On the resistance of an infinite square network of identical resistors – Theoretical and experimental comparison

A review of the theoretical approach for calculating the resistance between two arbitrary lattice points in an infinite square lattice (perfect and perturbed cases) is carried out using the lattice Green's function. We show how to calculate the resistance between the origin and any other site using the lattice Green's function at the origin, G_o (0, 0), and its derivatives. Experimental results are obtained for a finite square network consisting of 30 ×30 identical resistors, and a comparison with those obtained theoretically is presented.     

Network with Two Extra Interstitial Resistors

Investigation of the effective resistance between two arbitrary nodes of a resistor network is carried out when it is perturbed by introducing two extra interstitial resistors in the perfect lattice using lattice Green’s function. Analytical formula for the effective resistance of the perturbed lattice is given in terms of that for the unperturbed one. Numerical results for a square lattice are presented.     

Resistance Calculation for an Infinite Simple Cubic Lattice Application of Green's Function

It is shown that the resistance between the origin and any lattice point (l,m,n) in an infinite perfect Simple Cubic (SC) lattice is expressible rationally in terms of the known value of G ₀ (0,0,0). The resistance between arbitrary sites in an infinite SC lattice is also studied and calculated when one of the resistors is removed from the perfect lattice. The asymptotic behavior of the resistance for both the infinite perfect and perturbed SC lattice is also investigated. Finally, experimental results are obtained for a finite SC network consisting of 8×8×8 identical resistors, and a comparison with those obtained theoretically is presented.     

Small coupling (low temperature) expansions of nonabelian Yang-Mills fields on a lattice in temporal gauge

A systematic approach to large β expansions of nonabelian lattice gauge theories in temporal gauge is developed. The gauge fields are parameterized by a particular set of coordinates. The main problem is to define a regularization scheme for the infrared singularity that in this gauge appears in the Green's function in the infinite lattice limit. Comparison with exactly solvable two-dimensional models proves that regularization by subtraction of a naive translation invariant Green's function does not work. It suggests to use a Green's function of a half-space lattice first, to place the local observable in this lattice, and to let its distance from the lattice boundary tend to infinity at the end. This program is applied to the Wilson loop correlation function for the gauge group SU(2) which is calculated to second order in $\text{1}\text{β}$.     

Self-consistent calculations for the electronic structure of a vacancy in copper. A solution of the embedding problem

The charge density and the local density of states for a vacancy in Cu and for the first shell of Cu neighbours are calculated by the KKR-Green's function technique. The muffin-tin potentials for the vacancy and the neighbour shell atoms are determined self-consistently in the local density approximation of density functional theory. By the use of the proper host Green's function the embedding of this cluster of 13 perturbed muffin-tins into the infinite array of bulk Cu muffin-tin potentials is described rigorously, thus representing a solution of the embedding problem. The calculations demonstrate a rather large charge transfer of 1.1 electrons from the first neighbour shell to the vacancy.     

Field-theoretic approach to the properties of the solid state

The techniques of quantum field theory are used to investigate the thermodynamic ion displacement correlation function—or Green's function of the phonon field—in a crystal and especially in a metal. The structure of thermodynamic Green's functions is outlined and the method for solving for them at finite temperature is fully discussed.The analytic structure of the phonon Green's function is then considered. This function is shown to be bounded and invertible everywhere off the real axis; a spectral form is derived for its inverse. The symmetries imposed by the point group of the crystal are then discussed.Assuming small ionic oscillations, we find the inverse of the phonon Green's function as a linear function of the electronic contribution to the dielectric response function of the metal. This dielectric function is shown to be simply related to the longitudinal part of the conductivity tensor that gives the response of the electrons to the effective electric field in the metal. The assumption of translational invariance then leads to an explicit expression for the phonon Green's function in terms of this conductivity.The deformations in the lattice induced by an arbitrarily time varying external force are calculated in terms of the retarded phonon Green's function. In the static long wavelength limit the phonon Green's function yields the macroscopic elastic constants of the crystal. Their relation to the conductivity is exhibited, and several elastic constants are estimated. We also see that the complete phonon spectrum and the lifetimes of the phonon states may be calculated from this Green's function. A relation between the long wavelength acoustic attenuation in metals and the de conductivity is derived, which is in good agreement with recent experiments. Furthermore, the ions in a metal are shown to have a high-frequency oscillation along with the electrons, at essentially the electron plasma frequency.     

