Effective integral equation and vertex operator of the green function

A simple and rigorous formulation of effective integral equations for the Green functions is presented and a general formula for the effective vertex operator (or function) has been derived. As an illustration, this formalism has been applied to obtain (i) the effective Dyson equation and effective mass operator for the single-particle Green function, and (ii) the effective integral equation and effective interaction operator for the two-particle Green function as well as those for the particle-hole Green function.     

Automatic numerical evaluation of vacancy-mediated transport for arbitrary crystals: Onsager coefficients in the dilute limit using a Green function approach

Abstract

A general solution for vacancy-mediated diffusion in the dilute-vacancy/dilute-solute limit for arbitrary crystal structures is derived from the master equation. A general numerical approach to the vacancy lattice Green function reduces to the sum of a few analytic functions and numerical integration of a smooth function over the Brillouin zone for arbitrary crystals. The Dyson equation solves for the Green function in the presence of a solute with arbitrary but finite interaction range to compute the transport coefficients accurately, efficiently and automatically, including cases with very large differences in solute-vacancy exchange rates. The methodology takes advantage of the space group symmetry of a crystal to reduce the complexity of the matrix inversion in the Dyson equation. An open-source implementation of the algorithm is available, and numerical results are presented for the convergence of the integration error of the bare vacancy Green function, and tracer correlation factors for a variety of crystals including wurtzite (hexagonal diamond) and garnet.     

Influence of core excitations on the deuteron stripping reaction with the method of many-body green functions

Within the scope of many-body Green functions we derive a formally exact expression for the (d,p)-stripping amplitude, which resembles the DWBA for this reaction. However, the distorted wavefunctions in the entrance and exit channel are defined with respect to the most general expressions for the optical potentials, as they are given by the mass operators of corresponding Dyson equations for correlation functions. In the same way the transition operator is a complicated many-body quantity which explicitly depends on the deuteron energy and contains for example contributions coming from the core excitations of the target nucleus. In this paper we are mainly interested in the latter effect and a numerical estimation of a crude model for the many-body effects of the transition operator reveals no pronounced (sharply resonant) energy dependence. The essential difference of our paper to other publications on the same subject is that we are dealing with the “Dyson equation” approach developed recently for higher Green functions.     

Solutions of the Maxwell equations and photon wave functions

Properties of six-component electromagnetic field solutions of a matrix form of the Maxwell equations, analogous to the four-component solutions of the Dirac equation, are described. It is shown that the six-component equation, including sources, is invariant under Lorentz transformations. Complete sets of eigenfunctions of the Hamiltonian for the electromagnetic fields, which may be interpreted as photon wave functions, are given both for plane waves and for angular-momentum eigenstates. Rotationally invariant projection operators are used to identify transverse or longitudinal electric and magnetic fields. For plane waves, the velocity transformed transverse wave functions are also transverse, and the velocity transformed longitudinal wave functions include both longitudinal and transverse components. A suitable sum over these eigenfunctions provides a Green function for the matrix Maxwell equation, which can be expressed in the same covariant form as the Green function for the Dirac equation. Radiation from a dipole source and from a Dirac atomic transition current are calculated to illustrate applications of the Maxwell Green function.     

Self-consistent treatment of the Anderson model

TheN-fold degenerate Anderson single impurity model in the infiniteU limit is treated by means of the irreducible Green functions method. In this approach a derivation of an exact Dyson equation and an exact self-energy operator is possible. The necessity of introducing auxiliary fields, such as slave-bosons is avoided.     

Causal green function in relativistic quantum mechanics

The causal Green function or Feynman propagator for the free-field Klein-Gordon equation and related singular functions, defined as distributions, are related to the causal time-boundary data. Probability densities and amplitudes are defined in terms of the solutions of the Klein-Gordon equation for a complex scalar field interacting with an electromagnetic field. The convergence of the perturbation expansion of the solution of the Klein-Gordon equation for a charged scalar particle in an external field is shown for well-behaved electromagnetic potentials. Other relativistic wave equations are discussed briefly.     

