Advanced Green Functions Research Articles

Spin-2 Green’s functions on Kerr in radiation gauge

Abstract We construct retarded and advanced Green’s functions for gravitational perturbations in Kerr in an ingoing radiation gauge. Our Green’s functions have a frequency domain piece that has previously been obtained by Ori (2003 Phys. Rev. D 67) based on the Chrzanowski-Cohen-Kegeles metric reconstruction method. As is well known, this piece by itself is not sufficient to obtain an actual Green’s function. We show how to complete it with a piece based on a method by Green et al (2020 Class. Quantum Grav. 37). The completion piece has a completely explicit form in the time-domain and is supported on pairs of points on the same outgoing principal null geodesic which are in the appropriate causal order. We expect our Green’s functions to be useful for gravitational self-force calculations and other perturbation problems on Kerr spacetime.

Retarded and advanced Green functions for first-order differential equations

Retarded and advanced Green functions are examined through the lens of linear first-order ordinary differential equations. They are constructed for a finite interval with emphasis on the imposition of homogeneous Dirichlet boundary conditions at the interval ends. It is shown that both types of Green functions exhibit the property of facilitating solutions.

A Single-Layer Dual-Mesh Boundary Element Method for Multiscale Electromagnetic Modeling of Penetrable Objects in Layered Media

A surface integral representation of Maxwell's equations allows the efficient electromagnetic (EM) modeling of three-dimensional structures with a two-dimensional discretization, via the boundary element method (BEM). However, existing BEM formulations either lead to a poorly conditioned system matrix for multiscale problems, or are computationally expensive for objects embedded in layered substrates. This article presents a new BEM formulation which leverages the surface equivalence principle and Buffa-Christiansen basis functions defined on a dual mesh, to obtain a well-conditioned system matrix suitable for multiscale EM modeling. Unlike existing methods involving dual meshes, the proposed formulation avoids the double-layer potential operator for the surrounding medium, which may be a stratified substrate requiring the use of an advanced Green's function. This feature greatly alleviates the computational expense associated with the use of Buffa-Christiansen functions. Numerical examples drawn from several applications, including remote sensing, chip-level EM analysis, and metasurface modeling, demonstrate speed-ups ranging from 3x to 7x compared to state-of-the-art formulations.

Time-dependent spectral functions of the Anderson impurity model in response to a quench with application to time-resolved photoemission spectroscopy

We investigate several definitions of the time-dependent spectral function $A(\omega,t)$ of the Anderson impurity model following a quench and within the time-dependent numerical renormalization group method. In terms of the two-time retarded Green function $G^r(t_1,t_2)$, the definitions differ in the choice of the time variable $t$ with respect to $t_1$ and/or $t_2$. In a previous study [Nghiem {\it et al.} Phys. Rev. Lett. 119, 156601 (2017)], we investigated the spectral function, obtained from the Fourier transform of ${\rm Im}[G^r(t_1,t_2)]$ w.r.t. the time difference $t'=t_1-t_2$, with $t=t_2$. Here, we derivie expressions for the retarded Green function for the choices $t=t_1$ and the average time $t=(t_1+t_2)/2$, within the TDNRG approach. We compare and contrast the resulting $A(\omega,t)$ for the different choices of time reference. Expressions for the lesser, greater and advanced Green functions are also derived within TDNRG for all choices of time reference. The average time lesser Green function $G^<(\omega,t)$ is particularly interesting, as it determines the time-dependent occupied density of states $N(\omega,t)=G^<(\omega,t)/(2\pi i)$, a quantity that determines the photoemission current in time-resolved pump-probe photoemission spectroscopy. We present calculations for $N(\omega,t)$ for the Anderson model following a quench, and discuss the resulting time evolution of the spectral features, such as the Kondo resonance and high-energy peaks. We also discuss the issue of thermalization at long times for $N(\omega,t)$. Finally, we use the results for $N(\omega,t)$ to calculate the time-resolved photoemission current for the Anderson model following a quench (acting as the pump) and study the different behaviors that can be observed for different resolution times of a Gaussian probe pulse.

Efficient prediction of time- and angle-resolved photoemission spectroscopy measurements on a nonequilibrium BCS superconductor

We study how time- and angle-resolved photoemission (tr-ARPES) reveals the dynamics of BCS-type, s-wave superconducting systems with time-varying order parameters. Approximate methods are discussed, based on previous approaches to either optical conductivity or quantum dot transport, in order to enable computationally efficient prediction of photoemission spectra. One use of such predictions is to enable extraction of the underlying order parameter dynamics from experimental data, which is topical given the rapidly growing use of tr-ARPES in studying unconventional superconductivity. The methods considered model the two-time lesser Green's functions with an approximated lesser self-energy that describes relaxation by coupling of the system to two types of baths. The approach primarily used here also takes into consideration the relaxation of the excited states into equilibrium by explicitly including the level-broadening of the retarded and advanced Green's functions. We present equilibrium and non-equilibrium calculations of tr-ARPES spectrum from our model and discuss the signatures of different types of superconducting dynamics.

