Magnetic and electric phase transitions in the Hubbard model

Within the Hubbard model, two boson Green’s functions that describe the propagation of collective excitations of the electronic system—magnons (states with a single electron spin flip) and doublons (states with two electrons at one site of the crystal lattice)—are calculated for a Coulomb interaction of arbitrary strength and for an arbitrary electron concentration by applying a decoupling procedure to the double-time X-operator Green’s functions. It is found that the magnon and doublon Green’s functions are similar in structure and there is a close analogy between them. Instability of the paramagnetic phase with respect to spin ordering is investigated using the magnon Green’s function, and instability of the metallic phase to charge ordering is analyzed with the help of the doublon Green’s function. Criteria for the paramagnet-ferromagnet and metal-insulator phase transitions are found.     

The Continual Approach in the Yang-Mills Theory in the External Non-Abelian Field

The theory of Yang-Mills field in interaction with matter fields is considered in the presence of external gauge field. A closed expression for the generating functional of the Green functions is obtained, and a detailed analysis of the Green functions of the scalar, spinor, ghost and Yang-Mills fields is performed. The path-integral solution for all these Green functions is obtained, which includes the functional averaging over the classical trajectories in the space of commuting and anticommuting variables, the latter being anociated with the particle spin and isospin. For illustration an arbitrary Abelian-like external field is considered, as well as non-Abelian-like constant external field.     

General form of the full electromagnetic Green function in materials physics

In this article, we present the general form of the full electromagnetic Green function which is suitable for the application in bulk materials physics. In particular, we show how the seven adjustable parameter functions of the free Green function translate into seven corresponding parameter functions of the full Green function. Furthermore, for both the fundamental response tensor and the electromagnetic Green function, we discuss the reduction of the Dyson equation on the four-dimensional Minkowski space to an equivalent, three-dimensional Cartesian Dyson equation.     

Application of the Green's function method to ferrimagnetism

The Green function theory, which has been used hitherto for ferro- and antiferromagnetism, has been extended for the case of a two-sublattice ferrimagnet. Two parametrized Green functions are used corresponding to the two sublattices. The equations of motion are set up and the higher order Green functions are decoupled according toCallen's approximation. The functions are obtained by solving the simultaneous equations. The quasi-particle energies are evaluated from the singularities of the Green functions and the magnetization at low temperatures is found to obey theT ^3/2-law. These results obtained for the energy and magnetization agree well with those obtained by the conventional spin wave method.     

A renormalization procedure for the correlation function theory of finite Fermi systems

The quasiparticle renormalization of symmetrized correlation functions is treated in the framework of the double-time Green function theory of many body systems. The work is based on the Mori-theory of response functions transcribed for symmetrized correlation functions. For the specific example of finite Fermi systems it is shown that the physical situation assumed in the quasiparticle-quasihole renormalization of the many-time Green function theory allows to define an equivalent renormalization procedure for correlation functions. This procedure uses projection operator techniques and is therefore of purely algebraic nature.     

Integrals of the motion,green functions,and coherent states of dynamical systems

The connection between the integrals of the motion of a quantum system and its Green function is established. The Green function is shown to be the eigenfunction of the integrals of the motion which describe initial points of the system trajectory in the phase space of average coordinates and moments. The explicit expressions for the Green functions of theN-dimensional system with the Hamiltonian which is the most general quadratic form of coordinates and momenta with time-dependent coefficients is obtained in coordinate, momentum, and coherent states representations. The Green functions of the nonstationary singular oscillator and of the stationary Schrödinger equation are also obtained.     

Conserving approximations for response functions of the Fermi gas in a random potential

One- and two-electron Green functions are simultaneously needed to determine the responsefunctions of the electron gas in a random potential. Reliable approximations must retainconsistency between the two types of Green functions expressed via Ward identities so thattheir output is compliant with macroscopic symmetries and conservation laws. Such aconsistency is not directly guaranteed when summing nonlocal corrections to the local(dynamical) mean field. We analyze the reasons for this failure and show how the full Wardidentity can generically be implemented in the diagrammatic approach to the vertexfunctions without breaking the analytic properties of the self-energy. We use thelow-energy asymptotics of the conserving two-particle vertex determining the singular partof response and correlation functions to derive an exact representation of the diffusionconstant in terms of Green functions of the perturbation theory. We then calculateexplicitly the leading vertex corrections to the mean-field diffusion constant due tomaximally-crossed diagrams.     