Nonperturbative Time-Independent Green Function of Matrix Schrödinger Equation. General Formalism and Quasiclassical Representation

A Green function of time-independent multichannel Schrödinger equation is considered in matrix representation beyond a perturbation theory. Nonperturbative Green functions are obtained through the regular in zero and at infinity solutions of the multichannel Schrödinger equation for different cases of symmetry of the full Hamiltonian. The spectral expansions for the nonperturbative Green functions are obtained in simple form through multichannel wave functions. The developed approach is applied to obtain simple analytic equations for the Green functions and transition matrix elements for compound multipotential system within quasiclassical approximation. The limits of strong and weak interchannel interactions are studied.Alexander I. Pegarkov:On leave from Physics Faculty     

Various Full Green Functions in QED

We are interested in deriving various full Green functions through general Ward–Takahashi identities (WTIs) for quantized field theories. With the help of a postulate of gauge group parameter, the general local gauge transformation laws preserving the gauge-invariance of the generating functional itself of QED model have been established successfully. By using path-integral technique, the various WTIs with resulting anomaly terms are derived under the gauge transformations. The arising of Jacobian factor from the integration measure gives a viable possibility to express full Green function. As a consequence, the complete expressions of the full vector, the full axial-vector, the full tensor vertex functions and so on are presented respectively by solving the complete set of the WTIs in the momentum space without considering the constraint imposing any Ansatz. In addition, anomaly function also provides an effective means to judge the divergence of variant coupling currents on fields.     

Green’s function calculations of light nuclei

The influence of short-range correlations in nuclei was investigated with realistic nuclear force. The nucleon-nucleon interaction was renormalized with V_lowk technique and applied to the Green’s function calculations. The Dyson equation was reformulated with algebraic diagrammatic constructions. We also analyzed the binding energy of ⁴He, calculated with chiral potential and CD-Bonn potential. The properties of Green’s function with realistic nuclear forces are also discussed.     

Equations for the Green functions in quantum electrodynamics

Using the Freese-Matthews-Salam equations for chronological products of field operators, equations are written for Green functions of many electrons and photons. It is shown that in order to find any single Green function an infinite recursive system of these equations has to be solved. After adding terms containing the external electric current and external electromagnetic potential to the Lagrangian, this system is reduced to one equation containing functional derivatives of higher orders. It is shown that all relations and equations become much simpler when the definition of the Green function is appropriately changed.On leave from the Physical Institute of the Czechoslovak Academy of Sciences, Prague, Czechoslovakia.     

Non-Hermitian systems of Euclidean Lie algebraic type with real energy spectra

We study several classes of non-Hermitian Hamiltonian systems, which can be expressed in terms of bilinear combinations of Euclidean–Lie algebraic generators. The classes are distinguished by different versions of antilinear (PT)-symmetries exhibiting various types of qualitative behaviour. On the basis of explicitly computed non-perturbative Dyson maps we construct metric operators, isospectral Hermitian counterparts for which we solve the corresponding time-independent Schrödinger equation for specific choices of the coupling constants. In these cases general analytical expressions for the solutions are obtained in the form of Mathieu functions, which we analyze numerically to obtain the corresponding energy spectra. We identify regions in the parameter space for which the corresponding spectra are entirely real and also domains where the PT symmetry is spontaneously broken and sometimes also regained at exceptional points. In some cases it is shown explicitly how the threshold region from real to complex spectra is characterized by the breakdown of the Dyson maps or the metric operator. We establish the explicit relationship to models currently under investigation in the context of beam dynamics in optical lattices.     

Photon Green’s functions for a consistent theory of absorption and emission in nanostructure-based solar cell devices

The light-matter interaction in planar nanostructures with applications in photovoltaic devices is investigated by means of a microscopic quantum-kinetic theory based on the non-equilibrium Green’s function formalism. The Dyson and Keldysh equations for the Green’s functions of photons are solved numerically. The result is used to couple the optical and electronic degrees of freedom via respective self-energies. The numerical approach for the solution of the optical problem is verified against a standard transfer-matrix formalism and applied to the computation of absorption and emission characteristics in ultra-thin-absorber solar cells.     

Electron and Phonon in Neutron Scattering of Semiconductor Crystals

An interaction of electron with harmonic, localized and anharmonic fields has been taken to develop the theory of neutron scattering. The Fourier transformed electron Green’s function is evaluated by Zubarev equation of motion technique of quantum dynamics and Dyson equation approach. The expression of Debye-Waller factor (DWF) has been obtained from electron phonon linewidth. The study of its (DWF) effect on differential scattering cross-section through temperature, electron phonon coupling constant and excitations has been investigated in this approach.     

