As experimental probes have matured to observe ultrafast transient and high frequency responses of materials and devices, so to have the theoretical methods to numerically and analytically simulate time- and frequency-resolved transport. In this review article, we discuss recent progress in the development of the time-dependent and frequency-dependent non-equilibrium Green function (NEGF) technique. We begin with an overview of the theoretical underpinnings of the underlying Kadanoff-Baym equations and derive the fundamental NEGF equations in the time and frequency domains. We discuss how these methods have been applied to a variety of condensed matter systems such as molecular electronics, nanoscale transistors, and superconductors. In addition, we survey the application of NEGF in fields beyond condensed matter, where it has been used to study thermalization in ultra-cold atoms and to understand leptogenesis in the early universe. Throughout, we pay special attention to the challenges of incorporating contacts and interactions, as the NEGF method is uniquely capable of accounting for such features.
As experimental probes have matured to observe ultrafast transient and high frequency responses of materials and devices, so to have the theoretical methods to numerically and analytically simulate time- and frequency-resolved transport. In this review article, we discuss recent progress in the development of the time-dependent and frequency-dependent non-equilibrium Green function (NEGF) technique. We begin with an overview of the theoretical underpinnings of the underlying Kadanoff-Baym equations and derive the fundamental NEGF equations in the time and frequency domains. We discuss how these methods have been applied to a variety of condensed matter systems such as molecular electronics, nanoscale transistors, and superconductors. In addition, we survey the application of NEGF in fields beyond condensed matter, where it has been used to study thermalization in ultra-cold atoms and to understand leptogenesis in the early universe. Throughout, we pay special attention to the challenges of incorporating contacts and interactions, as the NEGF method is uniquely capable of accounting for such features.
Abstract We review one of the most versatile theoretical approaches to the study of time-dependent correlated quantum transport in nano-systems: the non-equilibrium Green’s function (NEGF) formalism. Within this formalism, one can treat, on the same footing, inter-particle interactions, external drives and/or perturbations, and coupling to baths with a (piece-wise) continuum set of degrees of freedom. After a historical overview on the theory of transport in quantum systems, we present a modern introduction of the NEGF approach to quantum transport. We discuss the inclusion of inter-particle interactions using diagrammatic techniques, and the use of the so-called embedding and inbedding techniques which take the bath couplings into account non-perturbatively. In various limits, such as the non-interacting limit and the steady-state limit, we then show how the NEGF formalism elegantly reduces to well-known formulae in quantum transport as special cases. We then discuss non-equilibrium transport in general, for both particle and energy currents. Under the presence of a time-dependent drive—encompassing pump–probe scenarios as well as driven quantum systems—we discuss the transient as well as asymptotic behavior, and also how to use NEGF to infer information on the out-of-equilibrium system. As illustrative examples, we consider model systems general enough to pave the way to realistic systems. These examples encompass one- and two-dimensional electronic systems, systems with electron–phonon couplings, topological superconductors, and optically responsive molecular junctions where electron–photon couplings are relevant.
In this thesis we present methodologies that have been developed to efficiently solve for the non-equilibrium Green’s Function (NEGF) of interacting electron systems, and demonstrate their applications to several different condensed matter systems. These solvers are based on the numerical integration of the equations of motion for the NEGF, which are known as the Kadanoff-Baym Equations (KBE). The KBE are a general set of equations which allow for the study of a wide range of systems that are of interest to condensed matter physicists, including superconductivity, nuclear matter, molecules and atoms, and qubits, among many others. They are also useful for solving impurity models and systems in the dynamical mean-field theory (DMFT) approximation through their ability to incorporate dynamical hybridization functions. The generality of these equations comes from their connection to time-dependent Many-Body Perturbation Theory (MBPT), which introduces non-Markovian dynamical correlations through the self-energy functional. MBPT ensures important conservation laws, such as those for density and energy, are obeyed no matter the type of time-dependent perturbations that are applied to the Hamiltonian. The non-Markovian correlations manifest in a kernel that couples the dynamics to the entire history of the system. This coupling leads to computational costs that become prohibitively expensive as longer-time dynamics are probed, this is especially true for systems which decohere slowly, leading to undamped dynamics and kernels. Due to the large amount of interest in non-equilibrium quantum dynamics, the availability of efficient solvers for the KBE are of great importance. This thesis will focus on the implementation and analysis of integration methods for the KBE, including compression algorithms which lower computational complexity and adaptive integration techniques which allow for resolution of dynamics which vary over wide ranges of timescales. Chapter 1 gives an introduction to quantum systems and the problems we will be studying. Chapter 2 provides a derivation of the NEGF and its representation on the Keldysh contour. Chapter 3 derives the KBE from MBPT and introduces the self- energy and its conservation laws. Chapter 4 introduces the numerical techniques used to integrate the KBE, as well as an analysis of its causality structure. In Chapter 5, we use compression techniques to lower the complexity of the KBE, and provide details on its implementation. In Chapter 6 we apply the compression algorithm to study the Higgs mode in a superconducting system. In Chapter 7, we couple our solver to an adaptive integrator and analyze its performance against fixed-timestep integrators. Finally, Chapter 8 provides a summary of the results and an outlook for future work on efficient solvers.