Nonperturbative Time-Independent Green Function of Matrix Schrödinger Equation. General Formalism and Quasiclassical Representation

A Green function of time-independent multichannel Schrödinger equation is considered in matrix representation beyond a perturbation theory. Nonperturbative Green functions are obtained through the regular in zero and at infinity solutions of the multichannel Schrödinger equation for different cases of symmetry of the full Hamiltonian. The spectral expansions for the nonperturbative Green functions are obtained in simple form through multichannel wave functions. The developed approach is applied to obtain simple analytic equations for the Green functions and transition matrix elements for compound multipotential system within quasiclassical approximation. The limits of strong and weak interchannel interactions are studied.Alexander I. Pegarkov:On leave from Physics Faculty     

Calculation of the Green function for the Laplace equation

A modification of the traditional method of calculating the Green function in a layered medium is suggested that allows one to substantially increase its accuracy. In addition, a technique for calculating the Green function that makes it possible to control the accuracy of the calculated potential is offered. This technique is based on the properties of the Bessel and Struve functions. An example of calculation using the suggested technique is illustrated. The results may be extended for a wide class of problems the solution to which requires calculation of the Green function for the Laplace equation in a layered medium.     

Regularization of the quantum field theory of charges and monopoles

A gauge-invariant regularization procedure for quantum field theories of electric and magnetic charges based on Zwanziger's local formulation is proposed. The bare regularized full Green functions of gauge-invariant operators are shown to be Lorentz invariant. This would have as a consequence the Lorentz invariance of the finite Green functions that might result after any reasonable subtraction, if such a subtraction can be found.     

Gauge invariant quark Green’s functions with polygonal Wilson lines

Properties of gauge invariant two-point quark Green’s functions, defined with polygonal Wilson lines, are studied. The Green’s functions can be classified according to the number of straight line segments their polygonal lines contain. Functional relations are established between the Green’s functions with different numbers of segments on the polygonal lines. An integrodifferential equation is obtained for the Green’s function with one straight line segment, in which the kernels are represented by a series of Wilson loop vacuum averages along polygonal contours with an increasing number of segments and functional derivatives on them. The equation is exactly solved in the case of two-dimensional QCD in the large-N _c limit. The spectral properties of the Green’s function are displayed.     

Solutions of the Maxwell equations and photon wave functions

Properties of six-component electromagnetic field solutions of a matrix form of the Maxwell equations, analogous to the four-component solutions of the Dirac equation, are described. It is shown that the six-component equation, including sources, is invariant under Lorentz transformations. Complete sets of eigenfunctions of the Hamiltonian for the electromagnetic fields, which may be interpreted as photon wave functions, are given both for plane waves and for angular-momentum eigenstates. Rotationally invariant projection operators are used to identify transverse or longitudinal electric and magnetic fields. For plane waves, the velocity transformed transverse wave functions are also transverse, and the velocity transformed longitudinal wave functions include both longitudinal and transverse components. A suitable sum over these eigenfunctions provides a Green function for the matrix Maxwell equation, which can be expressed in the same covariant form as the Green function for the Dirac equation. Radiation from a dipole source and from a Dirac atomic transition current are calculated to illustrate applications of the Maxwell Green function.     

Effective integral equation and vertex operator of the green function

A simple and rigorous formulation of effective integral equations for the Green functions is presented and a general formula for the effective vertex operator (or function) has been derived. As an illustration, this formalism has been applied to obtain (i) the effective Dyson equation and effective mass operator for the single-particle Green function, and (ii) the effective integral equation and effective interaction operator for the two-particle Green function as well as those for the particle-hole Green function.     

Generalized Green Functions and Current Correlations in the TASEP

We study correlation functions of the totally asymmetric simple exclusion process (TASEP) in discrete time with backward sequential update. We prove a determinantal formula for the generalized Green function which describes transitions between positions of particles at different individual time moments. In particular, the generalized Green function defines a probability measure at staircase lines on the space-time plane. The marginals of this measure are the TASEP correlation functions in the space-time region not covered by the standard Green function approach. As an example, we calculate the current correlation function that is the joint probability distribution of times taken by selected particles to travel given distance. An asymptotic analysis shows that current fluctuations converge to the Airy₂ process.     

