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.     

Calculation of the electroelastic Green’s function of the hexagonal infinite medium

The electroelastic 4 × 4 Green’s function of a piezoelectric hexagonal (transversely isotropic) infinitely extended medium is calculated explicitly in closed compact form ((73) ff. and (88) ff., respectively) by using residue calculation. The results can also be derived from Fredholm’s method [2]. In the case of vanishing piezoelectric coupling the derived Green’s function coincides with two well known results: Kröner’s expressions for the elastic Green’s function tensor [4] is reproduced and the electric part then coincides with the electric potential (solution of Poisson equation) which is caused by a unit point charge. The obtained electroelastic Green’s function is useful for the calculation of the electroelastic Eshelby tensor [16].     

Development of the self-consistent approximation and its application to the problem of magnetoelastic resonance in an inhomogeneous medium

Our previously proposed approximation involving both the first and second terms of the expansion of the vertex function is generalized to the system of two interacting wavefields of different physical nature. A system of self-consistent equations for the matrix Green’s function and matrix vertex function is derived. On the basis of this matrix generalization of the new self-consistent approximation, a theory of magnetoelastic resonance is developed for a ferromagnetic model, where the magnetoelastic coupling parameter ε(x) is inhomogeneous. Equations for magnetoelastic resonance are analyzed for one-dimensional inhomogeneities of the coupling parameter. The diagonal and off-diagonal elements of the matrix Green’s function of the system of coupled spin and elastic waves are calculated with the change in the ratio between the average value ε and rms fluctuation Δε of the coupling parameter between waves from the homogeneous case (ε ≠ 0, Δε = 0) to the extremely randomized case (ε = 0, Δε ≠ 0) at various correlation wavenumbers of inhomogeneities k _c. For the limiting case of infinite correlation radius (k _c = 0), in addition to approximate expressions, exact analytical expressions corresponding to the summation of all diagrams of elements of the matrix Green’s function are obtained. The results calculated for an arbitrary k _c value in the new self-consistent approximation are compared to the results obtained in the standard self-consistent approximation, where only the first term of the expansion of the vertex function is taken into account. It is shown that the new approximation corrects disadvantages of the Green’s functions calculated in the standard approximation such as the dome shape of resonances and bends on the sides of resonance peaks. The appearance of a fine structure of the spectrum in the form of a narrow resonance on the Green’s function of spin waves and a narrow antiresonance on the Green’s function of elastic waves, which was previously predicted in the standard self-consistent approximation, is confirmed. With an increase in the parameter k _c, the Green’s functions calculated in the standard and new approximations approach each other and almost coincide with each other at k _c/k ≥ 0.5. At the same time, the results of this work indicate that the new self-consistent approximation has a certain advantage for studying the problems of stochastic radiophysics in media with long-wavelength inhomogeneities (small k _c values), because it describes both the shape and width of peaks much better than the standard approximation.     

Approximate path integral solution for a Dirac particle in a deformed Hulthén potential

The problem of a Dirac particle moving in a deformed Hulthén potential is solved in the framework of the path integral formalism. With the help of the Biedenharn transformation, the construction of a closed form for the Green’s function of the second-order Dirac equation is done by using a proper approximation to the centrifugal term and the Green’s function of the linear Dirac equation is calculated. The energy spectrum for the bound states is obtained from the poles of the Green’s function. A Dirac particle in the standard Hulthén potential (q = 1) and a Dirac hydrogen-like ion (q = 1 and a → ∞) are considered as particular cases.     

Green’s function calculations of light nuclei

The influence of short-range correlations in nuclei was investigated with realistic nuclear force. The nucleon-nucleon interaction was renormalized with V_lowk technique and applied to the Green’s function calculations. The Dyson equation was reformulated with algebraic diagrammatic constructions. We also analyzed the binding energy of ⁴He, calculated with chiral potential and CD-Bonn potential. The properties of Green’s function with realistic nuclear forces are also discussed.     

