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.     

Diagram analysis of the Hubbard model: Stationarity property of the thermodynamic potential

The diagram approach proposed many years ago for the strongly correlated Hubbard model is developed with the aim to analyze the thermodynamic potential properties. A new exact relation between renormalized quantities such as the thermodynamic potential, the one-particle propagator, and the correlation function is established. This relation contains an additional integration of the one-particle propagator with respect to an auxiliary constant. The vacuum skeleton diagrams constructed from the irreducible Green’s functions and tunneling propagator lines are determined and a special functional is introduced. The properties of this functional are investigated and its relation to the thermodynamic potential is established. The stationarity property of this functional with respect to first-order variations of the correlation function is demonstrated; as a consequence, the stationarity property of the thermodynamic potential is proved.     

The classical limit of temperature Green's function expansions

Rules are obtained for calculating the classical limit of Green's function diagrammatic expansions. The classical cluster expansion is derived by calculating the classical limit of the exact Green's function. Other operators of interest in linear response theory may be calculated in the classical limit. The retarded real-time spin density correlation function, proportional to the magnetic susceptibility, is shown to be exactly proportional to the density in this limit. The relation of this work to other approaches is discussed.     

A green's function analysis of atomic and molecular (e, 2e) reactions

Investigation of Atomic and molecular (e, 2e) spectra will be discussed in terms of a Green's function approach. The energy, intensity and momentum distribution of energy levels observed by electron coincidence ionization spectroscopy, are directly related to the poles, pole strengths and generalized overlap amplitude of the one particle propagator or Green's function. The theoretical calculation of these observable quantities via the Green's function technique will be discussed. In particular, the position and intensity of satellite (or “shakeup”) lines, relative to the main lines, will be analysed in some detail.     

Exact solutions for spectra and Green's functions in random one-dimensional systems

For chains of harmonic oscillators with random masses a set of equations is derived, which determine the spatial Fourier components of the average one-particle Green's function. These equations are valid for complex values of the frequency. A relation between the spectral density and functions introduced by Schmidt is discussed. Exact solutions for this Green's function and the less complicated characteristics function-the analytic continuation into the complex frequency plane of the accumulated spectral density and the inverse localization length of the eigenfunctions-are derived for exponential distributions of the masses. For some cases the characteristic function is calculated numerically. For gamma distributions the equations are cast in the form of ordinary, higher order differential equations; these have been solved numerically for determining the characteristic function. For arbitrary mass distributions a cumulant expansion and a peculiar symmetry of the Green's function are discussed.The method is also applied to chains where the spring constants and/or the masses have random values. Also for these systems exact solutions are discussed; for exponential distributions, e.g., of both masses and spring constants the characteristic function is expressed in Bessel functions. The relation with certain random relaxation models is shown. Finally, X-Y Hamiltonians with random exchange constants and/or magnetic fields-or, equivalently, tight-binding electron models with diagonal and/or off-diagonal disorder-are considered. Here the Green's function does not depend on the wave number if the distribution of exchange constants is symmetric around the origin. New solutions for the characteristic function and Green's function are derived for a number of cases, including exponentially distributed magnetic fields and power law distributed exchange constants.     

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.     

Time-dependent radiation transport in a one-dimensional medium

We present an analytic solution of the time-dependent radiation transport problem in a one-dimensional, stationary and homogeneous medium of finite thickness. The solution is found by the method of images and is compared with an eigenfunction expansion. Previous conjectures about the structure of such an expansion are clarified. We also expand the Green's function of this problem in scattering orders.     

Biorthogonality and radiative transfer in finite slab atmospheres

It is shown that the exact solution of transfer problems of polarized light in finite slab atmospheres can be obtained from an eigenmode expansion, if there is a known set of adjoints defined appropriately to treat two-point, half-range boundary-value problems. The adjoints must obey a half-range biorthogonality relation.The adjoints are obtained in terms of Case's eigenvectors and the reflection or the transmission matrices. Half-range characteristic equations for the eigenvectors and their adjoints are derived, where the kernel functions of the integral operators are given by the boundary values of the source function matrix of the slab albedo problem. Spectral formulae are obtained for the surface Green's functions. A relationship is noted between the biorthogonality concept and some half-range forms of the transfer equation for the surface Green's functions and their adjoints. Linear and non-linear functional equations that are well known from an invariance approach, are derived from a new point of view. The biorthogonality concept offers the opportunity for a better understanding of mathematical structures and the nonuniqueness problem for solutions of such functional equations.     

