Time reversal symmetry is a fundamental property of many quantum mechanical systems. The relation between statistical physics and time reversal is subtle and not all statistical theories conserve this particular symmetry, most notably hydrodynamic equations and kinetic equations such as the Boltzmann equation. In this article it is shown analytically that quantum kinetic generalizations of the Boltzmann equation that are derived using the nonequilibrium Green functions formalism as well as all approximations that stem from $Φ$-derivable selfenergies are time reversal invariant.
The nonequilibrium dynamics of strongly-correlated fermions in lattice systems have attracted considerable interest in the condensed matter and ultracold atomic-gas communities. While experiments have made remarkable progress in recent years, there remains a need for the further development of theoretical tools that can account for both the nonequilibrium conditions and strong correlations. For instance, time-dependent theoretical quantum approaches based on the density matrix renormalization group (DMRG) methods have been primarily applied to one-dimensional setups. Recently, two-dimensional …
The nonequilibrium description of quantum systems requires, for more than two or three particles, the use of a reduced description to be numerically tractable. Two possible approaches are based on either reduced density matrices or nonequilibrium Green's functions (NEGF). Both concepts are formulated in terms of hierarchies of coupled equations---the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for the reduced density operators and the Martin-Schwinger-hierarchy (MS) for the Green's functions, respectively. In both cases, similar approximations are introduced to decouple the hierarchy…
Transport properties of strongly correlated quantum systems are of central interest in condensed matter, ultracold atoms and in dense plasmas. There, the proper treatment of strong correlations poses a great challenge to theory. Here, we apply a Nonequilibrium Green Functions approach using a lattice model as a basic system. This allow us to treat a finite spatially inhomogeneous system with an arbitrary nonequilibrium initial state. Placing all particles initially to one side of the system allows for a nonequilibrium study of diffusion. Strong correlation effects are incorporated via T-matrix…
The carrier generation in insulators subjected to strong electric fields is characterized by the Landau-Zener formula for the tunneling probability with a nonperturbative exponent. Despite its long history with diverse applications and extensions, study of nonequilibrium steady states and associated current response in the presence of the generated carriers has been mainly limited to numerical simulations so far. Here, we develop a framework to calculate the nonequilibrium Green's function of generic insulating systems under a DC electric field, in the presence of a fermionic reservoir. Using …
A nonequilibrium Green's functions (NEGF) approach for spatially inhomogeneous, strongly correlated artificial atoms is presented and applied to compute the time-dependent properties while starting from a (correlated) initial few-electron state at finite temperatures. In the regime of moderate to strong coupling, we consider the Kohn mode of a three-electron system in a parabolic confinement excited by a short pulsed classical laser field treated in dipole approximation. In particular, we numerically confirm that this mode is preserved within a conserving (e.g. Hartree-Fock or second Born) the…
The pair distribution function (PDF) is a key quantity for the analysis of correlation effects of a quantum system both in equilibrium and far from equilibrium. We derive an expression for the PDF in terms of the single-particle Green's functions---the solutions of the Keldysh/Kadanoff-Baym equations in the two-time plane---for a one- or two-component system. The result includes initial correlations and generalizes previous density matrix expressions from single-time quantum kinetic theory. Explicit expressions for the PDF are obtained in second Born approximation.
