Exact Green Function Research Articles

Some Exact Green Function Solutions for Non-Linear Classical Field Theories

We consider some non-linear non-homogeneous partial differential equations (PDEs) and derive their exact Green function solution as a functional Taylor expansion in powers of the source. The kind of PDEs we consider are dispersive ones where the exact solution of the corresponding homogeneous equations can have some known shape. The technique has a formal similarity with the Dyson–Schwinger set of equations to solve quantum field theories. However, there are no physical constraints. Indeed, we show that a complete coincidence with the statistical field model of a quartic scalar theory can be achieved in the Gaussian expansion of the cumulants of the partition function.

Wave amplitude gain within wedge waveguides through scattering by simple obstacles.

Wave confinement, e.g., in waveguides, gives rise to a huge number of distinct phenomena. Among them, amplitude gain is a recurrent and relevant effect in undulatory processes. Using a general purpose protocol to solve wave equations, the boundary wall method, we demonstrate that for relatively simple geometries, namely, a few leaky or opaque obstacles inside a θ wedge waveguide (described by the Helmholtz equation), one can obtain a considerable wave amplification in certain spatially localized regions of the system. The approach relies on an expression for the wedge waveguide exact Green's function in the case of θ=π/M (M=1,2,...), derived through the method of images allied to group theory concepts. The formula is particularly amenable to numerical calculations, greatly facilitating simulations. As an interesting by-product of the present framework, we are able to obtain the eigenstates of certain closed shapes (billiards) placed within the waveguide, as demonstrated for triangular structures. Finally, we briefly discuss possible concrete realizations for our setups in the context of matter and electromagnetic (for some particular modes and conditions) waves.

Exact Dirac-Bogoliubov-de Gennes Dynamics for Inhomogeneous Quantum Liquids.

We study inhomogeneous 1+1-dimensional quantum many-body systems described by Tomonaga-Luttinger-liquid theory with general propagation velocity and Luttinger parameter varying smoothly in space, equivalent to an inhomogeneous compactification radius for free boson conformal field theory. This model appears prominently in low-energy descriptions, including for trapped ultracold atoms, while here we present an application to quantum Hall edges with inhomogeneous interactions. The dynamics is shown to be governed by a pair of coupled continuity equations identical to inhomogeneous Dirac-Bogoliubov-de Gennes equations with a local gap and solved by analytical means. We obtain their exact Green's functions and scattering matrix using a Magnus expansion, which generalize previous results for conformal interfaces and quantum wires coupled to leads. Our results explicitly describe the late-time evolution following quantum quenches, including inhomogeneous interaction quenches, and Andreev reflections between coupled quantum Hall edges, revealing remarkably universal dependence on details at stationarity or at late times out of equilibrium.

Exact Green's functions for localized irreversible potentials

We study the quantum-mechanical problem of scattering caused by a localized obstacle that breaks spatial and temporal reversibility. Accordingly, we follow Maxwell's prescription to achieve a violation of the second law of thermodynamics by means of a momentum-dependent interaction in the Hamiltonian, resulting in what is known as Maxwell's demon. We obtain the energy-dependent Green's function analytically, as well as its meromorphic structure. The poles lead directly to the solution of the evolution problem, in the spirit of M. Moshinsky's work published in the 1950s. Symmetric initial conditions are evolved in this way, showing important differences between classical and wave-like irreversibility in terms of collapses and revivals of wave packets. Our setting can be generalized to other wave operators, e.g. electromagnetic cavities in a classical regime.

One-Shot GW Transport Calculations: A Charge-Conserving Solution.

Transport measurements are a common method of characterizing small systems in chemistry and physics. When interactions are negligible, the current through submicrometer structures can be obtained using the Landauer formula. Meir and Wingreen derived an exact expression for the current in the presence of interactions. This powerful tool requires knowledge of the exact Green's function. Alternatively, self-consistent approximations for the Green's function are frequently sufficient for calculating the current while crucially satisfying all conservation laws. We provide here yet another alternative, circumventing the high computational cost of these methods. We present expressions for the electric and thermal currents in which the lowest-order self-energy is summed to all orders (one-shot GW approximation). We account for both self-energy and vertex corrections such that current is conserved. Our formulas for the currents capture important features due to interactions and, hence, provide a powerful tool for cases in which the exact solution cannot be found.

