We consider a projection operator approach to the non-equilibrium Green function equation-of-motion (PO-NEGF EOM) method. The technique resolves problems of arbitrariness in truncation of an infinite chain of EOMs and prevents violation of symmetry relations resulting from the truncation (equivalence of left- and right-sided EOMs is shown and symmetry with respect to interchange of Fermi or Bose operators before truncation is preserved). The approach, originally developed by Tserkovnikov (1999 Theor. Math. Phys. 118 85) for equilibrium systems, is reformulated to be applicable to time-dependent non-equilibrium situations. We derive a canonical form of EOMs, thus explicitly demonstrating a proper result for the non-equilibrium atomic limit in junction problems. A simple practical scheme applicable to quantum transport simulations is formulated. We perform numerical simulations within simple models and compare results of the approach to other techniques and (where available) also to exact results.
We consider a projection operator approach to the non-equilibrium Green function equation-of-motion (PO-NEGF EOM) method. The technique resolves problems of arbitrariness in truncation of an infinite chain of EOMs and prevents violation of symmetry relations resulting from the truncation (equivalence of left- and right-sided EOMs is shown and symmetry with respect to interchange of Fermi or Bose operators before truncation is preserved). The approach, originally developed by Tserkovnikov (1999 Theor. Math. Phys. 118 85) for equilibrium systems, is reformulated to be applicable to time-dependent non-equilibrium situations. We derive a canonical form of EOMs, thus explicitly demonstrating a proper result for the non-equilibrium atomic limit in junction problems. A simple practical scheme applicable to quantum transport simulations is formulated. We perform numerical simulations within simple models and compare results of the approach to other techniques and (where available) also to exact results.
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 the equation of motion approach to the two-time Green's functions, conventional Tyablikov-type truncation of the chain of equations is rather arbitrary and apt to violate the analytical structure of Green's functions. Here, we propose a practical way to truncate the equations of motion using operator projection. The partial projection approximation is introduced to evaluate the Liouville matrix. It guarantees the causality of Green's functions, fulfills the time translation invariance and the particle-hole symmetry, and is easy to implement in a computer. To benchmark this method, we study the Anderson impurity model using the operator basis at the level of Lacroix approximation. Improvement over conventional Lacroix approximation is observed. The distribution of Kondo screening in the energy space is studied using this method.
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.
We present a non-perturbative Floquet-based non-equilibrium Green's function (NEGF) method to study electron transport in a quantum system driven simultaneously by multiple independent terms (multi-mode). We first derive the two-mode Floquet NEGF based on two-step transformations of the retarded-advanced Green's function from the Kadanoff-Baym equation. This derivation proceeds by elaborating on the expectation values of the number and current operators. The two-mode Floquet NEGF is then extended to cases with multiple drivings. The method is tested by investigating current suppression in the presence of two drivings. We show that an extra sinusoidal off-diagonal driving can cause substantial modification to the current suppression, provided careful selection of the driving frequency. Consequently, we expect that the established method has broad applications in a wide range of open quantum systems driven by complicated drivings.
The equations-of-motion (EOM) hierarchy satisfied by the Green functions of a quantum dot embedded in an external mesoscopic network is considered within a high-order decoupling approximation scheme. Exact analytic solutions of the resulting coupled integral equations are presented in several limits. In particular, it is found that at the particle-hole symmetric point the EOM Green function is temperature independent due to a discontinuous change in the imaginary part of the interacting self-energy. However, this imaginary part obeys the Fermi liquid unitarity requirement away from this special point, at zero temperature. Results for the occupation numbers, the density of states, and the local spin susceptibility are compared with exact Fermi liquid relations and the Bethe ansatz solution. The approximation is found to be very accurate far from the Kondo regime. In contrast, the description of the Kondo effect is valid on a qualitative level only. In particular, we find that the Friedel sum-rule is considerably violated, up to 30%, and the spin susceptibility is underestimated. We show that the widely used simplified version of the EOM method, which does not account fully for the correlations on the network, fails to produce the Kondo correlations even qualitatively.
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).