The method of nonequilibrium Green functions (NEGFs) constitutes a solid framework to describe closed and open quantum systems. It captures situations far from equilibrium including arbitrary perturbations and correlations - effects higher than first order in the binary interactions. On the contrary, in the absence of external fields, it reduces to the imaginary-time Matsubara formalism. The abilities (and successes) of the method originate from its self-consistency and the conserving character of many-body approximations that are introduced by diagram technique. On this account, NEGFs have, now for a long time, brought insight into the dynamics of nuclear matter, non-ideal quantum gases and (semiconductor) plasmas. However, applications to localized, finite and strongly interacting systems have emerged only within the recent decade. The dynamical properties follow from quantum kinetic equations - the two-time Kadanoff-Baym equations for the one-particle NEGF - the numerical solution of which is the central topic of this thesis. The non-Markovian structure of these equations thereby inhibits an unlimited, gradual propagation of the nonequilibrium Green function: Any self-consistent, future dynamics of the system is tightly linked to the (complete) past by non-local self-energies. Further, the success in obtaining solutions is being diminished if the system under investigation is not isotropic or inhomogeneous and, in addition, the spatial topology (e.g., the charge carrier or spin density) needs to be resolved in time. On top of this, small isolated systems are generally not candidates showing fast decoherence which offers little hope for smooth and well-damped integration kernels. Despite these difficulties, the thesis at hand is devoted to the computation of the two-time NEGF for finite, closed systems of which ensembles of electrons bound by nuclei forming atoms or molecules are the most natural representatives. Here, in the spirit of the well-studied one-dimensional helium atom, the electron dynamics in model atoms and molecules is addressed in ab initio fashion. To render calculations possible in the first place, (i) a novel hybrid representation of the NEGF is introduced based on the conception of finite elements (FE) and the discrete variable representation (DVR), and (ii) a parallel algorithm is developed that allows for an efficient, well scalable distributed-memory computation of the full two-time Green function. The first point attributes plenty of flexibility and makes the method readily adjustable to different Hamiltonians avoiding the need to, by hand or in intricate numerical manner, calculate the whole bunch of matrix elements of one- and two-particle operators. Also, in consequence, it drastically simplifies the evaluation of self-energies. The second point makes the NEGF approach ready for large-scale, high-performance computing facilities, the resources (of computer power: mainly memory!) of which are indispensable for (some of) the calculations presented. The applicability of the developed methods is demonstrated inter alia on the basis of helium, beryllium, hydrogen and lithium hydride modeled in one spatial dimension whereby electron-electron correlations are treated in the second(-order) Born approximation. After a thorough discussion on the self-consistent ground states, the thesis focuses on the electron dynamics in the linear response regime as well as in the presence of strong external fields. Overall, to assess the the performance of the second Born approximation, the results are compared to the Hartree-Fock approximation, on the one hand, and to the exact solution of the time-(in)dependent Schrödinger equation, on the other hand. In addition, aiming at the possibility of extended time-dependent calculations, the behavior of the generalized Kadanoff-Baym ansatz is discussed.
