Equations For Green Functions Research Articles

Method for breaking chains of equations for green's functions

This article presents a new method for closing infinite systems of equations for Green's functions. Its basis is a procedure for simultaneously splitting two higher-order Green's functions. There is a particular relationship between the three Green's functions that result. These Green's functions are self-consistent. Known Green's functions are used to reconstruct the corresponding correlation functions. An example is provided by the problem of magnetization for a spin system in the Heisenberg model with first-and secondorder approximations in the spin-spin interaction. The first-order approximation corresponds to the molecular-field approximation. The second-order computation corresponds to obtaining general relations determining the magnetization of a system.

Method of Solving Conformal Models inD-dimensional Space I

We study the Hilbert space of conformal field theory inD-dimensional space. The latter is shown to have model-independent structure. The states of matter fields and gauge fields form orthogonal subspaces. The dynamical principle fixing the choice of model may be formulated either in each of these subspaces or in their direct sum. In the latter case, gauge interactions are necessarily present in the model. We formulate the conditions specifying the class of models where gauge interactions are being neglected. The anomalous Ward identities are derived. Different values of anomalous parameters (D-dimensional analogs of a central charge, including operator ones) correspond to different models. The structure of these models is analogous to that of 2-dimensional conformal theories. Each model is specified byD-dimensional analog of null vector. The exact solutions of the simplest models of this type are examined. It is shown that these models are equivalent to Lagrangian models of scalar fields with a triple interaction. The values of dimensions of such fields are calculated, and the closed sets of differential equations for higher Green functions are derived.

Bridge-mediated electronic interactions: Differences between Hamiltonian and Green function partitioning in a non-orthogonal basis

An analysis of the partitioning (projection) technique is given with emphasis on non-orthogonal basis sets. The general expression for the effective Hamiltonian obtained via Löwdin partitioning of the Schrödinger equation is discussed in the context of semi-empirical theories and electron transfer matrix elements. Numerous pitfalls in calculations of matrix elements are pointed out. More importantly, it is shown that contrary to the case of an orthogonal basis, for a non-orthogonal basis Löwdin partitioning of the Schrödinger equation and partitioning of the Green function equation are not equivalent. The latter method provides a more general prescription for deriving effective Hamiltonians. Such Hamiltonians reproduce the full propagation in the partitioned subspace.

On the dynamics of the [formula omitted] Ising model in a transverse field

A new solution for the spin- 1 2 Ising model in an external transverse field for the case of three-dimensional nearest-neighbor interactions is obtained. The chain of equations for high-order Green functions is decoupled in the scope of symmetrized Tyablikov's scheme and consistently reduced to a closed equation system, which is solved exactly. This improves previous relevant solutions by short-range quantum correlation effects caused by spins involved in small compact clusters and suggests a new dynamic description of the paramagnetic phase of ferroelectrics, to which the model is directly applied.

Josephson and quasiparticle tunneling between anisotropically paired superconductors in the presence of externally applied voltage

Charge transport in tunnel junctions between anisotropically paired superconductors in the presence of externally applied voltage is investigated theoretically making use of the Eilenbergen equations for the quasiclassical Green functions. We found and described analitically characteristic singular points on the I-V curves which are specific for the case of anisotropic pairting. All four terms for the electric current are examined. Two of them describe ac Josephson effect and two correspond to the quasiparticle current (the last term occurs only for a time-dependent voltage). Various momentum dependences of the order parameter in the bulk of superconductors and different crystal orientations are considered.

