Averaged Green Function Research Articles

Scattering of a scalar wave from a two-dimensional randomly rough Neumann surface

Abstract We present a reciprocity and unitarity preserving formulation of the scattering of a scalar plane wave from a two-dimensional, randomly rough surface on which the Neumann boundary condition is satisfied. The theory is formulated on the basis of the Rayleigh hypothesis in terms of a single-particle Green's function G(q‖|k‖) for the surface electromagnetic waves that exist at the surface due to its roughness, where k‖ and q‖ are the projections on the mean scattering plane of the wave vectors of the incident and scattered waves, respectively. The specular scattering is expressed in terms of the average of this Green's function over the ensemble of realizations of the surface profile function (G(q‖|k‖)). The Dyson equation satisfied by (G(q‖|k‖)) is presented, and the properties of the solution are discussed, with particular attention to the proper self-energy in terms of which the averaged Green's function is expressed. The diffuse scattering is expressed in terms of the ensemble average of a two-p...

Effects of laser spatial incoherence with finite correlation length on the space-time behavior of backscattering instabilities.

The average space-time behavior of backscattering instabilities in the presence of a purely spatial incoherence of the laser beam is investigated analytically. It is found that the divergence of the strongly damped limit of the average backscattered light amplitude [cf. H. A. Rose and D. F. Dubois, Phys. Rev. Lett. 72, 2883 (1994)] must be replaced by the onset of an absolutelike instability characterized by a local, space dependent growth rate. The validity of the results is discussed in the regime where the breakdown of our model is caused by the onset of absolutely unstable hot spots. The time asymptotic behavior of the instability in this regime is computed heuristically. The average Green's function for the backscattered light is computed analytically in the case where the laser incoherence is characterized by an exponentially decaying correlation function.

Localization and the field intensity of surface-phonon polaritons.

We develop a numerical model for the calculation of the average field intensity of surface-phonon polaritons in the presence of random roughness. The average field intensity is related to the average two-particle Green function, which is computed within the framework of diagrammatic theory. The field of surface-phonon polaritons decays exponentially from the incident beam center on the surface. Particular emphasis is placed on examining effects of the localization of surface-phonon polaritons due to random roughness. It is found that the localization occurs in a limited frequency range near the maximum frequency of surface-phonon polaritons. The decaying length in the extended region is much larger than that in the localized region. In the extended region the field intensity of surface-phonon polaritons is usually smaller than the incident intensity, while in the localized region it can be enhanced. The numerical results demonstrate that localization effects can reduce the intensity enhancement by the order of ${10}^{3}$. Further, the intensity enhancement decreases as the roughness amplitude increases. We conclude that the localization of surface-phonon polaritons is due to the destructive interference between waves scattered from the rough surface.

Calculation of diffusivities of passive fields in turbulent media

Abstract The exact numerical and approximate analytical solutions of the simplest nonlinear integral equation with second order nonlinearity for the averaged Green function are presented. It is assumed that the turbulence is stationary, homogeneous, isotropic and incompressible. Numerous examples of turbulent spectra are considered (peak-like spectrum, spectra of Kolmogorov's type with different forms of “pumping” regions, stepwise spectra etc.). Special emphasis is given to investigating the case of so called “frozen” turbulence when the parameter ξ =u 0τ/R→∞ where uτ0,R 0 are characteristic velocity, lifetime and space scale of turbulent pulsations, respectively. It is shown that these solutions allow us to calculate the turbulent diffusivities accurately for arbitrary spectra with any values of the parameter ξ. The results take into account the possible helicity of turbulence concerned only with scalar passive fields (number density and temperature).

The effect of disorder on the Raman spectra of glasses

The effect of structural disorder on the first-order Raman scattering in glasses is studied by perturbation series approximations to the averaged Green function of an appropriately chosen model system. The proposed theoretical approach is applied to calculating the Raman spectra of Na2O-SiO2 glass systems on the basis of the disorder-modified spectrum of a Si2O7 dimer.

