The contour idea

Correlation effects in bistability at the nanoscale: Steady state and beyond

The possibility of finding multistability in the density and current of an interacting nanoscale junction coupled to semi-infinite leads is studied at various levels of approximation. The system is driven out of equilibrium by an external bias and the nonequilibrium properties are determined by real-time propagation using both time-dependent density functional theory (TDDFT) and many-body perturbation theory (MBPT). In TDDFT the exchange-correlation effects are described within a recently proposed adiabatic local density approximation (ALDA). In MBPT the electron-electron interaction is incorporated in a many-body self-energy which is then approximated at the Hartree-Fock (HF), second-Born,…

One-particle Green's function

In this chapter we get acquainted with the one-particle Green's function G , or simply the Green's function. The chapter is divided in three parts. In the first part (Section 6.1) we illustrate what kind of physical information can be extracted from the different Keldysh components of G . The aim of this first part is to introduce some general concepts without being too formal. In the second part (Section 6.2) we calculate the noninteracting Green's function. Finally in the third part (Sections 6.3 and 6.4) we consider the interacting Green's function and derive several exact properties. We also discuss other physical (and measurable) quantities that can be calculated from G and that are re…

Many-particle Green's functions

Diagrammatic expansion for positive density-response spectra: Application to the electron gas

In a recent paper [Phys. Rev. B 90, 115134 (2014)] we put forward a diagrammatic expansion for the self-energy which guarantees the positivity of the spectral function. In this work we extend the theory to the density response function. We write the generic diagram for the density-response spectrum as the sum of partitions. In a partition the original diagram is evaluated using time-ordered Green's functions (GF) on the left-half of the diagram, antitime-ordered GF on the right-half of the diagram and lesser or greater GF gluing the two halves. As there exist more than one way to cut a diagram in two halves, to every diagram corresponds more than one partition. We recognize that the most co…

In- and out-of-equilibrium {\em ab initio} theory of electrons and phonons

We lay down the {\em ab initio} many-body quantum theory of electrons and phonons in- and out-of-equilibrium at any temperature. We begin by addressing a fundamental issue concerning the {\em ab initio} Hamiltonian in the harmonic approximation, which we show must be determined {\em self-consistently} to avoid inconsistencies. After identifying the most suitable partitioning into a ``noninteracting'' and an ``interacting'' part we embark on the Green's function diagrammatic analysis. We single out key diagrammatic structures to carry on the expansion in terms of dressed propagators and screened interaction. The final outcome is the finite-temperature nonequilibrium extension of the Hedin eq…

Linear response theory: many-body formulation

Observation of an excitonic Mott transition through ultrafast core-cum-conduction photoemission spectroscopy

Time-resolved soft-X-ray photoemission spectroscopy is used to simultaneously measure the ultrafast dynamics of core-level spectral functions and excited states upon excitation of excitons in WSe$_2$. We present a many-body approximation for the Green's function, which excellently describes the transient core-hole spectral function. The relative dynamics of excited-state signal and core levels reveals a delayed core-hole renormalization due to screening by excited quasi-free carriers, revealing an excitonic Mott transition. These findings establish time-resolved core-level photoelectron spectroscopy as a sensitive probe of subtle electronic many-body interactions and an ultrafast electronic…

Equilibrium and nonequilibrium many-body perturbation theory: a unified framework based on the Martin-Schwinger hierarchy

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…

Ultra-nonlocality in density functional theory for photo-emission spectroscopy.

We derive an exact expression for the photo-current of photo-emission spectroscopy using time-dependent current density functional theory (TDCDFT). This expression is given as an integral over the Kohn-Sham spectral function renormalized by effective potentials that depend on the exchange-correlation kernel of current density functional theory. We analyze in detail the physical content of this expression by making a connection between the density-functional expression and the diagrammatic expansion of the photo-current within many-body perturbation theory. We further demonstrate that the density functional expression does not provide us with information on the kinetic energy distribution of…

Time-dependent Landauer—Büttiker formalism for superconducting junctions at arbitrary temperatures

We discuss an extension of our earlier work on the time-dependent Landauer– Büttiker formalism for noninteracting electronic transport. The formalism can without complication be extended to superconducting central regions since the Green’s functions in the Nambu representation satisfy the same equations of motion which, in turn, leads to the same closed expression for the equal-time lesser Green’s function, i.e., for the time-dependent reduced one-particle density matrix. We further write the finite-temperature frequency integrals in terms of known special functions thereby considerably speeding up the computation. Simulations in simple normal metal – superconductor – normal metal junctions…