Green’s functions and Hadamard parametrices for vector and tensor fields in general linear covariant gauges

We determine the retarded and advanced Green's functions and Hadamard parametrices in curved spacetimes for linearized massive and massless gauge bosons and linearized Einstein gravity with a cosmological constant in general linear covariant gauges. These vector and tensor parametrices contain additional singular terms compared with their Feynman/de Donder-gauge counterpart. We also give explicit recursion relations for the Hadamard coefficients, and indicate their generalization to $n$ dimensions. Furthermore, we express the divergence and trace of the vector and tensor Green's functions in terms of derivatives of scalar and vector Green's functions, and show how these relations appear as Ward identities in the free quantum theory.

Green's function representation of spin pumping effect

In this study, current pumping by an external potential is studied on the basis of the Keldysh Green's function method, and a pumping formula written in terms of retarded and advanced Green's functions is obtained. It is shown that pumping is essentially driven by a change of particle distribution before and after an external perturbation. The formula is used to study the spin pumping effect in the case of strong $s\ensuremath{-}d$ exchange interaction, and the driving field is identified to be the spin gauge field. At the lowest order in the precession frequency of magnetization, the spin gauge field works as a constant potential, and the system is shown to reduce to a static problem of spin current generation by a time-independent potential with off-diagonal spin components.

Acausality in nonlocal gravity theory

We investigate the nonlocal gravity theory by deriving nonlocal equations of motion using the traditional variation principle in a homogeneous background. We focus on a class of models with a linear nonlocal modification term in the action. It is found that the resulting equations of motion contain the advanced Green's function, implying that there is an acausality problem. As a consequence, a divergence arises in the solutions due to contributions from the future infinity unless the Universe will go back to the radiation dominated era or become the Minkowski spacetime in the future. We also discuss the relation between the original nonlocal equations and its biscalar-tensor representation and identify the auxiliary fields with the corresponding original nonlocal terms. Finally, we show that the acusality problem cannot be avoided by any function of nonlocal terms in the action.

A caveat on building nonlocal models of cosmology

Nonlocal models of cosmology might derive from graviton loop corrections to the effective field equations from the epoch of primordial inflation. Although the Schwinger-Keldysh formalism would automatically produce causal and conserved effective field equations, the models so far proposed have been purely phenomenological. Two techniques have been employed to generate causal and conserved field equations: either varying an invariant nonlocal effective action and then enforcing causality by the ad hoc replacement of any advanced Green's function with its retarded counterpart, or else introducing causal nonlocality into a general ansatz for the field equations and then enforcing conservation. We point out here that the two techniques access very different classes of models, and that neither one of them may represent what would actually arise from fundamental theory.

Exciton correlations and input–output relations in non-equilibrium exciton superfluids

The photoluminescence (PL) measurements on photons and the transport measurements on excitons are the two types of independent and complementary detection tools to search for possible exciton superfluids in electron–hole semi-conductor bilayer systems. In fact, it was believed that the transport measurements can provide more direct evidences on superfluids than the spectroscopic measurements. It is important to establish the relations between the two kinds of measurements. In this paper, using quantum Heisenberg–Langevin equations, we establish such a connection by calculating various exciton correlation functions in the putative exciton superfluids. These correlation functions include both normal and anomalous greater, lesser, advanced, retarded, and time-ordered exciton Green functions and also various two exciton correlation functions. We also evaluate the corresponding normal and anomalous spectral weights and the Keldysh distribution functions. We stress the violations of the fluctuation and dissipation theorem among these various exciton correlation functions in the non-equilibrium exciton superfluids. We also explore the input–output relations between various exciton correlation functions and those of emitted photons such as the angle resolved photon power spectrum, phase sensitive two mode squeezing spectrum and two photon correlations. Applications to possible superfluids in the exciton–polariton systems are also mentioned. For a comparison, using conventional imaginary time formalism, we also calculate all the exciton correlation functions in an equilibrium dissipative exciton superfluid in the electron–electron coupled semi-conductor bilayers at the quantum Hall regime at the total filling factor νT=1. We stress the analogies and also important differences between the correlations functions in the two exciton superfluid systems.

Past and Future Blurring at Fundamental Length Scale

We obtain the κ-deformed versions of the retarded and advanced Green functions and show that their causality properties are blurred in a time interval of the order of a length parameter q=1/(2κ). The functions also indicate a smearing of the light cone. These results favor the interpretation of q as a fundamental length scale below which the concept of a point in space-time should be substituted by the concept of a fuzzy region of radius q, as proposed long ago by Heisenberg.

Resonant States of Open Quantum Systems

We first show that quantum resonant states observe particle number conservation and hence are consistent with the probabilistic interpretation of quantum mechanics. We then present for a class of quantum open systems, a resonant-state expansion of the sum of the retarded and advanced Green's functions. The expansion is given purely in terms of all discrete eigenstates and does not contain any background integrals. Using the expansion, we argue that the Fano asymmetry of resonance peaks is interpreted as interference between discrete eigenstates. We microscopically derive the Fano parameters for several cases.