Green's function equations of motion for a many-fermion system: II. Inhomogeneous system at finite temperature

The equations of motion for many-time causal Green's functions are extended to an inhomogeneous many-fermion system at finite temperature. The boundary condition that the perturbation vanishes in the remote past and distant future (adiabatic hypothesis) is used to determine the unperturbed propagator. The temperature enters the theory only as a parameter. Thus there is no need for analytic continuations in the complex temperature-time plane. The theory is used to derive thermal Hartree-Fock theory and Wick's theorem at finite temperature. A linked cluster perturbation expansion at finite temperature is obtained by iterating the equations of motion, without unlinked disconnected diagrams even appearing. After integration over frequency, the present theory gives the perturbation theory rules in terms of global propagators that Baym and Sessler obtained from the imaginary-time theory.     

Generalized density expansion in the kinetic theory and the resistivity of liquid metals

For the system of electrons and immovable interacting centers an exact equation for averaged electron Green's function is formulated. The expansion of self-energy part over the one-particle t-matrices and explicit Green's functions is derived. It represents a kind of a generalized density series containing the correlation functions of the centres. In the low approximation over t-matrix, the transition probability (t)²S in the kinetic equation is obtained (S = the structure factor of centers).     

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.     

The intrinsical character of the electronic correlation in an electron gas

By introducing the phase transformation of electron operators, we map the equation of motion of an one-particle Green's function into that of a non-interacting one-particle Green's function where the electrons are moving in a time-depending scalar potential and pure gauge fields for a D-dimensional electron gas, and we demonstrate that the electronic correlation strength strongly depends upon the excitation energy spectrum and collective excitation modes of electrons. It naturally explains that the electronic correlation strength is strong in the one dimension, while it is weak in the three dimensions.     

Solution of the path integral for the H-atom

The Green's function of the H-atom is calculated by a simple reduction of Feynman's path integral to gaussian form.     

On the electronic energy spectrum of linear models for disordered systems

In this paper the basic relations for describing the electronic structure of linear models for monatomic and diatomic disordered systems are derived using a F.E.N. theoretical approach. Elegant three terms recurrence formulae between the wavefunctions at three successive atomic sites are established and their relation is discussed with recurrence formulae obtained previously by means of Green's function techniques.     

Quantized Dirac field in curved Riemann-Cartan background. I. Symmetry properties,Green's function

In the present series of papers, we study the properties of quantized Dirac field in curved Riemann-Cartan space, with particular attention on the role played by torsion. In this paper, we give, in the spirit of the original work of Weyl, a systematic presentation of Dirac's theory in curved Riemann-Cartan space. We discuss symmetry properties of the system, and derive conservation laws as direct consequences of these symmetries. Also discussed is conformal gauge symmetry, with torsion effectively playing the role of a conformal gauge field. To obtain short-distance behavior, we calculate the spinor Green's function, in curved Riemann-Cartan background, using the Schwinger-DeWitt method of proper-time expansion. The calculation corresponds to a generalization of DeWitt's calculation for a Riemannian background.     

A class of interface state models. I

The Green's function matching procedure of Garcia-Moliner and Rubio is applied to a class of one and three dimensional band models, based on separable Pseudopotentials, for which the Green's functions can be obtained in analytic form. Surface and interface states are obtained corresponding to the [100] and [110] surfaces for a simple cubic, single gap case.     

Linked cluster expansions in the equations of motion method II: Kinetic equations for the single-particle density matrix

The general method of paper I of this series is applied to derive kinetic equations (KE's), i.e. closed exact equations governing the time evolution of the single-particle density matrix. The short-memory approximation of these non-Markowian equations is formulated in such a way that it is valid even in strongly inhomogeneous systems. The c-number diagram expansion of the integral kernels of the KE's is obtained from the general rules of paper I. It is shown that certain secular divergent terms cancel each other. The diagrams decay into dynamic and correlational parts, the latter being given by cluster functions describing the correlations of the particles in the local equilibrium ensemble σ(t) which is formulated in terms of the single-particle density matrix and of the Hamiltonian. The appearance of the cluster functions is the most pronounced difference of our KE's in comparison with other KE's which are formulated in terms of the dynamics of isolated clusters of particles. It is argued that our KE's may be viewed as a highly summed version of these latter KE's and that the ultimate reason for this difference lies in the fact that in our theory the conservation of the average macroscopic energy is taken into account explicitly.     

A self-consistent method for treating point impurities with arbitrary range potentials

We propose a generalized Green's function technique for treating impurities whose potential has an arbitrary short, intermediate, or long-range nature. The approach exploits the recursion method, and can be formulated in a self-consistent way for point impurities.     

Exact and quasiclassical Green's functions of two-dimensional electron gas with Rashba–Dresselhaus spin–orbit interaction in parallel magnetic field

New exact and asymptotical results for the one particle Green's function of 2D electrons with combined Rashba–Dresselhaus spin–orbit interaction in the presence of in-plane uniform magnetic field are presented. A special case that allows an exact analytical solution is also highlighted. To demonstrate the advantages of our approach we apply the obtained Green's function to calculation of electron density and magnetization.