Wick Theorem for General Initial States

We present a compact and simplified proof of a generalized Wick theorem to calculate the Green's function of bosonic and fermionic systems in an arbitrary initial state. It is shown that the decomposition of the non-interacting $n$-particle Green's function is equivalent to solving a boundary problem for the Martin-Schwinger hierarchy; for non-correlated initial states a one-line proof of the standard Wick theorem is given. Our result leads to new self-energy diagrams and an elegant relation with those of the imaginary-time formalism is derived. The theorem is easy to use and can be combined with any ground-state numerical technique to calculate time-dependent properties.

Kadanoff-Baym approach to quantum transport through interacting nanoscale systems: from the transient to the steady-state regime

We propose a time-dependent many-body approach to study the short-time dynamics of correlated electrons in quantum transport through nanoscale systems contacted to metallic leads. This approach is based on the time-propagation of the Kadanoff-Baym equations for the nonequilibrium many-body Green's function of open and interacting systems out of equilibrium. An important feature of the method is that it takes full account of electronic correlations and embedding effects in the presence of time-dependent external fields, while at the same time satisfying the charge conservation law. The method further extends the Meir-Wingreen formula to the time domain for initially correlated states. We stu…

Vertex corrections for positive-definite spectral functions of simple metals

We present a systematic study of vertex corrections in the homogeneous electron gas at metallic densities. The vertex diagrams are built using a recently proposed positive-definite diagrammatic expansion for the spectral function. The vertex function not only provides corrections to the well known plasmon and particle-hole scatterings, but also gives rise to new physical processes such as generation of two plasmon excitations or the decay of the one-particle state into a two-particles-one-hole state. By an efficient Monte Carlo momentum integration we are able to show that the additional scattering channels are responsible for the bandwidth reduction observed in photoemission experiments on…

Electronic transport in molecular junctions : The generalized Kadanoff–Baym ansatz with initial contact and correlations

The generalized Kadanoff-Baym ansatz (GKBA) offers a computationally inexpensive approach to simulate out-of-equilibrium quantum systems within the framework of nonequilibrium Green's functions. For finite systems the limitation of neglecting initial correlations in the conventional GKBA approach has recently been overcome [Phys. Rev. B 98, 115148 (2018)]. However, in the context of quantum transport the contacted nature of the initial state, i.e., a junction connected to bulk leads, requires a further extension of the GKBA approach. In this work, we lay down a GKBA scheme which includes initial correlations in a partition-free setting. In practice, this means that the equilibration of the …

Correlation-driven sub-3 fs charge migration in ionised adenine

Sudden ionisation of a relatively large molecule can initiate a correlation-driven process dubbed charge migration, where the electron density distribution is expected to rapidly change. Capturing this few-femtosecond/attosecond charge redistribution represents the real-time observation of the electron correlation in the molecule. So far, there has been no experimental evidence of this process. Here we report on a time-resolved study of the correlation-driven charge migration process occurring in the bio-relevant molecule adenine after ionisation by a 15-35 eV attosecond pulse . We find that, the production of intact doubly charged adenine - via a shortly-delayed laser-induced second ionisa…

Dynamically screened vertex correction to $GW$

Diagrammatic perturbation theory is a powerful tool for the investigation of interacting many-body systems, the self-energy operator $\mathrm{\ensuremath{\Sigma}}$ encoding all the variety of scattering processes. In the simplest scenario of correlated electrons described by the $GW$ approximation for the electron self-energy, a particle transfers a part of its energy to neutral excitations. Higher-order (in screened Coulomb interaction $W$) self-energy diagrams lead to improved electron spectral functions (SFs) by taking more complicated scattering channels into account and by adding corrections to lower order self-energy terms. However, they also may lead to unphysical negative spectral f…

Phononic heat transport in the transient regime: An analytic solution

We investigate the time-resolved quantum transport properties of phonons in arbitrary harmonic systems connected to phonon baths at different temperatures. We obtain a closed analytic expression of the time-dependent one-particle reduced density matrix by explicitly solving the equations of motion for the nonequilibrium Green's function. This is achieved through a well-controlled approximation of the frequency-dependent bath self-energy. Our result allows for exploring transient oscillations and relaxation times of local heat currents, and correctly reduces to an earlier known result in the steady-state limit. We apply the formalism to atomic chains, and benchmark the validity of the approx…