Radiation from a dipole perpendicular to the interface between two planar semi-infinite magnetoelectric media

We consider two semi-infinite magnetoelectric media with constant dielectric permittivity separated by a planar interface, whose electromagnetic response is described by non-dynamical axion electrodynamics and investigate the radiation of a point-like electric dipole located perpendicularly to the interface. We start from the exact Green's function for the electromagnetic potential, whose far-field approximation is obtained using a modified steepest descent approximation. We compute the angular distribution of the radiation and the total radiated power finding different interference patterns, depending on the relative position dipole-observer, and polarization mixing effects which are all absent in the standard dipole radiation. They are a manifestation of the magnetoelectric effect induced by axion electrodynamics. We illustrate our findings with some numerical estimations employing realistic media as well as some hypothetical choices in order to illuminate the effects of the magnetoelectric coupling which is usually very small.

Acoustic scattering and the exact Green function.

Acoustic waves are scattered by a bounded three-dimensional sound-soft obstacle. The scattered field can be given explicitly in terms of the exact Green function, defined as the solution of the scattering problem when the incident waves are generated by a point source. Methods for constructing the exact Green function are described, with emphasis on a recent paper by Rother and Hawkins [J. Acoust. Soc. Am. 149, 2179-2188 (2021)]. These methods include boundary integral equations and null-field equations.

Stochastic gravitational waves from long cosmic strings

We compute the expected strain power spectrum and energy density parameter of the stochastic gravitational wave background (SGWB) created by a network of long cosmic strings evolving during the whole cosmic history. As opposed to other studies, the contribution of cosmic string loops is discarded and our result provides a robust lower bound of the expected signal that is applicable to most string models. Our approach uses Nambu-Goto numerical simulations, running during the radiation, transition and matter eras, in which we compute the two-point unequal-time anisotropic stress correlators. These ones act as source terms in the linearised equations of motion for the tensor modes, that we solve using an exact Green's function integrator. Today, we find that the rescaled strain power spectrum (k/ℋ0)2\U0001d4ab h peaks on Hubble scales and exhibits, at large wavenumbers, high frequency oscillations around a plateau of amplitude 100 (GU)2. Most of the high frequency power is generated by the long strings present in the matter era, the radiation era contribution being smaller.

Non-Markovian transients in transport across chiral quantum wires using space–time non-equilibrium Green functions

We study a system of two non-interacting quantum wires with fermions of opposite chirality with a point contact junction at the origin across which tunneling can take place when an arbitrary time-dependent bias between the wires is applied. We obtain the exact dynamical non-equilibrium Green function by solving Dyson’s equation analytically. Both the space–time dependent two and four-point functions are written down in a closed form in terms of simple functions of position and time. This allows us to obtain, among other things, the I–V characteristics for an arbitrary time-dependent bias. Our method is a superior alternative to competing approaches to non-equilibrium as we are able to account for transient phenomena as well as the steady state. We study the approach to steady state by computing the time evolution of the equal-time one-particle Green function. Our method can be easily applied to the problem of a double barrier contact whose internal properties can be adjusted to induce resonant tunneling leading to a conductance maximum. We then consider the case of a finite bandwidth in the point contact and calculate the non-equilibrium transport properties which exhibit non-Markovian behaviour. When a subsequently constant bias is suddenly switched on, the current shows a transient build up before approaching its steady state value in contrast to the infinite bandwidth case. This transient property is consistent with numerical simulations of lattice systems using time-dependent density matrix renormalization group suggesting thereby that this transient build up is merely due to the presence of a short distance cutoff in the problem description and not on the other details.

An asymptotic-preserving 2D-2P relativistic drift-kinetic-equation solver for runaway electron simulations in axisymmetric tokamaks

We propose an asymptotic-preserving (AP), uniformly convergent numerical scheme for the relativistic collisional Drift-Kinetic Equation (rDKE) to simulate runaway electrons in axisymmetric toroidal magnetic field geometries typical of tokamak devices. The approach is derived from an exact Green's function solution with numerical approximations of quantifiable impact, and results in a simple, two-step operator-split algorithm, consisting of a collisional Eulerian step, and a Lagrangian orbit-integration step with analytically prescribed kernels. The AP character of the approach is demonstrated by analysis of the dominant numerical errors, as well as by numerical experiments. We demonstrate the ability of the algorithm to provide accurate answers regardless of plasma collisionality on a circular axisymmetric tokamak geometry.

Cavitylike strong coupling in macroscopic waveguide QED using three coupled qubits in the deep non-Markovian regime

We introduce a three qubit waveguide QED system to mimic the cavity-QED strong coupling regime of a probe qubit embedded in atom-like mirrors, which was realized in recent experiments. We then extend this system into the deep non-Markovian regime and demonstrate the profound role that retardation plays on the dressed resonances, allowing one to significantly improve the polariton lifetimes (by many orders of magnitude), tune the resonances, as well as realize Fano-like resonances and simultaneous coupling to multiple cavity modes. Exact Green function solutions are presented for the spectral resonances, and the quantum dynamics are simulated using matrix product states, where we demonstrate additional control of the cavity-QED system using chiral probe qubits.