The dynamics of strongly correlated fermions following an external excitation reveals extremely rich collective quantum effects. Examples are fermionic atoms in optical lattices, electrons in correlated materials, and dense quantum plasmas. Presently, the only quantum-dynamics approach that rigorously describes these processes in two and three dimensions is nonequilibrium Green functions (NEGF). However, NEGF simulations are computationally expensive due to their $T^3$ scaling with the simulation duration $T$. Recently, $T^2$ scaling was achieved with the generalized Kadanoff--Baym ansatz (GKB…
We propose and apply the finite-element discrete variable representation to express the nonequilibrium Green's function for strongly inhomogeneous quantum systems. This method is highly favorable against a general basis approach with regard to numerical complexity, memory resources, and computation time. Its flexibility also allows for an accurate representation of spatially extended hamiltonians, and thus opens the way towards a direct solution of the two-time Schwinger/Keldysh/Kadanoff-Baym equations on spatial grids, including e.g. the description of highly excited states in atoms. As first…
In this contribution, we discuss the finite-element discrete variable representation (FE-DVR) of the nonequilibrium Green's function and its implications on the description of strongly inhomogeneous quantum systems. In detail, we show that the complementary features of FEs and the DVR allows for a notably more efficient solution of the two-time Schwinger/Keldysh/Kadanoff-Baym equations compared to a general basis approach. Particularly, the use of the FE-DVR leads to an essential speedup in computing the self-energies. As atomic and molecular examples we consider the He atom and the linear ver…
Suppression of rectification at metal--Mott-insulator interfaces, which is previously shown by numerical solutions to the time-dependent Schrödinger equation and experiments on real devices, is reinvestigated theoretically by nonequilibrium Green's functions. The one-dimensional Hubbard model is used for a Mott insulator. The effects of attached metallic electrodes are incorporated into the self-energy. A scalar potential originating from work-function differences and satisfying the Poisson equation is added to the model. For the electron density, we decompose it into three parts. One is obtai…
Nonperturbative dynamic theory has a particular advantage in studying the transport in a quantum impurity system in a steady state. Here, we develop a new approach for obtaining the retarded Green's function expressed in resolvent form. We use the Heisenberg picture to facilitate dynamic theory and propose a new systematic method of collecting the basis vectors spanning the Liouville space, which is the most crucial step in obtaining the resolvent Green's function. We obtain all the linearly independent basis vectors for studying the single-impurity Anderson models with one and two reservoirs.…
This work provides a unified theoretical treatment of the single and correlated double-electron emission from a general electronic system. Using Feshbach projection method, the states of interest are selected by the projection operator; the Feshbach-Schur map determines the effective Hamiltonian and the optical potential for the emitted electrons. On the other hand, the nonequilibrium Green's functions method is demonstrated to be a complementary approach and an explicit correspondence between both methods is established. For a self-contained exposition some results on single electron emission…
The zero-temperature single-particle Green's function of correlated fermion models with moderately large Hilbert-space dimensions can be calculated by means of Krylov-space techniques. The conventional Lanczos approach consists of finding the ground state in a first step, followed by an approximation for the resolvent of the Hamiltonian in a second step. We analyze the character of this approximation and discuss a numerically exact variant of the Lanczos method which is formulated in the time domain. This method is extended to get the nonequilibrium single-particle Green's function defined on …
We derive expressions for the zeroth and the first three spectral moment sum rules for the retarded Green's function and for the zeroth and the first spectral moment sum rules for the retarded self-energy of the inhomogeneous Bose-Hubbard model in nonequilibrium, when the local on-site repulsion and the chemical potential are time-dependent, and in the presence of an external time-dependent electromagnetic field. We also evaluate these moments for equilibrium, where all time dependence and external fields vanish. Unlike similar sum rules for the Fermi-Hubbard model, in the Bose-Hubbard model c…
We present a unified transport theory of hybrid structures, in which a confined normal state ($N$) sample is sandwiched between two leads each of which can be either a ferromagnet ($F$) or a superconductor ($S$) via tunnel barriers. By introducing a four-dimensional Nambu-spinor space, a general current formula is derived within the Keldysh nonequilibrium Green function formalism, which can be applied to various kinds of hybrid mesoscopic systems with strong correlations even in the nonequilibrium situation. Such a formula is gauge invariant. We also demonstrate analytically for some quantitie…
