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.     

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.     

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).     

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 $\text{1}\text{β}$.     

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).     

Surface and volume contributions to real and imaginary parts of the heavy-ion optical potential

Starting from the Feshbach expression for the optical potential, explicit formulae for the real and the imaginary parts of the optical potential between two heavy ions (HI's) are obtained. They are each composed of a volume and a surface term. The contributions to the volume term are calculated in two nuclear Fermi liquids which flow through each other starting from the realistic Reid soft core nucleon-nucleon (NN) interaction. Since the Fermi surface is formed by two spheres one obtains a complex Brueckner reaction matrix which is approximated by a complex, effective local interaction. It is used in a fully antisymmetrized double folding procedure to obtain the volume terms of the optical potential between the two HI's. The surface contributions are directly calculated in the collision of the two finite HI's. The collective surface vibrations (3^? octupole state and 2⁺, 4⁺ (T = 0) giant resonances for the ¹⁶O?¹⁶O collision) are included as intermediate states. This yields especially an imaginary contribution at the surface which reduces the transparency found with the volume terms alone. The method is applied to ¹⁶O?¹⁶O scattering at 83 and 332 MeV laboratory energy. The local approximations to the real and imaginary parts obtained in this way agree well with phenomenological fits. The surface terms improve the agreement of the differential cross section at 80 MeV where experimental data are available.     

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.     

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.     

Hydrogenic atom multiphonon ionization by intense fields with green's method

An expression for n-photon cross-section ionization of hydrogenic atoms by intense linearly polarized light is obtained using the second quantification formalism for the radiation and the Green's function method. It is shown that it reduces into Stobbe's formula in the one photon case.     

Application of the Green's function method to lattice vibrations in thin films

The Green's function method is used to investigate the lattice vibrations in a simple cubic thin film with nearest neighbour interactions. Appropriate Green's functions for the study of the surface waves for (001) and (011) orientations of the surface are determined. For a thin film with two parallel (001) free surfaces the atomic force tensors at the surface are modified in order to satisfy the rotational invariance condition. The surface waves are determined and a detailed discussion is made for the long wave-length limit k_y = 0 and k_x ? 1. For a given k_x there is a critical thickness below which only an antisymmetrical solution can exist. For the (011) orientation only the case of a single free surface is discussed.     

Higher order field equations II; Limit properties of green's functions

In a previous paper wave propagation was studied according to a sixth-order partial differential equation involving a complex mass M. The corresponding Yang-Feldman integral equations (indicated as SM-YF-equations), were formulated using modified Green's functions G^M_R(x) and G^M_A(x), which then incorporate the partial differential equation together with certain boundary conditions. In this paper certain limit properties of these modified Green's functions are derived: (a) It is shown that for |M| → ∞ the Green's functions G^M_R(x) and G^M_A(x) approach the Green's functions Δ_R(x) and Δ_A(x) of the corresponding KG-equation (Klein-Gordon equation). (b) It is further shown that the asymptotic behaviour of G^M_A(x) and G^M_A(x) is the same as of Δ_R(x) and Δ_A(x) - and also the same as for D_R(x) and D_A(x) for t→ ± ∞, where D_R and D_A are the Green n's functions for the KG-equation with mass zero. It is essential to take limits in the sense of distribution theory in both cases (a) and (b). The property (b) indicates that the wave propagation properties of the SM-YF-equations, the KG-equation with finite mass and the KG-equation with mass zero are closely related in an asymptotic sense.     

Self-energy radiative corrections in hydrogen-like systems

The one-photon self-energy radiative level shift of an electron in a Coulomb field is examined. An expression for the level shift which is suitable for direct numerical evaluation, for Z in the range 10–110, is obtained. It is based on the known Coulomb radial Green's functions and not on a power series expansion in Zα. In the following paper, the numerical evaluation of the level shift for the $\text{1S}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ state is described.     

Summation of perturbation series in the generalized Hubbard model

The cumulant expansion is proposed for the summation of perturbation series in the generalized Hubbard model, which considers strongly correlated d-electrons hybridized with nearly free conduction electrons. It is shown, that summation of the principle series for the Green's functions leads to the Kondo renormalization of d-level hybridized with s-band.     

The second Legendre transform of the P(φ)2 generating functional

We prove the existence of the second Legendre transform of the generating functional of connected Green's functions for even weakly coupled P(φ)₂ models. The transformed functional is the generator of two particle irreducible graphs.     

Ferromagnetic 3d-metals: On the definition of “local bands”

The basic concept of a picture for itinerant ferromagnetism is discussed. The central point is that a local exchange splitting and local moments, exist even above the transition temperature T_c. Transverse fluctuations (and not magnitude fluctuations, as in Stoner theory) are the dominant source for the phase transition to the paramagnetic state. The author's Green's function method is extended to the use of a full bandstructure including hybridization and general electron-electron interactions. Spin waves are also discussed.     

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.     

External gravitational interactions in quantum field theory

We consider the renormalization of Green's functions of λφ⁴ quantum field theory in an external gravitational field specified by the metric tensor g^μν(y). Green's functions Γ^(n,3) describing the interaction of j scalar particles to arbitrary order n in the gravitational field are shown to be made finite by the standard renormalizations of the flatspace theory and a renormalization of the coefficient of the improvement term in the action functional. These results in φ⁴ theory can be extended to all renormalizable field theories.     

Energy-dependent hopping in trans-polyacetylene: Experiment and theory for complex conductivity

Measurements of dc conductivity and both parts of the complex ac conductivity are reported for $\text{trans}$-polyacetylene. The imaginary part is obtained by identifying the residual capacitance of the material. The data is analysed in terms of hopping transport, modelled by rate equations. An energy-dependent transition rate with an appropriate density of states gives a good account of the data. This is compared with results obtained when Kivelson's energy-independent rate is used.     

Functional approach to the parametric derivatives of renormalized green's functions with multiple insertions

Functional derivation of formulae for parametric derivatives of renormalized Green's functions with multiple insertions of composite operators is presented. An application to the derivative with respect to the ultraviolet cutoff in the Pauli–Villars regularized ?⁴₄ theory is given.