The density of states and anisotropic optical absorption of a disordered polymer film

Motivated by a three-dimensional LDA band structure calculation for trans-polyacetylene (PA) we construct an effective two-dimensional model for a film of conjugated polymers. In real materials a non-perfect structural order is found. It has its origin in the finite conjugation length of single chain segments, impurities and crystalline defects. We model these effects through a fluctuating contribution to the interchain hopping amplitude. Using a supersymmetric path integral formalism we obtain the averaged one-particle Green function giving the electronic density of states (DOS) and furthermore the current correlation function determining the optical absorption coefficient and DC conductivity. As a consequence of the interchain coupling we find a broad shoulder at the band edge in the DOS and the optical absorption. The main effect of disorder is seen in a broadening of the interband transition peak and an increasing intraband absorption. The anisotropy in the absorption coefficient parallel and perpendicular to the chain direction allows us to fix the interchain coupling strength; the line shape contains information about the effective disorder strength.

Crystal surface with adsorbed impurities: Phonon frequencies and polarizations.

One considers a perfect crystal with a surface on which an incomplete layer of impurities or a layer of an alloy is adsorbed. A general method allowing one to calculate the phonon spectral density given by the imaginary part of the averaged Green function over the different configurations is presented. This function is decomposed into two parts. The first one gives the shift of phonon frequencies and the frequencies and polarizations of new surface phonons. The second part accounts for the broadening of the phonon lines which is dependent upon the statistical distribution of the adsorbaths. Quantitative results are given for the case of copper adatoms adsorbed on the sites of the (100) face of a copper crystal

Exact renormalization-group approach for the average Green functions of aperiodic lattices

We present a real-space renormalization-group approach to calculate the electronic average Green functions of one-dimensional aperiodic lattices. The characteristic of this approach is that it is only concerned with the global symmetry of the lattice under the renormalization-group transformation and the average Green function can be calculated in a simple manner to any degree of accuracy. The Thue-Morse and the Fibonacci lattices are studied in detail to illustrate the method.

Green's Function and Density of States of Fibonacci Quasicrystal

We present a new real-space decimation approach to calculate the electronic average Green's function and density of states of Fibonacci lattice. The approach only concerns with the deflation rule of the lattice. The average Green's function and the density of states can be calculated in a simple manner to any degree of accuracy.

Electronic properties of one-dimensional quasiperiodic lattices: Green's function renormalization group approach

Using the inflation-deflation symmetry, we have developed a new real-space decimation approach to study the electronic properties of one-dimensional quasiperiodic lattices. The key result is the construction of a compact renormalization group that allows simple calculation of the average Green's function and the average density of states to any degree. The Fibonacci and the generalized Fibonacci lattices are used to demonstrate the method. Numerical results for the average density of states of these lattices show a good agreement with the results obtained by other methods. This confirms the validity and the efficiency of the approach.

The Non-Linear Equations for the Green Function and Calculation of the Magnetic Field Turbulent Diffusivities and α-Effect

The exact numerical solution of the simplest non-linear equation from the hierarchy of non-linear equations for the averaged Green function shows that such solution allows to calculate the diffusivity and α-effect coefficient with a good accuracy for an arbitrary spectra of turbulence for all values of the characteristic parameter. It is derived also the improved equation describing the evolution of admixture fluctuations in a turbulent medium which takes into account the non-linear equation for the averaged Green function.

Localization lengths in disordered Peierls systems

We have studied the influence of bond- and site-type impurities on the ground state properties of one-dimensional Peierls systems. Using a functional integral formalism with both commuting and anticommuting variables we have calculated the averaged Green's function which determines the electronic density of states and localization length (Thouless formula). Some limiting cases can be solved analytically. To apply our model to doped polymers we derive the connection between doping concentration and disorder strengths Dj. For illustration we present the results with parameters appropriate for polyacetylene.

Some aspects of electron correlation, magnetism, and localization in spatially disordered systems

We consider a disordered Hubbard model for a system characterized by quenched liquid-like disorder, with correlation treated at the generalized Hartree–Fock level and the possibility of local magnetic moments introduced from the outset. A simple theory based on averaged Green functions is used to describe the properties of the system in the local moment domains in particular, and their evolution with number density and both structural and electronic parameters of relevance. A probabilistically based mean-field theory is then developed to address the localization characteristics of the HF pseudoparticle states, and the consequent disorder-induced metal–insulator transition. Three principal density domains of interest are identified: a low density insulator with local magnetic moments, a metallic phase with local moments at intermediate densities, and a higher density nonmagnetic metallic state. The theory is used to provide an interpretation of bulk experiments on expanded fluid alkali elements, with particular emphasis on the insulating and ‘‘dirty’’ metallic domains.

