The increasing interest in nonequilibrium effects in condensed matter theory motivates the adaption of diverse equilibrium techniques to Keldysh formalism. For methods based on multi-particle Green's or vertex functions, this involves a detailed knowledge of the real-time properties of those functions. In this review, we derive general properties of fermionic and bosonic multi-particle Green's and vertex functions for a stationary state described within Keldysh formalism. Special emphasis is put on the analytic properties associated with causality and on a detailed discussion of the Kubo–Martin–Schwinger conditions which characterize thermal equilibrium. Finally we describe how diagrammatic approximations and approximations within the functional renormalization group approach respect these properties.
The increasing interest in nonequilibrium effects in condensed matter theory motivates the adaption of diverse equilibrium techniques to Keldysh formalism. For methods based on multi-particle Green's or vertex functions, this involves a detailed knowledge of the real-time properties of those functions. In this review, we derive general properties of fermionic and bosonic multi-particle Green's and vertex functions for a stationary state described within Keldysh formalism. Special emphasis is put on the analytic properties associated with causality and on a detailed discussion of the Kubo–Martin–Schwinger conditions which characterize thermal equilibrium. Finally we describe how diagrammatic approximations and approximations within the functional renormalization group approach respect these properties.
Abstract Chapter 09 Nonequilibrium Green’s functions (NEGF), including coupled-correlated (C) single- and multi-particle Green’s functions, are defined as averages weighted with the time-development operator U(t0+τ,t0). Linear conductivity is exhibited as a two-particle equilibrium Green’s function (Kubo-type formulation). Admitting particle sources (S:η,η+) and non-conservation of number, the non-equilibrium multi-particle Green’s functions are constructed with numbers of creation and annihilation operators that may differ, and they may be derived as variational derivatives with respect to sources η,η+ of a generating functional eW=TrU(t0+τ,t0)CS/TrU(t0+τ,t0)C. (In the non-interacting case this yields the n-particle Green’s function as a permanent/determinant of single-particle Green’s functions.) These variational relations yield a symmetric set of multi-particle Green’s function equations. Cumulants and the Linked Cluster Theorem are discussed and the Random Phase Approximation (RPA) is derived variationally. Schwinger’s variational differential formulation of perturbation theories for the Green’s function, self-energy, vertex operator, and also shielded potential perturbation theory, are reviewed. The Langreth Algebra arises from analytic continuation of integration of products of Green’s functions in imaginary time to the real-time axis with time-ordering along the integration contour in the complex time plane. An account of the Generalized Kadanoff-Baym Ansatz is presented.
We present a unified framework for equilibrium and nonequilibrium many-body perturbation theory. The most general nonequilibrium many-body theory valid for general initial states is based on a time-contour originally introduced by Konstantinov and Perel'. The various other well-known formalisms of Keldysh, Matsubara and the zero-temperature formalism are then derived as special cases that arise under different assumptions. We further present a single simple proof of Wick's theorem that is at the same time valid in all these flavors of many-body theory. It arises simply as a solution of the equations of the Martin-Schwinger hierarchy for the noninteracting many-particle Green's function with appropriate boundary conditions. We further discuss a generalized Wick theorem for general initial states on the Keldysh contour and derive how the formalisms based on the Keldysh and Konstantinov-Perel'-contours are related for the case of general initial states.