A generating functional approach to the Hubbard model

The method of generating functional, suggested for conventional systems by Kadanoff and Baym, is generalized to the case of strongly correlated systems, described by the Hubbard X operators. The method has been applied to the Hubbard model with arbitrary value U of the Coulomb on-site interaction. For the electronic Green’s function constructed for Fermi-like X operators, an equation using variational derivatives with respect to the fluctuating fields has been derived and its multiplicative form has been determined. The Green’s function is characterized by two quantities: the self energy Σ and the terminal part Λ. For them we have derived the equation using variational derivatives, whose iterations generate the perturbation theory near the atomic limit. Corrections for the electronic self-energy Σ are calculated up to the second order with respect to the parameter W/U (W width of the band), and a mean field type approximation was formulated, including both charge and spin static fluctuations. This approximation is actually equivalent to the one used in the method of Composite Operators, and it describes an insulator-metal phase transition at half filling reasonably well. The equations for the Bose-like Green’s functions have been derived, describing the collective modes: the magnons and doublons. The main term in this equation represents variational derivatives of the electronic Green’s function with respect to the corresponding fluctuating fields. The properties of the poles of the doublon Green’s functions depend on electronic filling. The investigation of the special case n=1 demonstrates that the doublon Green’s function has a soft mode at the wave vector Q=(π,π,...), indicating possible instability of the uniform paramagnetic phase relatively to the two sublattices charge ordering. However this instability should compete with an instability to antiferromagnetic ordering. The generating functional method with the X operators could be extended to the other models of strongly correlated systems.     

Generating functional approach to the Green's functions diagrammatic technique for spin and pseudospin lattice systems: I. General theory

The theory of the irreducible many-point Green's functions, describing spin and pseudospin lattice systems, is formulated with the help of the generating functional approach. The diagrammatic technique for the generating functional is also developed. Special attention is paid to the construction and summation of the diagrammatic series for the one- and two-point Green's functions. Closed formulae for the one-point Green's function and the generalized Vaks-Larkin- Pikin equation are obtained. The $\text{1}\text{z}$ expansion scheme near the critical temperature of the order-disorder phase transition, is discussed, where z denotes the effective number of nearest- neighbours for a given site in a crystal lattice.     

Effect of impurities scattering potential on NMR relaxation rate in impure d-wave superconductors

The purpose of our research is to study the nuclear spin lattice relaxation rate of impure d-wave superconductors. We use the Green’s function method to derive the approximation equation of density of states including the impurity scattering potential. We can get the analytic equation of the nuclear spin lattice relaxation rate that contained the impurity scattering potential in case of weak scattering potential and strong scattering potential in the simple form as the power series of Δ(T) and T. The numerical calculations show that there is coherence peak in the weak impurity scattering potential but there is no peak in the strong impurity scattering potential.     

A cluster plus effective medium tight-binding study of SiOx systems

The valence band density of states of SiO_x amorphous systems is calculated using a tight-binding treatment. The Green's function method is applied to a small cluster embeddied in a suitable effective medium. An average of this Green's function is then performed assuming a statistical distribution of composition for the small clusters. The effective medium is treated as a Bethe lattice of the correct composition. The results are completely consistent with experiment if the Si O Si angle is assumed to decrease with x.     

Quantum interference in carbon nanotube intramolecular junction

The quantum conductance oscillations (QCOs) of the intramolecular junction (IMJ) composed of two single-wall carbon nanotubes (SWNTs) have been studied by using a π-orbital only tight-binding (TB) model and a Green’s function technique. It is found that in the IMJs in addition to the rapid QCO frequencies corresponding to the constituent tubes there exist also their sum frequencies. The slow QCO frequencies of the IMJ will be different from those of its corresponding two perfect tubes if they have different chiral angles.     