Superconductive states in the two-dimensional repulsive Hubbard model

The investigation of the two-dimensional repulsive Hubbard model is continued for the case in which the Fermi level is close to one of the saddle Van Hove points of the quasiparticle energy function. The Bethe-Salpeter equation for the two-particle scattering amplitude and the system of Dyson-Gor'kov equations for normal and anomalous Green functions are considered. The closeness of the Van Hove point to the Fermi level makes the investigation substantially simpler. A new method of evaluating the kernel of the equations, i.e., the oneloop polarization diagram, is suggested. It is shown that a nontrivial Cooper pairing (the superposition of the pairings with odd angular momenta) appears in the model if and only if the Fermi level is close to one of the Van Hove points. The temperature of the superconductive phase transition is maximal for some special mutual location of the Fermi level and the Van Hove point. Two different superconductive solutions (modes) are found which are antisymmetric functions in the momentum representation. These modes coexist in close vicinity of the doping parameter value corresponding to the intersection of the Van Hove point and the Fermi level. The values of the energy gap for these modes are, generally speaking, different (the two different gap values are observed in some experiments). Bibliography: 18 titles.

Collective modes in layered superconductors

Weakly damped collective oscillations in layered superconductors are examined theoretically. The calculation is based on kinetic equations for Green functions generalized to the case of layered superconductors with weak interlayer coupling. We show that weakly damped plasma oscillations of superconducting electrons can exist in such superconductors and calculate the spectrum of the mode. The oscillations are reminiscent of Josephson plasma oscillations in a superconducting tunnel junction. As the temperature approaches the critical one, these oscillations transform into Carlson-Goldman modes. Possibilities for detecting the plasma mode experimentally are discussed. We found distinctive features in the reflection coefficient for an electromagnetic wave incident at an arbitrary angle on a plane parallel to the layers. The presence of the mode changes also the dispersion relation for the waves propagating along a thin film of a layered superconductor.

Superconductivity mechanism of Ba 1− xK xBiO 3

Based on the singly occupied a iσ picture and the mechanism of charge fluctuation, the effective Hamiltonian of the two-band Hubbard model is derived by using the degeneracy perturbation theory for the noncuprate compound Ba 1− x K x BiO 3. By considering the next-neighbour pairing of two opposite spin holes in coordinate space, the Green's function equation of motion and the superconductivity equation are obtained. Furthermore, how the superexchange interaction and the hopping energy affect the next-neighbour pairing has been reasonably explained, and the superconductivity window is due to the Cooper pairing in coordinate space.

Kinetic equations for clean superconductors: Application to the flux flow hall effect

The kinetic equations for clean superconductors (l≫ξ) are derived. Expanding the equations for the time dependent Green functions in the quasiclassical parameter, the new contributions are found which contain the derivatives of the distribution functions with respect to the quasiparticle momentum. The transition from the ultra-clean case (no relaxation) to a relaxation-dominated behavior, for which the kinetic equations coincide with the usual quasiclassical approximation, occurs for the relaxation time of the order of ℏEF/Δ2. The kinetic equations can be used for various dynamic processes in superconductors including the flux-flow Hall effect. The derived equations, after necessary modifications for the p-wave pairing, are especially suitable for nonstationary problems in the theory of superfluidity of3He.

Scattering of edge states in quasi-one-dimensional periodic systems

We develop an integral equation Green's function method to study the transmission of waves (in two dimensions) through a continous potential in the presence of a perpendicular magnetic field. Motivated by recent experiments, the general formalism is then applied to compute the magnetoconductance of a two-dimensional system containing a finite number of barriers confined by a parabolic potential. When the magnetic field is strong and there are practically only one or two uncoupled edge states, we predict approximate periodic conductance oscillations in qualitative agreement with those observed. The periodic and the total number of oscillations within a miniband is determined by commensurability of the pertinent length scales (magnetic length, lattice constant and electron wave length) as well as by the size of the system. For weak magnetic fields, the number of edge states increases, but the effect of coupling between modes is evaluated and proves to be small.