On the problem of analyzing the dynamic properties of a layered half-space

An efficient method is developed for constructing the Green matrix functions of a layered inhomogeneous half-space. Matrix formulas convenient for programming are proposed, which make it possible to study the properties of a multilayered half-space with high accuracy. As an example of the problem of oscillations of a three-layer half-space, transformation of the dispersion characteristics of a three-layered medium is shown as a function of the relations of the mechanical and geometric parameters of its components. Study of the properties of the Green’s function of a medium with a low-velocity layered inclusion showed that each mode of a surface wave exists in a limited frequency range: in addition to the critical frequency of mode occurrence, the frequency of its disappearance exists—a frequency above which the mode is suppressed because of superposition of the zero of the Green’s function on its pole. A similar study conducted for a medium with a high-velocity layered inclusion has shown that in addition to the cutoff frequency (the frequency at which a surface wave propagating in the low-frequency range disappears), there is the frequency of its recurrent generation—the upper boundary of the “cutoff range” of the first mode. Beyond this range, the first mode propagates, and also the other propagating modes can appear. The critical relation of the geometric parameters of the medium determining the existence and boundaries of the cutoff range of a wave is established.     

Top quark polarization in polarized e+e− annihilation near threshold

Top quark polarization in e⁺e^{t -} annihilation into tt? is calculated for linearly polarized beams. The Green function formalism is applied to this reaction near threshold. The Lippmann—Schwinger equations for the S-wave and P-wave Green functions are solved numerically for the QCD chromostatic potential given by the two-loop formula for large momentum transfer and Richardson’s ansatz for intermediate and small momenta. S- P— wave interference contributes to all components of the top quark polarization vector. Rescattering of the decay products is considered. The mean values 〈nl〉 of the charged lepton four-momentum projections on appropriately chosen directions n in semileptonic top decays are proposed as experimentally observable quantities sensitive to top quark polarization. The results for 〈nl〉 are obtained including S- P— wave interference and rescattering of the decay products. It is demonstrated that for the longitudinally polarized electron beam a highly polarized sample of top quarks can be produced.     

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.     

Explicit time-domain approaches based on numerical Green’s functions computed by finite differences – The ExGA family

The present paper describes a new family of time stepping methods to integrate dynamic equations of motion. The scalar wave equation is considered here; however, the method can be applied to time-domain analyses of other hyperbolic (e.g., elastodynamics) or parabolic (e.g., transient diffusion) problems. The algorithms presented require the knowledge of the Green’s function of mechanical systems in nodal coordinates. The finite difference method is used here to compute numerically the problem Green’s function; however, any other numerical method can be employed, e.g., finite elements, finite volumes, etc. The Green’s matrix and its time derivative are computed explicitly through the range [0, Δt] with either the fourth-order Runge–Kutta algorithm or the central difference scheme. In order to improve the stability of the algorithm based on central differences, an additional matrix called step response is also calculated. The new methods become more stable and accurate when a sub-stepping procedure is adopted to obtain the Green’s and step response matrices and their time derivatives at the end of the time step. Three numerical examples are presented to illustrate the high precision of the present approach.     

The use of low-frequency noise in passive tomography of the ocean

A possible design of the mode tomography of the ocean with the use of a scheme requiring no expensive low-frequency radiators is considered. The design is based on the widely discussed method of estimating the Green’s function from the cross-coherence function of noise field received in a great number of observation points. The relationship between the Green’s function and the noise coherence function is derived from the Helmholtz-Kirchhoff integral. The use of the vertical multielement arrays composed of vector receivers is suggested to decrease the duration of noise signal accumulation required for a reliable determination of the Green’s function. The solution of the tomographic problem is based on the determination of the mode structure of acoustic field from the eigenvectors and eigenvalues of the cross-coherence matrix of the received noise field.     

Fractional Green’s Function for the Time-Dependent Scattering Problem in the Space-Time-Fractional Quantum Mechanics

Integral form of the space-time-fractional Schrödinger equation for the scattering problem in the fractional quantum mechanics is studied in this paper. We define the fractional Green’s function for the space-time fractional Schrödinger equation and express it in terms of Fox’s H-function and in a computable series form. The asymptotic formula of the Green’s function for large argument is also obtained, and applied to study the fractional quantum scattering problem. We get the approximate scattering wave function with correction of every order.     

Faddeev random phase approximation applied to molecules

The Faddeev Random Phase Approximation (FRPA) is a Green’s function method which couples collective degrees of freedom to the single particle motion by resumming an infinite number of Feynman diagrams. The Faddeev technique is applied to describe the two-particle-one-hole (2p1h) and two-hole-one-particle (2h1p) Green’s function in terms of non-interacting propagators and kernels for the particle-particle (pp) and particle-hole (ph) interactions. This results in an equal treatment of the intermediary pp and ph channels. In FRPA both the pp and ph phonons are calculated on the random phase approximation (RPA) level. In this work the equations that lead to the FRPA eigenvalue problem are derived. The method is then applied to atoms, small molecules and the Hubbard model, for which the ground state energy and the ionization energies are calculated. Special attention is directed to the RPA instability in the dissociation limit of diatomic molecules and in the Hubbard model. Several solutions are proposed to overcome this problem.     