Green's function formalism and quantum correction for finite-temperature baryon matter

A relativistic temperature Green's function formalism is used to find the one-loop quantum correction for infinite baryon matter with scalar and vector interactions. Effects on the equation of state and the baryon effective mass are analyzed.     

Comparison of the molecular cluster model with the phonon model for Jahn—Teller active impurities in crystals

For comparing the molecular cluster model with the phonon model in describing the Jahn-Teller interaction for magnetic impurities in crystals, the Jahn-Teller energy and the Ham reduction factors are expressed in terms of the phonon Green's function of the host crystal for an orbital triplet coupled to E_g modes of vibrations. Numerical results for the case of MgO lattice have been obtained using the phonon Green's function derived from the breathing shell model.     

Phase diagrams of a ferroelectric superlattice: A Green’s function technique study

A Fermi-type Green’s function method has been used to investigate the phase transition properties of a ferroelectric superlattice with two alternating materials on the basis of the transverse Ising model. By performing a higher-order decoupling to the equations of motion for the Green’s functions, the eigenfrequencies of the infinite ferroelectric superlattice are obtained. Moreover, we discuss the dependence of the phase diagrams on the interface coupling strength, the transverse field, and the thicknesses of two slabs. The comparison between the Green’s function technique and the usual mean-field approximation is illustrated.     

Statistical Theory of the Phonon Drag Force Acting on a Conduction Electron

Within the framework of microscopic fluctuation-dissipation theory, we obtain the stochastic equation describing the Brownian motion of an electron in the phonon field of the crystal lattice. An expression for the Green’s function of the phonon field is found in general form and for the case of linear phonon-variable interaction of an electron with the phonon field with allowance for the potential screening of crystal-lattice nuclei. An expression for the phonon drag acting on a conduction electron in the lattice field is found and analyzed with allowance for the interaction. Frequency dependence of the coefficient of the phonon drag acting on a conduction electron is studied and the contribution of the electron-phonon interaction to the effective mass of a charge carrier is determined.__________Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 48, No. 3, pp. 249–268, March 2005.     

Elastic continuum theory of bulk and surface-atom mean square displacements of hexagonal crystals

The mean square displacements of particles in a semi-infinite hexagonal elastic continuum with a free (0001) surface are calculated using a Green's function method in the high temperature limit.     

Study of Anderson localization by a continued-fraction method

The electron localization is studied for Anderson's tight-binding model of a disordered two-dimensional square lattice. For a large system of 10⁴ sites the averaged squared modulus $\text{| G}{}_{00}\text{|}{}^2$ of the Green's function is evaluated by a continued-fraction method. From this quantity, following Anderson's criterion, the energy of the mobility edge is found as a function of the degree of disorder. Also the Anderson transition is recognized.     

On the infinite summation in the theory of atomic multiphoton ionization

The infinite summation through the complete set of real unperturbed atomic states that appears in the Nth-order time-dependent perturbation theory when applied to multiphoton ionization is calculated by using the Green's function method. The cross-sections so obtained are in good agreement with other theoretical values published elsewhere.     

CPA theory for the giant Hall effect in disordered nanoscale systems

We present a theory based on the Green’s-function formalism to study the so-called giant Hall effect in disordered nanoscale systems in which the metallic sites build up percolating networks. The disordered Green’s functions are solved in the coherent potential approximation (CPA). This method is an extension of the Blackman-Esterling-Beck approach. The crucial point is that we are able to predict the quantum percolation threshold near which the Hall coefficient is enhanced by several orders of magnitude.     

Theory of multiphonon absorption in the wings of internal vibrational modes of impurities in ionic crystals

Green's function methods are applied to obtain the multiphonon absorption coefficient in crystals containing impurities with internal vibrational modes. The results imply that the shape of the wing absorption from such modes is determined by the host lattice density of states, but with a fall-off rate determined by the range of the lattice-impurity interaction potential.