Kadanoff-Baym Equations Research Articles

Accelerated gap collapse in a Slater antiferromagnet

We study the melting of long-range antiferromagnetic order in the Hubbard model after an interaction quench, using non-equilibrium dynamical mean-field theory. From previous studies, the system is known to quickly relax into a prethermal symmetry-broken state. Using a convergent truncation of the memory integrals in the Kadanoff Baym equations, we unravel the subsequent relaxation dynamics of this state over several orders of magnitude in time. At long times, the prethermal state can be characterized by a single slow variable which is related to the conduction band population. The dynamics of this variable does not follow the paradigmatic steady relaxation of pre-thermal states: It is highly nonlinear, with a pronounced speedup once the gap falls below a certain value. This behavior indicates that non-thermal order can be self-stabilized on some timescale. It is not reproduced using simple Fermi's golden rule estimate for the evolution of the conduction band population.

Non-equilibrium Quantum Brain Dynamics II: Formulation in [formula omitted] dimensions

This paper is concerned with an extension of the formalism of Quantum Brain Dynamics (QBD) to 3+1 dimensions, for which we derive evolution equations which involve water molecules’ electric dipoles and photon degrees of freedom. We obtain Schrödinger-like equations for coherent electric dipole fields, Klein–Gordon equations for coherent electric fields and Kadanoff–Baym equations for incoherent particles. We further consider aspects of thermodynamic equilibration with entropy production and super-radiance simultaneously derived from our system’s Lagrangian. We also show the presence of the Higgs mechanism and a tachyonic instability in the Klein–Gordon equations for coherent electric fields. Our approach describes non-equilibrium, non-secular, and entropy producing dynamics with conservation laws for the probability and the total energy. This representation of QBD describes non-equilibrium properties associated with memory formation in QBD.

Comparing the generalized Kadanoff-Baym ansatz with the full Kadanoff-Baym equations for an excitonic insulator out of equilibrium

We investigate out-of-equilibrium dynamics in an excitonic insulator (EI) with a finite momentum pairing perturbed by a laser-pulse excitation and a sudden coupling to fermionic baths. The transient dynamics of the excitonic order parameter is resolved using the full nonequilibrium Green's function approach and the generalized Kadanoff-Baym ansatz (GKBA) within the second-Born approximation. The comparison between the two approaches after a laser pulse excitation shows a good agreement in the weak and the intermediate photo-doping regime. In contrast, the laser-pulse dynamics resolved by the GKBA does not show a complete melting of the excitonic order after a strong excitation. Instead we observe persistent oscillations of the excitonic order parameter with a predominant frequency given by the renormalized equilibrium bandgap. This anomalous behavior can be overcome within the GKBA formalism by coupling to an external bath, which leads to a transition of the EI system towards the normal state. We analyze the long-time evolution of the system and distinguish decay timescales related to dephasing and thermalization.

Complete leading-order standard model corrections to quantum leptogenesis

Thermal leptogenesis, in the framework of the standard model with three additional heavy Majorana neutrinos, provides an attractive scenario to explain the observed baryon asymmetry in the universe. It is based on the out-of-equilibrium decay of Majorana neutrinos in a thermal bath of standard model particles, which in a fully quantum field theoretical formalism is obtained by solving Kadanoff-Baym equations. So far, the leading two-loop contributions from leptons and Higgs particles are included, but not yet gauge corrections. These enter at three-loop level but, in certain kinematical regimes, require a resummation to infinite loop order for a result to leading order in the gauge coupling. In this work, we apply such a resummation to the calculation of the lepton number density. The full result for the simplest “vanilla leptogenesis” scenario is by mathcal{O} (1) increased compared to that of quantum Boltzmann equations, and for the first time permits an estimate of all theoretical uncertainties. This step completes the quantum theory of leptogenesis and forms the basis for quantitative evaluations, as well as extensions to other scenarios.

