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.     

A comment on functional integrals and many-body systems

Green's functions for systems of non-interacting particles at T=0 are obtained through ab initio calculations using functional integral algorithms. It is shown in detail how the correct Green's functions are computed and how one recovers the structure of the ground state (which in the usual derivations is the starting point).     

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.     

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.     

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.     

Application of conventional equation of motion methods to the Tomonaga model

It is shown that the equation of motion technique provides a very concise way of calculating Green's functions for the Tomonaga-Luttinger model of a 1-d electron gas. The spectral function of the single-particle Green's function is worked out for the most general version of this model and for finite temperature. Extensions of the model are briefly discussed.     

A microscopic theory of the lambda transition,II

The previously proposed finite temperature field theory of the lambda transition based on the Schwinger functional method is investigated further. A systematic method for calculating the higher-order loop terms is presented by introducing the one-loop Green's functions, which are found to be a natural finite temperature extension of the Beliaev-Hugenholtz-Pines-Gavoret-Nozières zero-temperature Green's functions. The application of the finite temperature loop expansion to the dynamical properties is presented by calculating the retarded density correlation functions at the one-loop level. The result gives a microscopic basis for the form of the dynamical structure factor recently proposed by Woods and Svensson. From a general point of view, without using any approximations or model interactions, Goldstone's theorem for the lambda transition at finite temperature is presented.     

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.     

Generalized Wilson expansions in dimensional renormalization

Operator product expansions in the framework of dimensional regularization and renormalization are discussed. Following the definition of a subtraction operator for dimensionally regularized and points-split Green's functions, a generalized Wilson expansion pansion is proved. The terms of the expansion are normal products defined via dimensional renormalization, and the coefficients are doubly regularized with singularities in the physical dimension as the spacial separations of the product fields vanish, or at zero separations as the dimension of space-time becomes physical.     

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 $\text{1}\text{β}$.     

Geometric dependence of casimir energy and scalar green's functions

It is well known that the Casimir Energy depends on the geometry of the conducting cavity. In this note, scalar Green's functions are used to determine the Casimir energy's dependence on terms bilinear in the extrinsic curvatures of the cavity's surface, and thus to resolve the controversy over the Casimir energy's finiteness.     

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.     

On the energy-momentum tensor in gauge field theories

The energy-momentum tensor in spontaneously broken non-Abelian gauge field theories is studied. The motivation is to show that recent results on the finiteness and gauge independence of S-matrix elements in gauge theories extends to observable amplitudes for transitions in a gravitational field. Path integral methods and dimensional regularization are used throughout. Green's functions Γ_μν^(j)(q; p₁,…,p_j) involving the energy-momentum tensor and j particle fields are proved finite to all orders in perturbation theory to zero and first order in q, and finite to one loop order for general q. Amputated Green's functions of the energy momentum tensor are proved to be gauge independent on mass shell.     

Evaluation of the spin wave Green's functions for rutile structure Heisenberg antiferromagnets with exchange between ions both on the same and opposite sublattices

A straightforward method is presented for the evaluation of the spin wave Green's function appropriate to the Raman scattering in the xy and xz geometry in rutile structure Heisenberg antiferromagnets with exchange between ions both on the same and on opposite sublattices and arbitrary local anisotropy. The analytical asymptotic behaviour of the Green's functions near the singularities is explicitly given and the problem of the numerical evaluation of their real and imaginary parts is discussed. Tables of the imaginary parts, calculated at the points corresponding to a Gaussian quadrature procedure in the appropriate interval, are supplied on request.     

A note on Keldysh's perturbation formalism

The expansion theorem of quantum field theoy relating Heisenberg operators to asymptotic free-field operators is rewritten by means of the time-path technique, originally due to Schwinger, which to date has only found application in statistical mechanics. The theorem is combined with Bogoliubov's initial condition of vanishing correlations in the infinite past to rederive Keldysh's perturbation scheme for non-equilibrium statistical Green's functions.     

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.     

New many body perturbation method and the Hubbard model

By adapting the functional derivative method developed by Kadanoff and Baym to the Hubbard model, a new perturbation method is formulated. The unperturbed state is defined by the two equations which yield Hubbard's results, while the remainder is given by functional derivatives of the Green's functions which are shown to generate a complete perturbation series. Advantages of this method are discussed.     

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).     

Wave propagation according to higher order field equations I

A detailed study is made of wave propagation according to a sixth-order partial differential equation with complex masses proposed by Swieca and Marques, which presents a kind of generalized Klein-Gordon equation. The choice of definite Green's functions in the corresponding Yang-Feldman integral equation corresponds to a certain choice of boundary conditions for the allowed solutions of the corresponding partial differential equation. The advanced and retarded Green's functions used possess the anomalous feature of having non-zero values in the neighbourhoods of those, past or future parts of the light cone, for which traditional advanced and retarded Green's functions are zero. However, it is shown that a suitable averaging procedure provides the possibility of defining sets of functions, such that solutions of the Yang-Feldman equations belonging to this set possess the property that the future behaviour of the solution is determined by its asymptotic initial conditions. Certain features of the wave propagation, according to the equations considered, can be usefully compared with the properties of the solutions of the ordinary differential equation - and corresponding integral equation - which represents the equation of motion of a charged particle including the force for radiation reaction. The particle then has a certain “size”. Analogously the “non-local field equations” have solutions characterized by a certain “fundamental length” indicating the space-time distances for which averaging occurs. The admitted solutions of the field equations seem to represent a relativistic field with a “finite a number of degrees of freedom” within a finite volume.     

Dynamical pictures and causality in finite temperature Green's functions

Real-time finite temperature Green's functions are discussed on the basis of the definition of new dynamical pictures. Causality appears explicitly. Feynman diagrams are formally identical to the ones of zero temperature. The vanishing of disconnected diagrams follows naturally.