We study steady state transport through a double quantum dot array using the equation-of-motion approach to the nonequilibrium Green functions formalism. This popular technique relies on uncontrolled approximations to obtain a closure for a hierarchy of equations, however its accuracy is questioned. We focus on 4 different closures, 2 of which were previously proposed in the context of the single quantum dot system (Anderson impurity model) and were extended to the double quantum dot array, and develop 2 new closures. Results for the differential conductance are compared to those attained by a…
The nonequilibrium spectral properties of the Anderson impurity model with a chemical potential bias are investigated within a numerically exact real time quantum Monte Carlo formalism. The two-time correlation function is computed in a form suitable for nonequilibrium dynamical mean field calculations. Additionally, the evolution of the model's spectral properties are simulated in an alternative representation, defined by a hypothetical but experimentally realizable weakly coupled auxiliary lead. The voltage splitting of the Kondo peak is confirmed and the dynamics of its formation after a co…
The effect of electron-electron scattering on the equilibrium properties of few-electron quantum dots is investigated by means of nonequilibrium Green's functions theory. The ground and equilibrium state is self-consistently computed from the Matsubara (imaginary time) Green's function for the spatially inhomogeneous quantum dot system whose constituent charge carriers are treated as spin-polarized. To include correlations, the Dyson equation is solved, starting from a Hartree-Fock reference state, within a conserving (second order) self-energy approximation where direct and exchange contribut…
We present a generalized dynamical mean-field approach for the nonequilibrium physics of a strongly correlated system in the presence of a time-dependent external field. The Keldysh Green's function formalism is used to study the nonequilibrium problem. We derive a closed set of self-consistency equations in the case of a driving field with frequency Omega and wave vector q. We present numerical results for the local frequency-dependent Green's function and the self-energy for different values of the field amplitude in the case of a uniform external field using the iterated perturbation theory…
We develop a nonperturbative dynamical theory (NDT) to calculate the retarded Green's function under nonequilibrium conditions. The NDT is particularly useful for treating nonequilibrium transport problems in systems with strong correlation. We apply our NDT to the well-known single-impurity Anderson model at equilibrium to determine its feasibility. We then apply it to a nonequilibrium transport problem in a system with Kondo coupling. An Anderson model with two metallic reservoirs is studied to understand the phenomenon of Kondo-peak splitting in a single-electron transistor of mesoscopic si…
The operators of localized spins within a magnetic material commute at different sites of its lattice and anticommute on the same site, so they are neither fermionic nor bosonic operators. Thus, to construct diagrammatic many-body perturbation theory, the spin operators are usually mapped to the bosonic ones with Holstein-Primakoff (HP) transformation being the most widely used in magnonics and spintronics literature. However, to make calculations tractable, the square root of operators in the HP transformation is expanded into a Taylor series truncated to some low order. This poses a question…
Nonequilibrium Green Functions Simulations for Large Correlated Systems
Institute for Theoretical Physics and Astrophysics, Kiel University（基尔大学理论物理与天体物理研究所）；Kiel Nano, Surface and Interface Science KiNSIS, Kiel University（基尔大学基尔纳米、表面与界面科学KiNSIS）
The description of the dynamics of correlated electrons in quantum impurity models is typically described within the nonequilibrium Green function formalism combined with a suitable approximation. One common approach is based on the equation-of-motion technique often used to describe different regimes of the dynamic response. Here, we show that this approach may violate certain symmetry relations that must be fulfilled by the definition of the Green functions. These broken symmetries can lead to unphysical behavior. To circumvent this pathological shortcoming of the equation-of-motion approach…
We study the role of excited phonon populations in the relaxation rates of nonequilibrium electrons using a nonequilibrium Green's function formalism. The transient modifications in the phononic properties are accounted for by self-consistently solving the Dyson equation for the electron and phonon Green's functions. The pump induced changes manifest in both the electronic and phononic spectral functions. We find that the excited phonon populations suppress the decay rates of nonequilibrium electrons due to enhanced phonon absorption. The increased phonon occupation also sets the nonequilibriu…
We perform a perturbative analysis for the nonequilibrium Green functions of the spinless Falicov-Kimball model in the presence of an arbitrary external time-dependent but spatially uniform electric field. The conduction electron self-energy is found from a strictly truncated second-order perturbative expansion in the local electron-electron repulsion U. We examine the current at half-filling, and compare to both the semiclassical Boltzmann equation and exact numerical solutions for the contour-ordered Green functions from a transient-response formalism (in infinite dimensions) on the Kadanoff…