Current-induced atomic motion, structural instabilities, and negative temperatures on molecule-electrode interfaces in electronic junctions

Molecule-electrode interfaces in molecular electronic junctions are prone to chemical reactions, structural changes, and localized heating effects caused by electric current. These can be exploited for device functionality or may be degrading processes that limit performance and device lifetime. We develop a nonequilibrium Green's function based transport theory in which the central region atoms and, more importantly, atoms on molecule-electrode interfaces are allowed to move. The separation of time-scales of slow nuclear motion and fast electronic dynamics enables the algebraic solution of the Kadanoff-Baym equations in the Wigner space. As a result, analytical expressions for dynamical corrections to the adiabatically computed Green's functions are produced. These dynamical corrections depend not only on the instantaneous molecular geometry but also on the nuclear velocities. To make the theoretical approach fully self-consistent, the same time-separation approach is used to develop expressions for the adiabatic, dissipative, and stochastic components of current-induced forces in terms of adiabatic Green's functions. Using these current induced forces, the equation of motion for the nuclear degrees of freedom is cast in the form of a Langevin equation. The theory is applied to model molecular electronic junctions. We observe that the interplay between the value of the spring constant for the molecule-electrode chemical bond and electronic coupling strength to the corresponding electrode is critical for the appearance of structural instabilities and, consequently, telegraphic switching in the electric current. The range of model parameters is identified to observe structurally stable molecular junctions as well as various different kinds of current-induced telegraphic switching. The interfacial structural instabilities are also quantified based on current noise calculations.

Quantum Boltzmann equation for bilayer graphene

A-B stacked bilayer graphene has massive electron and hole-like excitations with zero gap in the nearest-neighbor hopping approximation. In equilibrium, the quasiparticle occupation approximately follows the usual Fermi-Dirac distribution. In this paper we consider perturbing this equilibrium distribution so as to determine DC transport coefficients near charge neutrality. We consider the regime $\beta |\mu| \lesssim 1$ (with $\beta$ the inverse temperature and $\mu$ the chemical potential) where there is not a well formed Fermi surface. Starting from the Kadanoff-Baym equations, we obtain the quantum Boltzmann equation of the electron and hole distribution functions when the system is weakly perturbed out of equilibrium. The effect of phonons, disorder, and boundary scattering for finite sized systems are incorporated through a generalized collision integral. The transport coefficients, including the electrical and thermal conductivity, thermopower, and shear viscosity, are calculated in the linear response regime. We also extend the formalism to include an external magnetic field. We present results from numerical solutions of the quantum Boltzmann equation. Finally, we derive a simplified two-fluid hydrodynamic model appropriate for this system, which reproduces the salient results of the full numerical calculations.

Non-Equilibrium Quantum Electrodynamics in Open Systems as a Realizable Representation of Quantum Field Theory of the Brain.

We derive time evolution equations, namely the Klein–Gordon equations for coherent fields and the Kadanoff–Baym equations in quantum electrodynamics (QED) for open systems (with a central region and two reservoirs) as a practical model of quantum field theory of the brain. Next, we introduce a kinetic entropy current and show the H-theorem in the Hartree–Fock approximation with the leading-order (LO) tunneling variable expansion in the 1st order approximation for the gradient expansion. Finally, we find the total conserved energy and the potential energy for time evolution equations in a spatially homogeneous system. We derive the Josephson current due to quantum tunneling between neighbouring regions by starting with the two-particle irreducible effective action technique. As an example of potential applications, we can analyze microtubules coupled to a water battery surrounded by a biochemical energy supply. Our approach can be also applied to the information transfer between two coherent regions via microtubules or that in networks (the central region and the reservoirs) with the presence of quantum tunneling.