Fourier transform approach to nonperiodic boundary value problems in porous conductive media

AbstractIn this article, we develop an extension of the Fourier transform solution method in order to solve conduction equation with nonperiodic boundary conditions (BC). The periodic Lippmann–Schwinger equation for porous materials is extended to the case of non‐periodicity with relevant source terms on the boundary. The method is formulated in Fourier space based on the temperature as unknown, using the exact periodic Green function and form factors to describe the boundaries. Different types of BC: flux, temperature, mixed and combined with periodicity can be treated by the method. Numerical simulations show that the method does not encounter convergence issues due to the infinite contrast and yields accurate results for both local fields and effective conductivity.

Double versus single junction: From tunneling to contact regime in a double quantum dot molecule system

In this work a theoretical study of the charge transport through a nanostructure composed by a double quantum dot was made by treating the system in two different ways: as single junction (SJ) within the scanning tunneling microscopy theory and a double junction (DJ) within the usual non equilibrium Green's function theory approach. In the first case (SJ), the three blocks – contact-device-contact – is treated as a single junction by ‘breaking’ the nanostructure into two parts and considering each one as belonging to an ‘extended contact’. In the DJ case, the usual treatment with two contacts-related broadening and the exact Green's function describing the propagation inside the system was made. Proceeding in this way, for a minimum two-level model system, we show that the results obtained with the SJ and DJ treatment are identical despite they start with a different general formulae for transmission. Finally, both treatments were used with a minimal model for a biphenyl system in an asymmetric sample-biphenyl-tip configuration and analyzed in terms of a SJ point of view bringing another perspective to the main features of this system.

Exact Green's function approach to RKKY interactions

The Green's function~(GF) of two localized magnetic moments embedded in the electron gas is calculated exactly. The electrons are treated in the effective mass approximation and the magnetic moments are coupled with electrons by a delta-like~$s-d$ interaction. The resulting GF is obtained by the exact summation of the Born series using a generalization of the method developed by Slater-Koster and Ziman to non-commuting spin operators with the use of the Woodbury identities. The exact GF crucially depends on the value of the one-electron GF at the origin, denoted as~$g_0$. The Born series is convergent only if~$g_0$ is finite, which holds for electrons in parabolic energy bands in~$1D$, but not in~$2D$ and~$3D$. In the general case, the exact GF includes nonlinear combination of localized spins operators. A method of calculating matrix elements of these operators is given. For spins~$S_a=S_b = 1/2$ the exact GF is expressed as a linear combination of components of~$\hS_a, \hS_b$, and the exact range function~${\cal J}(r)$ is obtained as a double integral over analytical expression. For electron energy~$E=0$ and~$J g_0/2 = 2$ or~$J g_0/2=-2/3$ the range function and GF are singular. Poles of GF occur in the vicinities of singularity points and the resulting energies of bound states are calculated. There are three regimes of~$J$. For small $J$ the range function resembles RKKY one: it has the same period~$\pi/k_F$, the same decay character and a slightly different amplitude, usually within a few percent. This regime occurs for most frequently in the nature. For~$|J|$ comparable to~$|g_0|^{-1}$ the exact range function differs qualitatively from RKKY one,. For large $|J|$ the exact range function oscillates with the same period and power-like decay as the usual RKKY function but it has much lower amplitude decaying with growing~$|J|$.

Possibility for inhibited spontaneous emission in electromagnetically open parity–time-symmetric guiding structures

Remote sensing and manipulation of quantum emitters are functionalities of significant practical importance in quantum optics. Unfortunately, these abilities are considered as fundamentally challenging in systems of inhibited spontaneous emission. The reason is intimately related to the common perception that, in order to nullify the spontaneous emission decay rate, the system has to be electromagnetically closed, meaning that all loss channels should be avoided, including radiation. However, since radiation is prohibited in these systems, far-field sensing and by reciprocity, also far-field manipulation are considered impossible. Here, we suggest a possible solution to this challenge and theoretically propose an electromagnetically open system that may exhibit a complete inhibition of spontaneous emission while supporting guiding waves. This peculiar functionality is achieved through a feedback wave mechanism that is found in parity-time-symmetric structure. The analysis is based on an exact Green's function derivation as well as full wave simulations involving a realistic design for the sake of future experimental validation.