Distinguishing Majorana Zero Modes from Impurity States through Time-Resolved Transport

We study time-resolved charge transport in a superconducting nanowire using time-dependent Landauer-B{\"u}ttiker theory. We find that the steady-state Majorana zero-bias conductance peak emerges transiently accompanied by characteristic oscillations after a bias-voltage quench. These oscillations are absent for a trivial impurity state that otherwise shows a very similar steady-state signal as the Majorana zero mode. In addition, we find that Andreev bound states or quasi-Majorana states in the topologically trivial bulk phase can give rise to a zero-bias conductance peak, also retaining the transient properties of the Majorana zero mode. Our results imply that (1) time-resolved transport m…

The generalized Kadanoff-Baym ansatz with initial correlations

Within the non-equilibrium Green's function (NEGF) formalism, the Generalized Kadanoff-Baym Ansatz (GKBA) has stood out as a computationally cheap method to investigate the dynamics of interacting quantum systems driven out of equilibrium. Current implementations of the NEGF--GKBA, however, suffer from a drawback: real-time simulations require {\em noncorrelated} states as initial states. Consequently, initial correlations must be built up through an adiabatic switching of the interaction before turning on any external field, a procedure that can be numerically highly expensive. In this work, we extend the NEGF--GKBA to allow for {\em correlated} states as initial states. Our scheme makes i…

Ultrafast dynamics of adenine following XUV ionization

JPhys photonics 4, 034003 (2022). doi:10.1088/2515-7647/ac6ea5 special issue: "Focus on Nanophotonics and Biophotonics for Biomedical and Environmental Applications"

Diagrammatic expansion for positive spectral functions beyond GW : Application to vertex corrections in the electron gas

We present a diagrammatic approach to construct self-energy approximations within many-body perturbation theory with positive spectral properties. The method cures the problem of negative spectral functions which arises from a straightforward inclusion of vertex diagrams beyond the GW approximation. Our approach consists of a two-steps procedure: we first express the approximate many-body self-energy as a product of half-diagrams and then identify the minimal number of half-diagrams to add in order to form a perfect square. The resulting self-energy is an unconventional sum of self-energy diagrams in which the internal lines of half a diagram are time-ordered Green's functions whereas those…

Linear response theory: preliminaries

Time-dependent Landauer-Büttiker formula for transient dynamics

We solve analyti ally the Kadano Baym equations for a nonintera ting jun tion onne ted to an arbitrary number of nonintera ting wide-band terminals. The initial equilibrium state is properly des ribed by the addition of an imaginary tra k to the time ontour. From the solution we obtain the time-dependent ele tron densities and urrents within the jun tion. The nal results are analyti expressions as a fun tion of time, and therefore no time propagation is needed either in transient or in steady-state regimes. We further present and dis uss some appli ations of the obtained formulae. peerReviewed

Real-time switching between multiple steady-states in quantum transport

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First-principles nonequilibrium Green's-function approach to transient photoabsorption: Application to atoms

We put forward a first-principle NonEquilibrium Green's Function (NEGF) approach to calculate the transient photoabsorption spectrum of optically thin samples. The method can deal with pump fields of arbitrary strength, frequency and duration as well as for overlapping and nonoverlapping pump and probe pulses. The electron-electron repulsion is accounted for by the correlation self-energy, and the resulting numerical scheme deals with matrices that scale quadratically with the system size. Two recent experiments, the first on helium and the second on krypton, are addressed. For the first experiment we explain the bending of the Autler-Townes absorption peaks with increasing the pump-probe d…

Comparative study of many-body perturbation theory and time-dependent density functional theory in the out-of-equilibrium Anderson model

We study time-dependent electron transport through an Anderson model. The electronic interactions on the impurity site are included via the self-energy approximations at Hartree-Fock (HF), second Born (2B), GW, and T-matrix levels as well as within a time-dependent density functional (TDDFT) scheme based on the adiabatic Bethe-ansatz local density approximation (ABALDA) for the exchange-correlation potential. The Anderson model is driven out of equilibrium by applying a bias to the leads, and its nonequilibrium dynamics is determined by real-time propagation. The time-dependent currents and densities are compared to benchmark results obtained with the time-dependent density matrix renormali…

Charge dynamics in molecular junctions: nonequilibrium Green's function approach made fast

