Screening of classical charges in quantum Coulomb systems.

The imaginary-time Green's function is a building block of various numerical methods for correlated electron systems. Recently, it was shown that a model-independent compact orthogonal representation of the Green's function can be constructed by decomposing its spectral representation. We investigate the performance of this so-called intermediate representation (IR) from several points of view. First, we develop an efficient algorithm for computing the IR basis functions of arbitrary high degree. Second, for two simple models, we study the number of coefficients required to represent the Green's function within a given tolerance. We show that the number of coefficients grows only as $O(ln\ensuremath{\beta})$ for fermions and converges to a constant for bosons as temperature $T=1/\ensuremath{\beta}$ decreases. Third, we show that this remarkable feature is ascribed to the properties of the physically constructed basis functions. The fermionic basis functions on the real-frequency axis have features whose width is scaled as $O(T)$, which is consistent with the low-$T$ properties of quasiparticles in a Fermi liquid state. On the other hand, the properties of the bosonic basis functions are consistent with those of spin/orbital susceptibilities at low $T$. These results demonstrate the potential wide applications of the IR to calculations of correlated systems.

Exchange energy and specific heat of the electron gas at low temperature

We develop a causal optimization method that ensures causality in numerical calculations of Green's functions in interacting electron systems. Our method removes noncausality of numerical data by finding causal functions closest to the data. By testing our method with an exactly calculable model and applying it to practical dynamical mean-field calculations, we find that intermediate-frequency behaviors of Green's functions are determined solely by causality, and noncausal statistical errors are removed very efficiently. Furthermore, we demonstrate that numerical calculations of the physical branch of the Luttinger-Ward functional can be stabilized by ensuring causality of the noninteracting Green's function. Our method and findings provide a basis for improving stability and efficiency of numerical simulations of quantum many-body systems.

I compute the spectral functions of a 1D Holstein polaron moving in a modulated potential, to examine the effects of electron-phonon interactions on band gaps. The imaginary time Green function is computed using diagrammatic quantum Monte Carlo, which exactly sums the diagrammatic series. From the imaginary time Green function, spectral functions are computed. The electron-phonon interaction flattens the electronic dispersion and leads to an increase in the gap at momentum k = π/2. At strong coupling, polaron sidebands form in the gap. These results demonstrate the strong effect that electron-phonon interactions can have on band gaps.

A coordinate-independent formulation of the chiral-symmetric pion theory is developed using a dreibein field which introduces a redundant gauge field into the field equations. There is a particular gauge, where the field variables transform linearly under chiral transformations and the field equations reduce to those of the Sugawara-model. Quantization of Sugawara's field equations is obtained in terms of Green's functions. The field equations for the coordinate-independent Green's functions are translated into equations for coordinate Green's functions by introduction of the coordinates as auxiliary variables. The latter field equations incorporate the additional Feynman graphs discovered before.

Ab initio electron propagator methods are developed to study electronic properties of molecular systems with strong electron-electron and electron-phonon interactions. For the calculation of electron Green's functions we apply a canonical small polaron transformation that intrinsically contains strong electron-phonon effects. In the transformed Hamiltonian, the energy levels for the noninteracting particles are shifted down by the relaxation (solvation) energies. The Coulomb integrals are also renormalized by the electron-phonon interaction. For certain values of the electron-phonon coupling constants, the renormalized Coulomb integrals can be negative which implies the attraction between two electrons. Within the small polaron transformation we develop a diagrammatic technique for the calculation of electron Green's function in which the electron-phonon interaction is already included into the multiple phonon correlation functions. Since the decoupling of the phonon correlation functions is impossible, and therefore, a Wick's theorem for such correlation functions is invalid, there is no Dyson equation for the electron Green's function. To find the electron Green's function, we use different approximations. One of them is a link-cluster approximation that includes diagonal transitions for the renormalized zeroth Green's function. In the linked-cluster approach the Dyson equation is derived in the most general case, where the self-energy operator is an arbitrary functional (not only in the Hartree-Fock approximation). It is shown that even a Hartree-Fock electron (hole) is not a particle any longer. It is a quasiparticle with a finite lifetime that depends on energy of particle and hole states in different ways. As a consequence of this, a standard description of a Hartree-Fock approximation in terms of wave functions becomes inappropriate in this problem. To challenge the linked-cluster approximation we develop a different approach: a sequential propagation approximation where scattering events occur only for sequential transitions. A self-consistent Hartree-Fock equation for a four-index Green's function matrix is derived. In conclusion, the proposed schemes can be considered for future method developments for quantum chemical calculations for large molecules with strong nonadiabatic effects, e-e correlated electron transfer reactions, and electron transport in molecular transport junctions.

The Feynman rules for the Yang-Mills field, originally derived by Feynman and DeWitt from $S$-matrix theory and the tree theorem, are here derived as a consequence of field theory. Our starting point is the gauge-independent, path-dependent formalism which we proposed earlier. The path-dependent Green's functions in this theory are expressed in terms of auxiliary, path-independent Green's functions in such a way that the path-dependence equation is automatically satisfied. The formula relating the path-dependent to the auxiliary Green's functions is similar to the classical formula relating the path-dependent field variables to the potentials. By using a notation similar but not identical to Schwinger's functional notation, the infinite set of equations satisfied by the Green's function can be replaced by a single equation. When the equation for the auxiliary Green's functions of electromagnetism is solved in a perturbation series, the usual Feynman rules result. For the Yang-Mills field, however, one obtains extra terms; such terms correspond precisely to the closed loops of fictitious scalar particles introduced by Feynman, DeWitt, and Faddeev and Popov.