Extremely short optical pulse in a system of nanotubes with adsorbed hydrogen

In this Letter we address the system of carbon nanotubes with adsorbed hydrogen, which is a problem of particular practical importance. Based on the periodic Anderson model we describe the electronic subsystem in such a system, so that employing the method of Green functions allowed us to obtain the dispersion law for electrons. In the low-temperature limit we investigated a joint dynamics of electrons and electromagnetic field. The effective equation, which describes the propagation of ultrashort optical pulses, has been derived. We analyze the solutions of this equation and their dependence on the parameters of the problem for a two-dimensional CNT system.     

Phase factors representation for electromagnetic scattering from arough-surface perfect conductor

The surface magnetic field (current) integral equation for a plane in an average rough interface between a vacuum and a perfect conductor is considered. This equation is presented in a form that contains the random elevations only as exponential functions with purely imaginary exponents. Iterations of this equation lead to a solution in powers of multiplicity of the Kirchhoff scattering, and each iterative term contains the random elevations only in exponents. Because the modulus of such functions is equal to unity, we may expect that convergence of such series is independent of the Rayleigh parameter. All mean values can be expressed directly in terms of joint characteristic functions of elevations in different points of a rough surface. To develop this representation of the scattering problem we use the three-dimensional Green function in terms of the one-dimensional Fourier integral with respect to z.     

ON THE DYSON EQUATION AND THE WARD-TAKAHASHI IDENTITIES OF THE CLOSED TIME PATH GREEN'S FUNCTIONS

The Dyson equation satisfied by the closed time path Green's function of the or-der parameters is considered.The transport equation for the number density of thequasi-particles is written down in a general but simple form.Using the path integ-ral formulation for the generating functional of these Green's functions,the Ward-Takahashi identities are deduced.     

RPA equations for 2-particle-1-hole states

Within the Dyson equation approach to higher correlation functions we derive RPA-like equations for the 2-particle-1-hole problem. The equations can be entirely expressed by the RPA sub-solutions of the particle-hole and particle-particle problem. In this way our equations are very well suited for weak or intermediate coupling situations.     

Gauge parameter dependence in the background field gauge and the construction of an invariant charge

By using the enlarged BRS transformations we control the gauge parameter dependence of Green functions in the background field gauge. We show that it is unavoidable — also if we consider the local Ward identity — to introduce the normalization value ξ₀ of the gauge parameter ξ. The dependence of Green functions on ξ₀ is governed by a partial differential equation in a similar manner as the dependence on the normalization point κ is governed by the RG equation. By modifying the Ward identity we are able to construct in 1-loop order a gauge parameter independent combination of 2-point vector and background vector functions. By explicit construction of the next orders we show that this combination can be used to construct a gauge parameter independent RG-invariant charge. However, it is seen that this RG-invariant charge does not satisfy the differential equation of the normalization value ξ₀ of the gauge parameter, and, hence, is not ξ₀-independent as required.     

各向异性介质三维电磁响应模拟的Ho-GEBA算法

本文基于积分方程法研究并建立了一种模拟横向同性介质中任意各向异性异常 体三维电磁响应的高阶广义扩展Born近似(Ho-GEBA)算法. 首先利用逐次迭代技术给出积分方程的广义级数展开解, 为保证其收敛性, 引入一种各向异性条件下满足压缩映射的迭代算子. 然后利用异常体区域分解技术, 并结合扩展Born近似原理, 得到各向异性介质三维电磁响应的Ho-GEBA解. 为提高效率, 计算过程中采用并矢Green函数的解析表达式. 最后通过数值计算实例对比验证了本文算法的有效性. 关键词： 高阶广义扩展Born近似 积分方程 电磁模拟 解析Green函数     

Theory of high-temperature superconductivity in cuprates

A microscopic theory of electronic spectrum and superconducting pairing in the high-temperature cuprate superconductors is presented. The theory is based on consideration of strong electron correlations within the Bogolyubov polar model. The Dyson equation is derived by using the equation of motion method for the thermodynamic Green functions in terms of the Hubbard operators. The self-energy is evaluated in the noncrossing approximation for electron scattering on spin and charge fluctuations induced by kinematic interaction. The theory demonstrates that a strong Coulomb repulsion results in the anomalous electronic spectrum and unconventional (d-wave) superconducting pairing with high T _c mediated by the antiferromagnetic exchange and spin fluctuations.