate methods of the description of many-body systems out of equilibrium. Here we will mention some of them. 1.1. Challenges, open questions, techniques Proper description of many-body character of systems, which represents a real • challenge already in equilibrium. In addition there is a problem with consistency of used approximations: to ensure this consistency we have to check conservation laws, which is often not an easy task; Int. J. Mod. Phys. B 2014.28. Downloaded from www.worldscientific.com Formulation and incorporation of nontrivial initial conditions into the formalism; • Understanding of different nonequilibrium regimes and their description from • short to long times evolution; Influence of external fields on time evolution; • Time evolution of open systems: formulation, what is the system and its sur- by INSTITUTE OF PHYSICS THE CZECH ACADEMY SCIENCES on 08/25/14. For personal use only. • rounding representing reservoirs and proper treatment of interactions between these two parts, loss of quantum coherences and dissipation processes. These problems are still far from their complete solutions. Due to complexity of problems and technical difficulties involved, several complementary approaches have been developed, which are dealing with various aspects of the above prob- lems in more or less details. These are the following techniques: time dependent density functional theory (TDDFT),59–69 time dependent dynamical mean-field theory (TDMFT)70–74 and various versions of density matrix renormalization group 1430013-2 July 8, 2014 13:58 WSPC/Guidelines-IJMPB S0217979214300138 Electron systems out of equilibrium (DMRG) techniques.75–80 We will not follow these “competitive” techniques to solve above mentioned key problems of nonequilibrium dynamics here. 1.2. Nonequilibrium Green’s functions In this review we will address problems of nonequilibrium statistical physics via NGFs techniques.81–131 As we will see later on this approach is based on methods of quantum field the- ory, and is able to deal with many important problems of statistical physics17–27 and generalization of equilibrium many-body techniques8–14 to nonequilibrium systems. This approach has been used for such diverse nonequilibrium systems as particles in plasmas,119,120 electrons, spins and phonons in various condensed matter systems like metals, semiconductors, superconductors and mesoscopic systems132–175 or nu- clei106,128,176–181 as it is well documented also in the special volume dedicated to workshops and conferences about the NGF techniques.111–115 It enables to describe nonequilibrium extended systems as well as mesoscopic (nanoscopic) systems, which have to be treated like open systems. The time evolution of observables can be cal- culated in various nonequilibrium regimes. In particular, the NGF formalism can also be conveniently used for the description of various steady state and equilib- rium situations. NGF have been used not only for calculations of nonequilibrium occupation numbers, currents, current densities, but they have been also general- ized to provide noise characteristics.182–186 They have been able to describe such different processes like decay of initial correlations, dynamics of formation of cor- relations or quasiparticles, various transient and transport regimes, fast electron and spin dynamics, quantum coherence and decoherence processes, thermalization, physics of nonequilibrium cold atoms, e.g., dynamics of bosonic and fermionic sys- tems in traps. Nowadays computers enable to solve complicated NGF equations for simplified, but often quite realistic models. Over the recent years many numerical calculations of the NGF equations have been performed.111–115,120,129,187–192 Int. J. Mod. Phys. B 2014.28. Downloaded from www.worldscientific.com 1.2.1. Note on the NGF history The text of the article will not follow the historical developments of the NGF by INSTITUTE OF PHYSICS THE CZECH ACADEMY SCIENCES on 08/25/14. For personal use only. techniques. It is, however, useful to mention several key figures and lines of early developments of the NGF technique. The beginning of the NGF technique is related to Julian Schwinger at the end of the forties followed by works of his school rep- resented e.g., by Martin, Kadanoff, Baym, Korenman, Craig, Horing. Later on two streams of developments have been related to the influential articles and the book written by Kadanoff and Baym85 on the one side and Keldysh89 and his followers on the other side. The reader can find many interesting details about early developments of the NGF techniques in the following references.81–101 1430013-3 July 8, 2014 13:58 WSPC/Guidelines-IJMPB S0217979214300138 V. Spiˇcka,ˇ B. Velick´y& A. Kalvov´a 1.3. Topics We may now formulate more precisely the subjects of this review: to overview the possibilities of the NGF techniques to describe nonequilibrium behavior of many- body systems either of bulk (very large) or small sizes for the whole time evolution of the system: from short to long times. The general problems of nonequilibrium statistical physics mentioned above are mirrored in the following topics, which will be discussed in this article within the NGF frame. 1.3.1. Formulation of transport theory and NGF The first topic of this paper is to formulate demands on transport theory to motivate the NGF approach to nonequilibrium systems, see Sec. 2. We will introduce NGF in more detail in the following Sec. 3. An important aspect of the theory are controlled approximations. We will briefly mention related conservation laws like particle number and energy conservation, Ward identities and their generalization for nonequilibrium situations at the end of the section. 1.3.2. Initial conditions The second topic deals with the task how to incorporate a nontrivial initial condition to the NGF formalism and implementation of the NGF techniques in the case of a fast transient process starting at a finite initial time and induced by a nonstationary initial condition and/or by an external field turned on at the initial time.