SUPERCONDUCTIVITY MECHANISM OF NON-CUPRATE SUPERCONDUCTORS

Based on the singly occupied a-picture and the mechanism of charge fluctuation, the effective Hamiltonian of two-band Hubbard model is derived by using the degeneracy perturbation theory for the non-cuprate comoound Ｂａ1-xＫxＢｉＯ3. By considering the next neighbour pairing of two opposite spin holes in coordinate space, the Green's function equation of motion and superconductivity equation are obtained. Furthermore, how the superexchange interaction and the hopping energy affect the next neighbour pairing has been explained reasonably, and the superconductivity window is due to the Cooper pairing in coordinate space.

Signal restoration after transmission through a nonuniform LCRG line

In the present paper, signal restoration after transmission through a nonuniform LCRG line is treated by means of a compact Green function approach. These Green functions are characteristic of the nonuniform line and independent of the incident signal. The compact Green functions map the signal which is received after transmission through a nonuniform LCRG line to split components of the voltage at an arbitrary point on the nonuniform line. Partial differential equations for the compact Green functions are derived together with initial, boundary and jump conditions. A convolution integral with the compact Green function as the kernel is used to restore the original signal. Numerical results for the compact Green functions and for the restored signal are presented. It is noted that for transmission lines with considerable losses, good restoration can be obtained instantaneously by means of a simple RC circuit.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the electron spectrum of the Hubbard model including interactions with local anharmonic vibrations

This paper presents a self-consistent calculation of the single-electron excitation spectrum within the framework of an Hubbard model including interaction with pseudospin degrees of freedom due to the anharmonic vibrations of the apex oxygen (O(4)) in high-temperature superconductors of the YBaCuO-type (Müller model). The spectra are calculated for different electron concentrations using Green's function equations with a decoupling analogous to a Hubbard-I approximation. We use an approach which is based on the exact solution of the single-site problem. In contrast to the Hubbard model, the spectra show additional subbands due to the interaction of the electrons with the anharmonic vibrations (pseudospins). Those interactions result in the destruction of the electron-hole symmetry and the system can show hole conductivity even at half filling.

Time domain Green function technique for a point source over a dissipative stratified half‐space

In this paper the time domain scattering problem of a stratified dissipative half‐space excited by a point source is considered. Both the direct and inverse scattering problem are treated. The dissipative wave equation is reduced to an equation in one spatial dimension by spatial Fourier series and a Hankel transform. The transformed wave equation is then solved by a combination of a time domain wave‐splitting technique and a Green function technique. The Green functions represent the internal fields inside the stratified medium. The splitting is expressed in terms of a Neumann operator K which has an explicit expression as a convolution integral with a Bessel function kernel. Based upon this wave splitting, Green functions can be defined and a set of first order partial differential equations for Green functions are derived. The equations are well suited for a numerical treatment and a number of numerical examples for both the direct and inverse problems are presented. A brief discussion of the invariant imbedding method applied to the direct and inverse scattering problem is also given. It is shown that one‐sided reflection data are sufficient to reconstruct the phase velocity and the dissipation coefficient simultaneously by using two different values of the Hankel transform parameter.

A dynamic Green's function theory for a highly excited direct gap semiconductor

Abstract The problem of highly excited semiconductors is treated self-consistently with respect to carrier as well as photon dynamics by a new Green's function (GF) technique. With the help of this method the shortcoming of quasi-static considerations has been overcome by a new concept of separation of fast and slowly moving observables. The inhomogeneities of the GF equations of motion are solutions of kinetic equations in the slow time scale. In this concept the total spectral weight function for conduction band electrons is obtained in a femtosecond time scale of the laser pulse duration.

Frequency and Time Domain Green Function Technique for Nonuniform LCRG Transmission Lines with Frequency-Dependent Parameters

The reflection and transmission of waves on a nonuniform LCRG transmission line of finite length with frequency-dependent parameters are considered. A Green function technique combined with wave splitting is used both in the frequency domain and the time domain. In the frequency domain ordinary differential equations (ODEs) for the time-harmonic Green functions are obtained. In the time domain the transmission line model is established through a set of dispersion kernels. Partial differential equations (PDEs) for the time-domain Green functions are derived. Numerical results for transient reflected, transmitted and internal currents are presented. Furthermore, it is noted that a special case of the present formalism can be used to determine the reflected, transmitted and internal fields for normal plane wave incidence on a stratified dispersive and dissipative slab.