Impurity problems for steady-state nonequilibrium dynamical mean-field theory

The mapping of steady-state nonequilibrium dynamical mean-field theory from the lattice to the impurity is described in detail. Our focus is on the case with current flow under a constant dc electric field of arbitrary magnitude. In addition to formulating the problem via path integrals and functional derivatives, we also describe the distribution-function dependence of the retarded and advanced Green's functions. Our formal developments are exact for the Falicov–Kimball model. We also show how these formal developments are modified for more complicated models (like the Hubbard model).

Steady-State Nonequilibrium Density of States of Driven Strongly Correlated Lattice Models in Infinite Dimensions

An exact formalism for calculating the retarded and advanced Green's functions of strongly correlated lattice models in a uniform electric field is derived within dynamical mean-field theory. To illustrate the method, we solve for the nonequilibrium density of states of the Hubbard model in both the metallic and Mott-insulating phases at half-filling (with an arbitrary strength electric field) by employing the approximate numerical renormalization group as the impurity solver. This general approach can be applied to any strongly correlated lattice model in the limit of large dimensions.

AN ITERATION SCHEME FOR CALCULATING TRANSPORT PROPERTIES OF MOLECULAR SYSTEMS

An iteration method of functional theory is used to solve Poisson equation self-consistently in order to determine the retarded and advanced Green's function of a molecular system. The approach is demonstrated using a system with a spherical potential. The retarded and advanced Green's functions are calculated and the determined current–voltage results compare well with experimental data for the system involving Coulomb potential. The approach is expected to find wide applications in general systems.

Quasiparticle band structure of infinite hydrogen fluoride and hydrogen chloride chains

We study the quasiparticle band structure of isolated, infinite (HF)(infinity) and (HCl)(infinity) bent (zigzag) chains and examine the effect of the crystal field on the energy levels of the constituent monomers. The chains are one of the simplest but realistic models of the corresponding three-dimensional crystalline solids. To describe the isolated monomers and the chains, we set out from the Hartree-Fock approximation, harnessing the advanced Green's function methods local molecular orbital algebraic diagrammatic construction (ADC) scheme and local crystal orbital ADC (CO-ADC) in a strict second order approximation, ADC(2,2) and CO-ADC(2,2), respectively, to account for electron correlations. The configuration space of the periodic correlation calculations is found to converge rapidly only requiring nearest-neighbor contributions to be regarded. Although electron correlations cause a pronounced shift of the quasiparticle band structure of the chains with respect to the Hartree-Fock result, the bandwidth essentially remains unaltered in contrast to, e.g., covalently bound compounds.

Ab initioelectron propagator theory of molecular wires. II. Multiorbital terminal description

Correlated, ab initio electron propagator methodology may be applied to the calculation of electrical current through a molecular wire. A new theoretical formalism is developed for the calculation of retarded and advanced Green functions in terms of the electron propagator matrix for a bridge molecule. The calculation of the current requires integration in a complex half-plane for a trace that involves terminal and Green function matrices that may have any rank. Because the latter arrays have poles represented by matrices, an alternative expression is developed in terms of ordinary poles which are (n-1)-fold degenerate or nondegenerate. For an arbitrary number of terminal orbitals, the analytical expression for the current is given in terms of pole strengths, poles, and terminal matrix elements of the electron propagator, i.e., the parameters that are found in the output of numerical calculations.

Explicit formulations for advanced Green's functions with built-in boundaries

This paper amplifies upon a previously presented BEM formulation where the two-dimensional logarithmic fundamental solution is transformed so as to automatically accommodate rectangular boundaries with fixed boundary conditions. Explicit derivations are presented using conformal mapping. Computational examples and comparisons with the standard procedure illustrate the advantages of the method.

Correlated Ab initio electron propagators in the study of molecular wires

AbstractCorrelated ab initio electron propagators provide a systematic approach to treating correlation effects in the study of molecular wires. Some preliminary formal and computational steps toward the application of such methods to electron transport problems in nanoscience are presented. Retarded and advanced Green functions are expressed in terms of a general electron propagator for a bridge molecule. Terminal and Green function matrices of any rank may be employed. An analytical expression for the current depends on correlated electron binding energies, pole strengths, and Dyson orbitals that may be produced with readily available software. Calculations on benzene dithiolate between gold atoms illustrate the capabilities of this approach. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Ab initioelectron propagator theory of molecular wires: I. Formalism

Ab initio electron propagator methodology may be applied to the calculation of electrical current through a molecular wire. A new theoretical approach is developed for the calculation of the retarded and advanced Green functions in terms of the electron propagator matrix for the bridge molecule. The calculation of the current requires integration in a complex half plane for a trace that involves terminal and Green's-function matrices. Because the Green's-function matrices have complex poles represented by matrices, a special scheme is developed to express these "matrix poles" in terms of ordinary poles. An expression for the current is derived for a terminal matrix of arbitrary rank. For a single terminal orbital, the analytical expression for the current is given in terms of pole strengths, poles, and terminal matrix elements of the electron propagator. It is shown that Dyson orbitals with high pole strengths and overlaps with terminal orbitals are most responsible for the conduction of electrical current.