Timescale separation solution of the Kadanoff-Baym equations for quantum transport in time-dependent fields

The interaction with time-dependent external fields, especially the interplay between time-dependent driving and quantum correlations, changes the familiar picture of electron transport through nanoscale systems. Although the exact solution of the problem of AC quantum transport of noninteracting electrons has been known for more than two decades, the treatment of correlated particles presents a significant theoretical challenge. In this paper, using the perturbative separation of fast electron tunnelling and slow driving time-scales, we developed a practical approach for time-dependent quantum transport with nonequilibrium Green's functions. The fast electronic dynamics is associated with relative time whilst the slow driving is related to the central time in the Green's functions. The ratio of characteristic electron tunneling time over the period of harmonic driving is used as a small parameter in the theory to obtain a convergent time-derivative expansions of the Green's functions. This enables the algebraic solution of the Kadanoff-Baym equations in Wigner space. Consequently, we produced analytical expressions for dynamical corrections to advanced, retarded, and lesser Green's functions, as well as an improved expression for AC electric current. The method developed is applicable to the general case of multi-channel electron transport through a correlated central region. The theory is applied to different transport scenarios: time-dependent transport through a driven single-resonant level is compared to exact results; and electron transport through a molecular junction described by the Holstein model with a time-oscillating voltage bias is also investigated.

Nuclear response functions with low-momentum interactions

Linear density response functions for nuclear matter are calculated with separable 1S0 and 3S1 2-nucleon interactions obtained from experimental phase-shifts by inverse scattering techniques. It has been shown that results of nuclear matter binding energy Brueckner calculations with these potentials agree closely with those using realistic Bonn interactions. It has also been shown that low-momentum versions give binding energies (nearly) independent of momentum cut-offs $\Lambda \geq \sim 2$ fm-1. Shown here is that the $\Lambda = 2$ rank-one version of the separable interaction (for the S-states) yields a near agreement between a second-order and an all-order (Brueckner) binding energy calculation. Second order calculations of response functions are then made with this version using a method that Kwong and Bonitz used for the Coulomb gas. It is based on time-evolving two-time Green’s functions by Kadanoff-Baym (KB) equations. The effect of correlations, going beyond the conventional HF+RPA method, is included by “dressing” the Green’s function propagators with time-dependent complex self-energies in addition to the real HF-field. It would be of interest to see if the calculated response is also, like the binding energy, independent of cut-offs larger than $\sim 2$ fm-1. That would require using separable potentials of a higher rank, not accommodated by the present computer program. Some results with a 1.5 fm-1 cut-off are however also shown below together with those with a 2.0 fm-1 cut-off. The KB-equations are time-evolved until the nuclear medium is fully correlated. The time-dependent density response to an external perturbation is then registered and Fourier transformed to obtain energy-dependent density response functions. Most previous calculations have been made with in-medium effective (e.g., Gogny or Skyrme) interactions. The medium dependence is in the present calculations contained in the second-order terms.

Beyond the Generalized Kadanoff–Baym Ansatz

The study of electron transport is easier if the Kadanoff–Baym equations (KBE) are simplified using the Generalized Kadanoff–Baym Ansatz (GKBA). For molecular bridges, GKBA is empirically known to safely work for weak and flat tunneling functions. It fails, if either of the conditions is not satisfied. The case in point is a molecular bridge formed by an Anderson type local center treated in the mean field and linked by tunneling junctions to two ferromagnetic electrodes whose tunneling functions simulate nickel with complex sd structure. Transient magnetic currents under a constant galvanic bias between electrodes are invoked by sudden switching on of both junctions. We consider three tasks: To establish quantitative criteria for the validity of the Ansatz; to develop a practically tractable correction to the Ansatz working beyond its validity range; to obtain Generalized Master Equations (GMEs) for the one‐electron distribution following from both GKBA and its corrected form avoiding thus KBE. All three points are resolved by treating first the stationary (non‐equilibrium) limit and transferring the results to finite times. The corrections to the Ansatz are obtained as a stationary approximation to the vertex part of the exact reconstruction equations whose free term is the standard GKBA.