We study electron transmission through two impurities in a narrow quantum wire by solving Dyson’s equations for single electron Green functions. We have verified that, for the delta-function potential of two impurities, the Green function can be factorized into a product of the ‘free’ Green function and current transmission amplitude. Meanwhile Green function and current transmission amplitude obey Fisher-Lee’s relation. An analytical expression of the electron transmission amplitude for intrasubband and intersubband is obtained as a function of Fermi energy and the distance between two impurities. The resonant behavior of the current transmission amplitude are detail discussed.

The method of generating functional is generalized to the case of strongly correlated systems, and applied to the Hubbard model. For the electronic Green's function constructed for Hubbard operators, an equation using variational derivatives with respect to the fluctuating fields has been derived and its multiplicative form has been determined. Corrections for the electronic self-energy 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. The equations for the Bose-like Green's functions have been derived, describing the collective modes: the magnons and doublons. 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=(pi,pi,...), 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.

A systematic method is presented for deriving the Thomas-Fermi equation for an atom and the quantum corrections from the many-body description. The novel feature of the method is that it does not require any a priori assumptions about the assignment of electrons to fully occupied single-particle states or about the distribution of electrons in phase space, but shows instead that the distribution which is usually assumed, or derived from the assumption of fully occupied single particle states, is a direct consequence of specifying that the many particle system is in its ground state. The procedure used in the derivation is the expansion of the mixed position-momentum representation of the Green's function in a series of powers of $\ensuremath{\hbar}$. The lowest order term is found to correspond with the Thomas-Fermi density. The form of the higher order terms, which are to be considered as corrections to zeroth order term, depends on the approximations made in the many-body equations for obtaining the Green's function. This paper deals only with the Hartree-Fock approximation, but the methods presented here allow generalization to other approximations which can include correlation effects.

Time-dependent multiparticle correlation functions and response functions are formulated with the functional derivatives in the grand-canonical ensemble. Equations for the single-particle Green's functions are the Dyson equations describing scattering of the single particles by density-fluctuation excitations. Equations for the density-fluctuation excitations are truncated with a dynamic version of the convolution approximation to triple correlations. A closed set of equations for the Green's functions and excitations are thereby obtained, which constitutes a dynamic hypernetted-chain scheme for the degenerate electron liquids. The formalism is extended also to the description of the spin-dependent response and correlations.

In this paper the time domain scattering problem of a stratified dissipative half‐space excited by a point source is considered. Both the direct and inverse scattering problem are treated. The dissipative wave equation is reduced to an equation in one spatial dimension by spatial Fourier series and a Hankel transform. The transformed wave equation is then solved by a combination of a time domain wave‐splitting technique and a Green function technique. The Green functions represent the internal fields inside the stratified medium. The splitting is expressed in terms of a Neumann operator K which has an explicit expression as a convolution integral with a Bessel function kernel. Based upon this wave splitting, Green functions can be defined and a set of first order partial differential equations for Green functions are derived. The equations are well suited for a numerical treatment and a number of numerical examples for both the direct and inverse problems are presented. A brief discussion of the invariant imbedding method applied to the direct and inverse scattering problem is also given. It is shown that one‐sided reflection data are sufficient to reconstruct the phase velocity and the dissipation coefficient simultaneously by using two different values of the Hankel transform parameter.

Recently, a series of noncentrosymmetric superconductors has been a subject of considerable interest since the discovery of superconductivity in CePt_3Si. In noncentrosymmetric materials, the degeneracy of bands is lifted in the presence of spin-orbit coupling. This will bring about new effects in the Kondo effect since the band degeneracy plays an important role in the scattering of electrons by localized spins. We investigate the single-impurity Kondo problem in the presence of spin-orbit coupling. We examine the effect of spin-orbit coupling on the scattering of conduction electrons, by using the Green's function method, for the s-d Hamiltonian, with employing a decoupling procedure. As a result, we obtain a closed system of equations of Green's functions, from which we can calculate physical quantities. The Kondo temperature T_K is estimated from a singularity of Green's functions. We show that T_K is reduced as the spin-orbit coupling constant \alpha is increased. When 2\alpha k_F is comparable to or greater than k_BT_K(\alpha=0), T_K shows an abrupt decrease as a result of the band splitting. This suggests a Kondo collapse accompanied with a sharp decrease of T_K. The log T-dependence of the resistivity will be concealed by the spin-orbit interaction.

The method of the derivation of two-particle Green's functions in non-equilibrium matter is developed. The closed set of equations for the vertex functions and also for the two-particle Green's functions is obtained by means of the summation of the series of irreducible diagrams. The solution of such equations completely defines the two-particle Green's functions in matter.

Nuclear response theory beyond the one-loop approximation is formulated for the case of finite temperature. For this purpose, the time blocking approximation to the time-dependent part of the in-medium nucleon-nucleon interaction amplitude is adopted for the thermal (imaginary-time) Green's function formalism. We found that introducing a soft blocking, instead of a sharp blocking at zero temperature, brings the Bethe-Salpeter equation to a single-frequency variable equation also at finite temperatures. The method is implemented self-consistently in the framework of quantum hadrodynamics and designed to connect the high-energy scale of heavy mesons and the low-energy domain of nuclear medium polarization effects in a parameter-free way. In this framework, we investigate the temperature dependence of dipole spectra in the even-even nuclei ^{48}Ca and ^{100,120,132}Sn with a special focus on the giant dipole resonance's width problem and on the low-energy dipole strength distribution.