The progress in micro-fabrication technologies makes it possible to address individual, tailored, mesoscopic structures with sizes on a nanometer to micrometer scale. The transport properties of such quasi zero- or one-dimensional systems are considerably influenced by the electron-electron interaction. In theoretical studies this often becomes apparent in the emergence of divergences in a perturbative treatment of the interaction. A successful approach may then be given by renormalization group (RG) methods which are based on the idea of treating energy scales successively from high to low. In the functional RG (fRG) an exact infinite hierarchy of flow equations for multi-particle Green or vertex functions is derived, that allows for systematic approximations on different levels. In numerous studies the fRG has been found to be a powerful tool for the investigation of low dimensional Fermi systems. Usually it is formulated within Matsubara formalism which applies to a system in thermal equilibrium. A central topic of this thesis is the implementation of the fRG within Keldysh formalism, providing the possibility to treat equilibrium and non-equilibrium situations on equal footing. The resulting Keldysh fRG is used in this thesis to explore the transport properties of quantum wires with several barriers and of the single impurity Anderson model. The thesis starts with an introduction to Keldysh formalism and to related concepts. Then the structure of the Green and vertex functions of Keldysh formalism is investigated. This includes studying the consequences of causality, characterizing thermal equilibrium, and formulating corresponding relations for the multi-particle functions. After that, the implementation of the fRG within Keldysh formalism is presented in detail. The flow equations for the vertex functions are derived via their diagrammatic expansion. Two especially useful flow parameters are determined: one of them cuts off the imaginary frequencies that appear as poles of the Fermi functions of the reservoirs. The second flow parameter is an artificially enhanced reservoir hybridization; on one hand it has the advantage to preserve the fluctuation dissipation theorem in equilibrium even in frequency dependent approximations; on the other hand, its use can be more cumbersome. One of the models studied in this thesis is a tight-binding chain with nearest-neighbour interaction. At weak and intermediate interactions it belongs to the class of Luttinger liquids (LLs) whose low energy properties are characterized by power laws. The transport properties of LLs have been studied experimentally for example on carbon nanotubes. However, the presence of transport barriers at the contacts make the interpretation of the experimental data difficult. A theoretical treatment with the fRG permits a rather large flexibility to model the barrier configuration. In this thesis the fRG is first used in its equilibrium Matsubara implementation in order to analyse the temperature dependence of the linear conductance in the presence of two, three or four barriers. It is shown that temperature induces an averaging over quantum phases of propagation, leading to combinations of the power law exponents of the individual barriers. An experimental set-up that allows to verify this behaviour is proposed. In a second step the Keldysh version of the fRG is applied to the problem of transport through a quantum wire at finite bias voltage. For a wire with two high contact barriers it is found that the non-equilibrium particle distribution in the wire is responsible for the appearance of a two-fold power law in the density of states. An additional weakly attached probe lead can be used to make the two different power laws visible in the conductance. The second model which is analysed is the one of the single Anderson impurity, often considered as a paradigm for the investigation of strong local electron correlations. The latter become noticeable in the appearance of the Kondo scale, an energy exponentially small in the interaction strength, that characterizes the low energy physics of the model in the regime of spin fluctuations. On the basis of this model an advanced frequency dependent approximation for the Keldysh fRG is developed in this thesis. Such an approximation can account for subleading effects of the interaction, namely the fact that two-particle interactions can generate decay rates for single-particle states. The derivation of a consistent approximation scheme is presented, that turns out to yield a good description of the model at moderate interaction strength.
We investigate nonequilibrium properties of the single-impurity Anderson model by means of the functional renormalization group (fRG) within Keldysh formalism. We present how the level broadening $\ensuremath{\Gamma}/2$ can be used as flow parameter for the fRG. This choice preserves important aspects of the Fermi-liquid behavior that the model exhibits in case of particle-hole symmetry. An approximation scheme for the Keldysh fRG is developed which accounts for the frequency dependence of the two-particle vertex in a way similar but not equivalent to a recently published approximation to the equilibrium Matsubara fRG. Our method turns out to be a flexible tool for the study of weak to intermediate on-site interactions $U\ensuremath{\lesssim}3\ensuremath{\Gamma}$. In equilibrium we find excellent agreement with numerical RG results for the linear conductance at finite gate voltage, magnetic field, and temperature. In nonequilibrium, our results for the current agree well with time-dependent density-matrix RG. For the nonlinear conductance as function of the bias voltage, we propose reliable results at finite magnetic field and finite temperature. Furthermore, we demonstrate the exponentially small scale of the Kondo temperature to appear in the second-order derivative of the self-energy. We show that the approximation is, however, not able to reproduce the scaling of the effective mass at large interactions.
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.
Quantum many-body systems are a central feature of condensed matter physics, relevant to important, modern research areas such as ultrafast light-matter interactions and quantum information. This book offers detailed coverage of the contour Green's function formalism – an approach that can be successfully applied to solve the quantum many-body and time-dependent problems present within such systems. Divided into three parts, the text provides a structured overview of the relevant theoretical and practical tools, with specific focus on the Schwinger-Keldysh formalism. Part I introduces the mathematical frameworks that make use of Green's functions in normal phase states. Part II covers fermionic superfluid phases with discussion of topics such as the BCS-BEC crossover and superconducting systems. Part III deals with the application of the Schwinger-Keldysh formalism to various topics of experimental interest. Graduate students and researchers will benefit from the book's comprehensive treatment of the subject matter and its novel arrangement of topics.