Non-equilibrium Green's function theory for non-adiabatic effects in quantum transport: Inclusion of electron-electron interactions.

Non-equilibrium Green's function theory for non-adiabatic effects in quantum transport [Kershaw and Kosov, J. Chem. Phys. 147, 224109 (2017) and J. Chem. Phys. 149, 044121 (2018)] is extended to the case of interacting electrons. We consider a general problem of quantum transport of interacting electrons through a central region with dynamically changing geometry. The approach is based on the separation of time scales in the non-equilibrium Green's functions and the use of the Wigner transformation to solve the Kadanoff-Baym equations. The Green's functions and correlation self-energy are non-adiabatically expanded up to the second order central time derivatives. We produce expressions for Green's functions with non-adiabatic corrections and a modified formula for electric current; both depend not only on instantaneous molecular junction geometry but also on nuclear velocities and accelerations. The theory is illustrated by the study of electron transport through a model single-resonant level molecular junction with local electron-electron repulsion and a dynamically changing geometry.

Nuclear slabs with Green’s functions: mean field and short-range correlations

Computational difficulties aside, nonequilibrium Green’s functions appear ideally suited for investigating the dynamics of central nuclear reactions. Many particles actively participate in those reactions. At the two energy extremes for the collisions, the limiting cases of the Green’s function approach have been successful: the time-dependent Hartree–Fock theory at low energy and Boltzmann equation at high. The strategy for computational adaptation of the Green’s function to central reactions is discussed. The strategy involves, in particular, incremental progression from one to three dimensions to develop and assess approximations, discarding of far-away function elements, use of effective interactions and preparation of initial states for the reactions through adiabatic switching. At this stage we concentrate on inclusion of correlations in one dimension, where relatively few approximations are needed, and we carry out reference calculations that can benchmark approximations needed for more dimensions. We switch on short-range interactions generating the correlations adiabatically in the Kadanoff–Baym equations to arrive at correlated ground states for uniform matter. As the energy of the correlated matter does not quite match the expectations for nuclear matter, we add mean field to arrive at the match in energy. From there on, we move to finite systems. In switching on the correlations, we observe emergence of extended tails in momentum distributions and evolution of single particle occupations away from 1 to 0.

Non-adiabatic corrections to electric current in molecular junctions due to nuclear motion at the molecule-electrode interfaces.

We present quantum electron transport theory that incorporates dynamical effects of motion of atoms on electrode-molecule interfaces in the calculations of the electric current. The theory is based on non-equilibrium Green's functions. We separate time scales in the Green's functions as fast relative time and slow central time. The derivative with respect to the central time serves as a small parameter in the theory. We solve the real-time Kadanoff-Baym equations for molecular Green's functions using Wigner representation and keep terms up to the second order with respect to the central time derivatives. Molecular Green's functions and consequently the electric current are expressed as functions of molecular junction coordinates as well as velocities and accelerations of molecule-electrode interface nuclei. We apply the theory to model a molecular system and study the effects of non-adiabatic nuclear motion on molecular junction conductivity.

Molecular junctions and molecular motors: Including Coulomb repulsion in electronic friction using nonequilibrium Green's functions

We present a theory of molecular motors based on the Ehrenfest dynamics for nuclear coordinates and the adiabatic limit of the Kadanoff-Baym equations for current-induced forces. Electron-electron interactions can be systematically included through many-body perturbation theory, making the nonequilibrium Green's function formulation suitable for first-principles treatments of realistic junctions. The method is benchmarked against simulations via real-time Kadanoff-Baym equations, finding an excellent agreement. Results on a paradigmatic model of a molecular motor show that correlations can change dramatically the physical scenario by, e.g., introducing a sizable damping in self-sustained van der Pol oscillations.