Analytical study of non-linear transport across a semiconductor-metal junction

In this paper we study analytically a one-dimensional model for a semiconductor-metal junction. We study the formation of Tamm states and how they evolve when the semi-infinite semiconductor and metal are coupled together. The non-linear current, as a function of the bias voltage, is studied using the non-equilibrium Green’s function method and the density matrix of the interface is given. The electronic occupation of the sites defining the interface has strong non-linearities as a function of the bias voltage due to strong resonances present in the Green’s functions of the junction sites. The surface Green’s function is computed analytically by solving a quadratic matrix equation, which does not require adding a small imaginary constant to the energy. The wave function for the surface states is given.     

A method for calculating above-threshold multiphoton processes in atoms: Polarizability of alkali-metal and noble gas atoms

A technique for calculating above-threshold bound-bound transitions of an atomic optical electron, based on the use of generalized Sturm expansion of the Green’s function for a potential with a Coulomb asymptotics has been developed within the model-potential method. The dynamic polarizabilities of alkali-metal and noble gas atoms are calculated in the frequency range above the ionization threshold.     

Quasiclassical formalism and the Andreev equation

In this article I shall make explicit the connection between the quasiclassical Green’s function and the Andreev quasiparticle energies and wavefunctions. The physical meaning of the components of the Green’s function is elucidated.     

Critical temperature of metallic hydrogen sulfide at 225-GPa pressure

The Eliashberg theory generalized for electron—phonon systems with a nonconstant density of electron states and with allowance made for the frequency behavior of the electron mass and chemical potential renormalizations is used to study T _c in the SH₃ phase of hydrogen sulfide under pressure. The phonon contribution to the anomalous electron Green’s function is considered. The pairing within the total width of the electron band and not only in a narrow layer near the Fermi surface is taken into account. The frequency and temperature dependences of the complex mass renormalization ReZ(ω), the density of states N(ε) renormalized by the electron—phonon interactions, and the electron—phonon spectral function obtained computationally are used to calculate the anomalous electron Green’s function. A generalized Eliashberg equation with a variable density of electron states has been solved. The frequency dependence of the real and imaginary parts of the order parameter in the SH₃ phase has been obtained. The value of T _c ≈ 177 K in the SH₃ phase of hydrogen sulfide at pressure P = 225 GPa has been determined by solving the system of Eliashberg equations.     

Elastic interaction of point defects in cubic and hexagonal crystals

The elastic interaction of two point defects in cubic and hexagonal structures has been considered. On the basis of the exact expression for the tensor Green’s function of the elastic field obtained by the Lifschitz–Rozentsveig for a hexagonal medium, an exact formula for the interaction energy of two point defects has been obtained. The solution is represented as a function of the angle of their relative position on the example of semiconductors such as III-nitrides and α-SiC. For the cubic medium, the solution is found on the basis of the Lifschitz–Rozentsveig Green’s tensors corrected by Ostapchuk, in the weak-anisotropy approximation. It is proven that the calculation of the interaction energy by the original Lifschitz–Rozentsveig Green’s tensor leads to the opposite sign of the energy. On the example of the silicon crystal, the approximate solution is compared with the numerical solution, which is represented as an approximation by a series of spherical harmonics. The range of applicability of the continual approach is estimated by the quantum mechanical calculation of the lattice Green’s function.     

The correspondence between Casimir pressure and energy in one-dimensional field models

We propose a method of calculation of Casimir pressure using the Green function for one-dimensional case. This method yields the renormalized pressure if an external field is absent, otherwise it permits us to calculate the dependence of pressure at one boundary on the other boundary’s coordinate. The calculated pressure permits one to obtain the Casimir energy for the systems under consideration.     

Thermodynamics of a quantum gas and a two-particle Green’S function

It has been shown with the use of the virial theorem and the equation of motion for the single-particle Green’s function that the thermodynamic properties of a single-component quantum gas beyond the perturbation theory are fully determined by the two-particle Green’s function Moscow Power Engineering Institute.