Transport properties for random walks in disordered one-dimensional media: Perturbative calculation around the effective-medium approximation.

A perturbation method for the calculation of transport properties in disordered one-dimensional media is presented. It permits considering cases of strong and weak disorder and different initial conditions in a unified way, giving new results and reproducing known ones. The effective-medium approximation (EMA) appears as the zeroth-order step of our diagrammatic scheme. The method is applied to the random-barrier and random-trap models. The range of validity of the EMA is established. Exact results beyond the EMA are obtained for the low-frequency behavior of the frequency-dependent diffusion coefficient and the modified Burnett coefficient associated with the average Green's function and with the response function of the problem.

The density of states in the Anderson model at weak disorder: A renormalization group analysis of the hierarchical model

We study the density of states in a hierarchical approximation of the Anderson tight-binding model at weak disorder using a renormalization group approach. Since the Laplacian term in our model is hierarchical, the renormalization group transformations act essentially on the local potential distribution and the energy. Technically, we use the supersymmetric replica trick and study the averaged Green's function. Starting with a Gaussian distribution with small variance, we find that the density of states is analytic as soon as the variance of the potential is turned on, except possibly near the band edge, where we can show this only forα>√2, which corresponds tod>4. Moreover, it is perturbatively close to the free one, except near the eigenvalues of the (hierarchical) Laplacian, where it is given (up to perturbative corrections) by the rescaled potential distribution.

Soluble theories for the density of states of a spatially disordered two-level tight-binding model

An analysis is given for the configurationally averaged Green functions of a random multi-level tight-binding model characterised by quenched liquid-like disorder, using graph-theoretical methods. An exact self-consistency equation for the average diagonal Green function matrix, G(z), is derived, from which follow the partial densities of states (DOS). From the exact description, various approximate theories for G(z) may be developed systematically. The authors examine in particular three tractable theories for the case of a two-band system: the Hubbard approximation, the Matsubara-Toyozawa approximation and the single super-chain approximation (SSCA) which is equivalent to the effective medium approximation (EMA) of Roth (1974, 1976). With Yukawa transfer matrix elements the SSCA/EMA is solved analytically by exploiting direct analogies with the theory of classical binary liquid mixtures. For all three theories the material parameter dependence of the DOS is examined systematically and comparatively, with particular regard to band overlap effects which may lead to a metal-insulator transition.

Density of states in a disordered solid, Borel-Pad� analysis

Wegner's gauge-invariant model of the localization problem is treated on ad-dimensional lattice. Borel-Pade methods are used to analyze the perturbation series for the averaged one-particle Green function. Results for the density of states are obtained.

Calculation of averaged Green functions for the Gaussian unitary ensemble

Averages of special Green functions (1- and 2-point function) for the Gaussian unitary ensemble which appear in the context of stochastic quantum systems are obtained by a new technique. The calculation is based on the representation of the 1- and 2-point function by an integral over graded variables. The present approach, in contrast to previous ones, does not employ a Hubbard Stratonovich transformation and a saddle point approximation; the integration is performed directly and yields analytical expressions also for finite dimension of the ensemble matrices. The limit of infinite dimension and the connection with correlation functions is briefly discussed.

The density of states of a spatially disordered tight-binding model

The authors give an exact analysis of the configurationally averaged Green functions for a random tight-binding model characterised by quenched liquid-like disorder, using a graph-theoretical method originally applied by Wertheim to a problem in classical dielectric theory. The structural characteristics of the system are incorporated fully. They derive a formally exact self-consistency equation for the averaged diagonal Green function G(z), from which follows the density of states. A systematic derivation and critical discussion of various approximate theories for G(z) is given. They also show that extension to include site-diagonal disorder is straightforward for single-site theories. Finally, an illustrative calculation of the density of states is carried out for the low-density domain.

The influence of disorder on the electronic structure of conjugated polymers

The averaged quasi-classical Green functions are used to study the influence of impurities on the electronic structure of conjugated polymers. It is found that the electronic gap and the size of the dimerisation are decoupled in contrast to what was found in previous investigations. This leads to a situation where the polymer still can be dimerised without a gap in the electronic spectrum. Consequences for non-linear excitations (kinks and polarons) are discussed.