Modeling Plasmon-Induced Hot-Carrier Transfer

In this issue of Chem , Zhang et al. use first-principle simulations (i.e., no empirically adjustable parameters) to elucidate the detailed mechanisms of plasmon-induced hot-carrier transfer at a heterojunction interface between a plasmonic gold nanomaterial and a semiconductor. Together with earlier work by Long and Prezhdo, this study shows how the donor-acceptor interaction at the interface determines the mechanism by which plasmonic excitations lead to hot-electron transfer at interfaces.

Spectral properties from Matsubara Green's function approach: Application to molecules

We present results for many-body perturbation theory for the one-body Green's function at finite temperatures using the Matsubara formalism. Our method relies on the accurate representation of the single-particle states in standard Gaussian basis sets, allowing to efficiently compute, among other observables, quasiparticle energies and Dyson orbitals of atoms and molecules. In particular, we challenge the second-order treatment of the Coulomb interaction by benchmarking its accuracy for a well-established test set of small molecules, which includes also systems where the usual Hartree-Fock treatment encounters difficulties. We discuss different schemes how to extract quasiparticle properties and assess their range of applicability. With an accurate solution and compact representation, our method is an ideal starting point to study electron dynamics in time-resolved experiments by the propagation of the Kadanoff-Baym equations.

Many-body perturbation theory and non-perturbative approaches: screened interaction as the key ingredient

Many-body perturbation theory is often formulated in terms of an expansion in the dressed instead of the bare Green’s function, and in the screened instead of the bare Coulomb interaction. However, screening can be calculated on different levels of approximation, and it is important to define what is the most appropriate choice. We explore this question by studying a zero-dimensional model (so called ‘one-point model’) that retains the structure of the full equations. We study both linear and non-linear response approximations to the screening. We find that an expansion in terms of the screening in the random phase approximation is the most promising way for an application in real systems. Moreover, by making use of the nonperturbative features of the Kadanoff–Baym equation for the one-body Green’s function, we obtain an approximate solution in our model that is very promising, although its applicability to real systems has still to be explored.

Quantum quench of the Sachdev-Ye-Kitaev model

We describe the non-equilibrium dynamics of the Sachdev-Ye-Kitaev models of fermions with all-to-all interactions. These provide tractable models of the dynamics of quantum systems without quasiparticle excitations. The Kadanoff-Baym equations show that the final state is thermal, and their numerical analysis appears consistent with a thermalization rate proportional to the absolute temperature of the final state. We also obtain an exact analytic solution of the non-equilibrium dynamics in the large $q$ limit of a model with $q$ fermion interactions: in this limit, the thermalization of the fermion Green's function is instantaneous.

Comment on “On the unphysical solutions of the Kadanoff-Baym equations in linear response: Correlation-induced homogeneous density-distribution and attractors”

In a recent Rapid Communication [A. Stan, Phys. Rev. B \textbf{93}, 041103(R) (2016)], the reliability of the Keldysh--Kadanoff--Baym equations (KBE) using correlated selfenergy approximations applied to linear and nonlinear response has been questioned. In particular, the existence of a universal attractor has been predicted that would drive the dynamics of any correlated system towards an unphysical homogeneous density distribution regardless of the system type, the interaction and the many-body approximation. Moreover, it was conjectured that even the mean-field dynamics would be damped. Here, by performing accurate solutions of the KBE for situations studied in that paper, we prove these claims wrong being caused by numerical inaccuracies.

Backreaction effects on nonequilibrium spectral function

We show how to compute the spectral function for a scalar theory in two different scenarios: one which disregards backreaction, i.e. the response of the environment to the external particle, and the other one where backreaction is considered. The calculation was performed using the Kadanoff–Baym equation through the Keldysh formalism. When backreaction is neglected, the spectral function is equal to the equilibrium one, which can be represented as a Breit–Wigner distribution. When backreaction is introduced we observed a damping in the spectral function of the thermal bath. Such behavior modifies the damping rate for particles created within the bath.