ON REPRESENTATION OF QUANTUM ACTION FUNCTIONAL

A representation for QA functional is constructed. The representation can be formulated in terms of compact (one-particle—irreducible) Feynman diagrams with special correspondence rules. The main properties are: (1) the representation of the exact vertex requires only a finite number of diagrams, (2) manifest symmetry with respect to permutations of particles. That enables one to develop consistent calculational methods. The resulting set of equations for the exact Green functions and vertices can open a way to approximate nonperturbative solutions.

N-PARTICLE PROBLEM IN QUANTUM FIELD THEORY AND THE FUNCTIONAL LEGENDRE TRANSFORMS

The Bethe–Salpeter equations for the n-particle irreducible Green functions in quantum field theory are derived. Their kernels are expressed in terms of the functional Legendre transforms, and n-particle irreducibility of the perturbation theory graphs for the kernel of the n-particle equation is proved. The method for obtaining the Faddeev–Yakubovski equations in quantum field theory is demonstrated for the three-particle case.

EXACTLY SOLVABLE CONFORMALLY INVARIANT QUANTUM FIELD MODELS IN D DIMENSIONS

A class of exactly solvable models of the conformally invariant quantum field theory in D dimensions is proposed. It is shown that in any conformal theory of the field φ(x) with the scale dimension d there exists an infinite collection of the tensor fields Ps of the ranks and the dimensions ds=d+s independently of the type of interaction. These tensor fields appear in the product Tµν(x1)φ(x2) operator expansion where Tµν is the energy-momentum tensor. The fields Ps are analogues to the certain superpositions of the secondary fields of D=2 models. The existence of the fields Ps follows from the structure of the Ward identities for the energy-momentum tensor conformally invariant Green functions. Each model of the above-mentioned class is defined by the operator equation Ps(x)=0. The method of solving these models is proposed. Some of the models coincide with the certain lagrangian models. The method allows us to obtain closed differential equations for each Green function of fundamental and composite fields, and also algebraic equations for scale dimensions of the fields. The derivation of all these equations is based on the energy momentum conformal Ward identities’ specific property. Detailed analysis of the Ward identities is given in the paper. It is shown that each D&gt;2-model involves a special scalar field PD−2 with the scale dimension dP=D−2. This field is an analogue of the central charge of D=2 models and becomes the constant coinciding with the central charge when D=2. A new class of D=2 models with broken infinite parametric symmetry is obtained. In each model the closed differential equations for the highest Green functions are derived and the central charge is calculated. The general method of solving the D≥2 models is illustrated on examples of Thirring and Wess-Zumino-Witten models and on a trivial model in four-dimension space.

Electronic states in trans-polyacetylene with bond-type impurities in the coherent-potential approximation.

Disorder effects on the electronic states in trans-polyacetylene are investigated in the case of bond-type impurities which give rise to backward scatterings of electrons. The generalized Takayama--Lin-Liu--Maki model is studied with the help of the coherent-potential approximation (CPA). The dimerization of the lattice is assumed to be uniform and is represented by an order parameter \ensuremath{\Delta}. The CPA gives equations for electron Green functions. They are solved numerically, while \ensuremath{\Delta} is determined as to give minimum energy. We find three solutions at low impurity concentrations, one of which is stable solution, another is metastable, while the third is unstable. In the stable solution, which survives for higher impurity concentrations, impurity bands are not formed. In the metastable solution, the conduction and valence bands intrude into the gap region, indicating the development of impurity bands. Concentration dependences of the order parameters, energy gaps, and the critical concentration for the metastable solution are derived.