The influence of relative source-receiver positions on the recovery of P- and S-waves and energy contributions by GFSNC-binary-operator

Recently, the possibility of gathering relevant information from correlations of movements due to seismic noise has been demonstrated. Such is the case of the Green's function that is widely known as one mathematical solution of the wave equation and commonly used to represent particle displacements given an excitation source. That is the reason of its importance in the field of geophysics and seismology. From seismic noise correlations, it is also possible to obtain not only the Green's function, but the energy contributions of the wave types that are propagating in an elastic medium. For instance, in a two-dimensional (2D) medium, only the movement of compressional (P-) and shear (S-) waves takes place, each one providing fixed amounts of energy. A formulation that allows both, recovering of the Green's function from seismic noise correlations also called GFSNC binary-operator and, the gathering of related energy contributions of P- and S-waves for several representations of materials and models, is applied in this paper. As consequence of this study, it has been shown that the number of sources and their relative distribution with respect the observation points, are essential parameters for an accurate recovery of the exact Green's function. It is also proved that in all cases, P-waves have less energy in proportion to S-waves and even more, the P-wave energy contribution vanishes for Poisson's ratios close to 0.5. Having Poisson's ratios of ν={0.1,0.2,0.3,0.4}, P-waves contribute with approximately 30, 27, 22 and 14% of the total energy, while S-waves with 69, 72, 77 and 85%, respectively. These results agree with theoretical data and might be of much help to understand the origin of wave amplifications, seismic acquisition designs with artificial sources and modeling of the seismic response by potential earthquakes.

The gauged Thirring model in thermodynamic equilibrium

We study the gauged Thirring model (also known as Kondo model) in thermodynamic equilibrium using the Matsubara-Fradkin-Nakanishi formalism. In this formulation, both the temperature and the chemical potential are kept to be nonvanishing. Starting from the field equations, we write down the Dyson-Schwinger-Fradkin equations, the Ward-Fradkin-Takahashi identities, and expressions for the thermodynamical generating functional. We find the partition function of the theory and study some key features of its exact two-point Green functions, including the Landau-Khalatnikov/Fradkin transformations and some limiting cases of interest as well. In particular, we show that we can recover results from both the Schwinger and the Thirring models from the Kondo model in thermodynamic equilibrium.

Positivity of the Spectral Densities of Retarded Floquet Green Functions.

Periodically driven nonequilibrium many-body systems are interesting because they have quasi-energy spectra, which can be tailored by controlling the external driving fields. We derive the general spectral representation of retarded Green functions in the Floquet regime, thereby generalizing the well-known Lehmann representation from equilibrium many-body physics. The derived spectral Floquet representation allows us to prove the non-negativity of spectral densities and to determine exact spectral sum rules, which can be employed to benchmark the accuracy of approximations to the exact Floquet many-body Green functions.

Thermal contribution of unstable states

Within the framework of the Lee model, we analyze in detail the difference between the energy derivative of the phase shift and the standard spectral function of the unstable state. The fact that the model is exactly solvable allows us to demonstrate the construction of these observables from various exact Green functions. The connection to a formula due to Krein, Friedal, and Lloyd is also examined. We also directly demonstrate how the derivative of the phase shift correctly identifies the relevant interaction contributions for consistently including an unstable state in describing the thermodynamics.

Ion Impact Induced Ultrafast Electron Dynamics in Finite Graphene‐Type Hubbard Clusters

Strongly correlated systems of fermions have an interesting phase diagram arising from the Hubbard gap. Excitation across the gap leads to the formation of doubly occupied lattice sites (doublons) which offers interesting electronic and optical properties. Moreover, when the system is driven out of equilibrium interesting collective dynamics may arise that are related to the spatial propagation of doublons. Here, a novel mechanism that was recently proposed by the authors [Balzer et al., Phys. Rev. Lett. 121, 267602 (2018)] is verified by exact diagonalization and nonequilibrium Green functions (NEGF) simulations – fermionic doublon creation by the impact of energetic ions. The formation of a nonequilibrium steady state with homogeneous doublon distribution is reported. The effect should be particularly important for correlated finite systems, such as graphene nanoribbons, and directly observable with fermionic atoms in optical lattices. It is demonstrated that doublon formation and propagation in correlated lattice systems can be accurately simulated with NEGF. In addition to two‐time results, single‐time results within the generalized Kadanoff–Baym ansatz (GKBA) with Hartree–Fock propagators (HF‐GKBA) is presented. Finally systematic improvements of the GKBA that use correlated propagators (correlated GKBA) and a correlated initial state are discussed.