The framework of nonequilibrium Green functions (NEGFs) is developed to accurately describe interacting quantum systems evolving in the presence of strongly exciting ultra-fast fields.At its core stands the selfconsistent determination of the selfenergy, a functional of the single-particle Green function, which incorporates all relevant many-body physics and, via diagrammatic techniques, guarantees the conservation of the constants of motion.Over the last decades, NEGFs have been successfully applied to a broad range of domains ranging from homogeneous systems such as non-ideal quantum gases and plasmas over condensed matter to highly inhomogeneous systems such as atoms and molecules photo-ionized by laser-fields or colliding nuclear matter.Only recently, the first accurate simulation of the expansion dynamics of clouds of ultra-cold fermionic atoms in a two-dimensional optical lattice could be achieved using NEGFs.The single-particle Green function is the solution of a coupled set of kinetic equations.Its intrinsic dependence on two independent time-arguments, ranging over the complete past of the system, not only enables the determination of all singleparticle observables but also provides access to important two-particle observables and spectral information.With this coupling to the history, though, comes the practical necessity to apply physically motivated approximation techniques to the structure of the selfenergy and the Green functions, to achieve numerical tractability.This thesis is devoted to provide a thorough overview of the possible approximation techniques including perturbative and non-perturbative approaches.Starting from the exact coupled set of equations of motion for the Green functions, the expressions for all selfenergy approximations are derived in the most general basis representation, i.e., for interaction tensor elements w ijkl .From this, the corresponding expressions in a diagonal basis, where the interaction can be described as a matrix w ij , and for the special case of the Hubbard model, where the interaction is just given as a scalar U , are deduced.
The method of nonequilibrium Green functions (NEGFs) constitutes a solid framework to describe closed and open quantum systems. It captures situations far from equilibrium including arbitrary perturbations and correlations - effects higher than first order in the binary interactions. On the contrary, in the absence of external fields, it reduces to the imaginary-time Matsubara formalism. The abilities (and successes) of the method originate from its self-consistency and the conserving character of many-body approximations that are introduced by diagram technique. On this account, NEGFs have, now for a long time, brought insight into the dynamics of nuclear matter, non-ideal quantum gases and (semiconductor) plasmas. However, applications to localized, finite and strongly interacting systems have emerged only within the recent decade. The dynamical properties follow from quantum kinetic equations - the two-time Kadanoff-Baym equations for the one-particle NEGF - the numerical solution of which is the central topic of this thesis. The non-Markovian structure of these equations thereby inhibits an unlimited, gradual propagation of the nonequilibrium Green function: Any self-consistent, future dynamics of the system is tightly linked to the (complete) past by non-local self-energies. Further, the success in obtaining solutions is being diminished if the system under investigation is not isotropic or inhomogeneous and, in addition, the spatial topology (e.g., the charge carrier or spin density) needs to be resolved in time. On top of this, small isolated systems are generally not candidates showing fast decoherence which offers little hope for smooth and well-damped integration kernels. Despite these difficulties, the thesis at hand is devoted to the computation of the two-time NEGF for finite, closed systems of which ensembles of electrons bound by nuclei forming atoms or molecules are the most natural representatives. Here, in the spirit of the well-studied one-dimensional helium atom, the electron dynamics in model atoms and molecules is addressed in ab initio fashion. To render calculations possible in the first place, (i) a novel hybrid representation of the NEGF is introduced based on the conception of finite elements (FE) and the discrete variable representation (DVR), and (ii) a parallel algorithm is developed that allows for an efficient, well scalable distributed-memory computation of the full two-time Green function. The first point attributes plenty of flexibility and makes the method readily adjustable to different Hamiltonians avoiding the need to, by hand or in intricate numerical manner, calculate the whole bunch of matrix elements of one- and two-particle operators. Also, in consequence, it drastically simplifies the evaluation of self-energies. The second point makes the NEGF approach ready for large-scale, high-performance computing facilities, the resources (of computer power: mainly memory!) of which are indispensable for (some of) the calculations presented. The applicability of the developed methods is demonstrated inter alia on the basis of helium, beryllium, hydrogen and lithium hydride modeled in one spatial dimension whereby electron-electron correlations are treated in the second(-order) Born approximation. After a thorough discussion on the self-consistent ground states, the thesis focuses on the electron dynamics in the linear response regime as well as in the presence of strong external fields. Overall, to assess the the performance of the second Born approximation, the results are compared to the Hartree-Fock approximation, on the one hand, and to the exact solution of the time-(in)dependent Schrödinger equation, on the other hand. In addition, aiming at the possibility of extended time-dependent calculations, the behavior of the generalized Kadanoff-Baym ansatz is discussed.
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).