The conductance through a mesoscopic system of interacting electrons coupled to two adjacent leads is conventionally derived via the Keldysh nonequilibrium Green's function technique, in the limit of noninteracting leads [see Y. Meir \emph{et al.}, Phys. Rev. Lett. \textbf{68}, 2512 (1991)]. We extend the standard formalism to cater for a quantum dot system with Coulombic interactions between the quantum dot and the leads. The general current expression is obtained by considering the equation of motion of the time-ordered Green's function of the system. The nonequilibrium effects of the intera…
This article presents an overview on recent progress in the theory of nonequilibrium Green functions (NEGF). NEGF, presently, are the only \textit{ab-initio} quantum approach that is able to study the dynamics of correlations for long times in two and three dimensions. However, until recently, NEGF simulations have mostly been performed with rather simple selfenergy approximations such as the second-order Born approximation (SOA). While they correctly capture the qualitative trends of the relaxation towards equilibrium, the reliability and accuracy of these NEGF simulations has remained open, …
The time evolution in quantum many-body systems after external excitations is attracting high interest in many fields. The theoretical modeling of these processes is challenging, and the only rigorous quantum-dynamics approach that can treat correlated fermions in two and three dimensions is nonequilibrium Green functions (NEGF). However, NEGF simulations are computationally expensive due to their $T^3$-scaling with the simulation duration $T$. Recently, $T^2$-scaling was achieved with the generalized Kadanoff--Baym ansatz (GKBA), for the second-order Born (SOA) selfenergy, which has substanti…
We present a nonequilibrium strong-coupling approach to inhomogeneous systems of ultracold atoms in optical lattices. We demonstrate its application to the Mott-insulating phase of a two-dimensional Fermi-Hubbard model in the presence of a trap potential. Since the theory is formulated self-consistently, the numerical implementation relies on a massively parallel evaluation of the self-energy and the Green's function at each lattice site, employing thousands of CPUs. While the computation of the self-energy is straightforward to parallelize, the evaluation of the Green's function requires the …
We study the high- and low-voltage properties of the out-of-equilibrium Anderson model for quantum dots, using a functional method in the Keldysh formalism. The Green's function at the impurity site can be regarded as a functional of a nonequilibrium distribution function. The dependence of the Green's function on the bias voltage V and temperature T arises through the nonequilibrium distribution function. From this behavior as a functional, it is shown that the nonequilibrium Green's function at high-voltage limit is identical to the equilibrium Green's function at high-temperature limit. Thi…
This is an introductory chapter on how to calculate nonequilibrium Green's functions via dynamical mean-field theory for the Autumn School on Correlated Electrons: Many-Body Methods for Real Materials, 16-20 September 2019, Forschungszentrum Juelich. It is appropriate for graduate students with a solid state physics and advanced quantum mechanics background.
We propose a description of nonequilibrium systems via a simple protocol that combines exchange-correlation potentials from density functional theory with self-energies of many-body perturbation theory. The approach, aimed to avoid double counting of interactions, is tested against exact results in Hubbard-type systems, with respect to interaction strength, perturbation speed and inhomogeneity, and system dimensionality and size. In many regimes, we find significant improvement over adiabatic time dependent density functional theory or second Born nonequilibrium Green's function approximations…
We derive an exact mapping from the action of nonequilibrium dynamical mean-field theory (DMFT) to a single-impurity Anderson model (SIAM) with time-dependent parameters, which can be solved numerically by exact diagonalization. The representability of the nonequilibrium DMFT action by a SIAM is established as a rather general property of nonequilibrium Green functions. We also obtain the nonequilibrium DMFT equations using the cavity method alone. We show how to numerically obtain the SIAM parameters using Cholesky or eigenvector matrix decompositions. As an application, we use a Krylov-based…
Magnetization reversal is a well-studied problem with obvious applicability in computer hard-drives. One can accomplish a magnetization reversal in at least one of two ways: application of a magnetic field, or through a spin current. The latter is more amenable to a fully quantum mechanical analysis. We formulate and solve the problem whereby a spin current interacts with a ferromagnetic Heisenberg spin chain, to eventually reverse the magnetization of the chain. Spin-flips are accomplished through both elastic and inelastic scattering. A consequence of the inelastic scattering channel, when i…
In this work, we explore the possibility of emergent nonequilibrium steady states arising from the electric-field-driven Mott insulator via the Keldysh-Floquet dynamical mean field theory (DMFT), which can determine the fully-interacting, nonequilibrium steady-state Green's functions with the noninteracting counterparts as an input to the DMFT self-consistency loop. Unlike the retarded component, obtaining the lesser Green's function for the noninteracting system presents an important obstacle since the thermalization of the noninteracting system still requires a precise understanding of the d…