Green Functions for the Wrong-Sign Quartic

It has been shown that the Schwinger-Dyson equations for non-Hermitian theories implicitly include the Hilbert-space metric. Approximate Green functions for such theories may thus be obtained, without having to evaluate the metric explicitly, by truncation of the equations. Such a calculation has recently been carried out for various PT-symmetric theories, in both quantum mechanics and quantum field theory, including the wrong-sign quartic oscillator. For this particular theory the metric is known in closed form, making possible an independent check of these approximate results. We do so by numerically evaluating the ground-state wave-function for the equivalent Hermitian Hamiltonian and using this wave-function, in conjunction with the metric operator, to calculate the one- and two-point Green functions. We find that the Green functions evaluated by lowest-order truncation of the Schwinger-Dyson equations are already accurate at the 6% level. This provides a strong justification for the method and a motivation for its extension to higher order and to higher dimensions, where the calculation of the metric is extremely difficult.     

Exploring correlations in the stochastic Yang–Mills vacuum

The correlation lengths of nonperturbative-nonconfining and confining stochastic background Yang–Mills fields are obtained by means of a direct analytic path-integral evaluation of the Green functions of the so-called one- and two-gluon gluelumps. Numerically, these lengths turn out to be in a good agreement with those known from the earlier, Hamiltonian, treatment of such Green functions. It is also demonstrated that the correlation function of nonperturbative-nonconfining fields decreases with the deviation of the path in this correlation function from the straight-line one.     

Green functions of the anisotropic Heisenberg model

The Green functions of the anisotropic Heisenberg model are studied by a method which was applied previously to the reduced density matrices. Integral equations are used to prove the existence of the infinite volume limit of the Green functions, and some analyticity properties with respect to the fugacity (or magnetic field), the potentials, and the complex times.Research supported by the National Science Foundation.     

Green functions of vortex operators

We study the euclidean Green functions of the 't Hooft vortex operator, primarily for abelian gauge theories. The operator is written in terms of elementary fields, with emphasis on a form in which it appears as the exponential of a surface integral. We explore the requirement that the Green functions depend only on the boundary of this surface. The Dirac veto problem appears in a new guise. We present a two-dimensional “solvable model” of a Dirac string, which suggests a new solution of the veto problem. The renormalization of the Green functions of the abelian Wilson loop and abelian vortex operator is studied with the aid of the operator product expansion. In each case, an overall multiplication of the operator makes all Green functions finite; a surprising cancellation of divergences occurs with the vortex operator. We present a brief discussion of the relation between the nature of the vacuum and the cluster properties of the Green functions of the Wilson and vortex operators, for a general gauge theory. The surface-like cluster property of the vortex operator in an abelian Higgs theory is explored in more detail.     

Green functions of scalar particles in stochastic fields

A variational method of evaluating functional integrals is proposed. This method is used to investigate the asymptotic behavior of the scalar-particle Green functions in stochastic fields. The equations for the Green functions in Euclidean space in stochastic fields are written. The solutions of these equations are represented in the form of a functional integral and then they are averaged over Gaussian stochastic fields. The variational method formulated above is used to evaluate the asymptotic behavior of these Green functions. The following equations are considered in this paper: a stochastic contribution to the mass of a scalar particle, a gauge stochastic field, and a weak stochastic contribution to the flat metric of Euclidean space.     

Error control in the perfectly matched layer based multilevel fast multipole algorithm

The development of efficient algorithms to analyze complex electromagnetic structures is of topical interest. Application of these algorithms in commercial solvers requires rigorous error controllability. In this paper we focus on the perfectly matched layer based multilevel fast multipole algorithm (PML-MLFMA), a dedicated technique constructed to efficiently analyze large planar structures. More specifically the crux of the algorithm, viz. the pertinent layered medium Green functions, is under investigation. Therefore, particular attention is paid to the plane wave decomposition for 2-D homogeneous space Green functions in very lossy media, as needed in the PML-MLFMA. The result of the investigations is twofold. First, upper bounds expressing the required number of samples in the plane wave decomposition as a function of a preset accuracy are rigorously derived. These formulas can be used in 2-D homogeneous (lossy) media MLFMAs. Second, a more heuristic approach to control the error of the PML-MLFMA’s Green functions is presented. The theory is verified by means of several numerical experiments.