Real-time Green's function simulations of molecular junctions (open quantum systems) are typically performed by solving the Kadanoff-Baym equations (KBE). The KBE, however, impose a serious limitation on the maximum propagation time due to the large memory storage needed. In this work we propose a simplified Green's function approach based on the Generalized Kadanoff-Baym Ansatz (GKBA) to overcome the KBE limitation on time, significantly speed up the calculations, and yet stay close to the KBE results. This is achieved through a twofold advance: first we show how to make the GKBA work in open systems and then construct a suitable quasi-particle propagator that includes correlation effects …

Kadanoff-Baym approach to time-dependent quantum transport in AC and DC fields

We have developed a method based on the embedded Kadanoff-Baym equations to study the time evolution of open and inhomogeneous systems. The equation of motion for the Green's function on the Keldysh contour is solved using different conserving many-body approximations for the self-energy. Our formulation incorporates basic conservation laws, such as particle conservation, and includes both initial correlations and initial embedding effects, without restrictions on the time-dependence of the external driving field. We present results for the time-dependent density, current and dipole moment for a correlated tight binding chain connected to one-dimensional non-interacting leads exposed to DC …

MBPT for the Green's function

Conserving approximations: two-particle Green's function

Image charge dynamics in time-dependent quantum transport

In this work we investigate the effects of the electron-electron interaction between a molecular junction and the metallic leads in time-dependent quantum transport. We employ the recently developed embedded Kadanoff-Baym method [Phys. Rev. B 80, 115107 (2009)] and show that the molecule-lead interaction changes substantially the transient and steady-state transport properties. We first show that the mean-field Hartree-Fock (HF) approximation does not capture the polarization effects responsible for the renormalization of the molecular levels neither in nor out of equilibrium. Furthermore, due to the time-local nature of the HF self-energy there exists a region in parameter space for which …

Time-dependent Landauer-Büttiker formula: Application to transient dynamics in graphene nanoribbons

In this work we develop a time-dependent extension of the Landauer-B\"uttiker approach to study transient dynamics in time-dependent quantum transport through molecular junctions. A key feature of the approach is that it provides a closed integral expression for the time-dependence of the density matrix of the molecular junction after switch-on of a bias or gate potential which can be evaluated without the necessity of propagating individual single-particle orbitals. This allows for the study of time-dependent transport in large molecular systems coupled to wide band leads. As an application of the formalism we study the transient dynamics of zigzag and armchair graphene nanoribbons of diff…

Fast Green’s Function Method for Ultrafast Electron-Boson Dynamics

The interaction of electrons with quantized phonons and photons underlies the ultrafast dynamics of systems ranging from molecules to solids, and it gives rise to a plethora of physical phenomena experimentally accessible using time-resolved techniques. Green's function methods offer an invaluable interpretation tool since scattering mechanisms of growing complexity can be selectively incorporated in the theory. Currently, however, real-time Green's function simulations are either prohibitively expensive due to the cubic scaling with the propagation time or do neglect the feedback of electrons on the bosons, thus violating energy conservation. We put forward a computationally efficient Gree…

Time-resolved photoabsorption in finite systems: A first-principles NEGF approach

We describe a first-principles NonEquilibrium Green’s Function (NEGF) approach to time-resolved photoabsortion spectroscopy in atomic and nanoscale systems. The method is used to highlight a recently discovered dynamical correlation effect in the spectrum of a Krypton gas subject to a strong ionizing pump pulse. We propose a minimal model that captures the effect, and study the performance of time-local approximations versus time-nonlocal ones. In particular we implement the time-local Hartree-Fock and Markovian second Born (2B) approximation as well as the exact adiabatic approximation within the Time-Dependent Density Functional Theory framework. For the time-nonlocal approximation we ins…

Time-dependent Landauer-B\"uttiker formalism for superconducting junctions at arbitrary temperatures

We discuss an extension of our earlier work on the time-dependent Landauer--B\"uttiker formalism for noninteracting electronic transport. The formalism can without complication be extended to superconducting central regions since the Green's functions in the Nambu representation satisfy the same equations of motion which, in turn, leads to the same closed expression for the equal-time lesser Green's function, i.e., for the time-dependent reduced one-particle density matrix. We further write the finite-temperature frequency integrals in terms of known special functions thereby considerably speeding up the computation. Numerical simulations in simple normal metal -- superconductor -- normal m…

Time-resolved core-level photoemission data of tungsten diselenide

Pump-probe core-level photoemission spectroscopy data of tungsten diselenide (WSe2) measured using an electron momentum microscope at the FLASH Free-electron laser.