Higher-order approximations for the particle-particle propagator

A new type of approximations for many-body Green's functions proposed recently is applied to the particle-particle (pp) propagator for anN-particle fermion system. The new approach which is referred to as the algebraic diagrammatic construction (ADC) is based on an exact resummation of the perturbation series for the pp-propagator in terms of a simple algebraic form introducing energy-independent effective interaction matrix elements and transition amplitudes. These effective quantities are represented by perturbation expansions and can be determined consistently through a given ordern of perturbation theory by comparing the algebraic form with the diagrammatic perturbation series of the pp-propagator through ordern. By this procedure one obtaines a systematic set of approximation schemes (ADC(n)) that represent infinite partial summations for the pp-propagator being complete throughnth order of perturbation theory. The explicit ADC equations forn=1 and 2 are presented and discussed. Comparison is made with the particle-particle random phase approximation (RPA). It is demonstrated that the second-order ADC scheme constitutes an essential step beyond the RPA which is consistent only through first order.     

TFD extension of a self-consistent RPA to finite temperatures

The self-consistent RPA (SCRPA) developed by Schuck and coauthors is extended to finite temperatures. The corresponding equations are derived by using the formalism of thermofield dynamics. The intrinsic energy of a system is calculated as the expectation value of the Hamiltonian with respect to a T-dependent thermal vacuum state for a thermal-phonon operator. A nonvanishing number of thermal quasiparticles in the vacuum state are assumed. By virtue of the assumption, the thermal Hartree-Fock (HF) equations appear to be coupled to the equations of motion for phonon variables. The thermal occupation numbers are also calculated in a consistent way with the energies of the HF quasiparticles. The approximation is applied to the two-level Lipkin model. Advantages of the thermal SCRPA (TSCRPA) are most obvious at temperatures near the phase-transition point. In the TSCRPA, the phase transition occurs at lower T than in other approximations. Moreover, within the TSCRPA, a statistical behavior of the Lipkin model is described with an appropriate accuracy at any T even if the HF transformation parameter is kept fixed at a value corresponding to the “spherical” phase of the HF field.     

Effects of dirac sea on giant resonance states

There are two approximations in relativistic models which keep the continuity equation of the baryon current without renormalization of the divergence. One is the no-free-term approximation (NFA) which neglects the divergent terms, but keeps the Pauli blocking terms coming from nucleon-antinucleon excitations in the RPA correlation functions. The other is the no-sea approximation (NSA) where antiparticle states are assumed to be empty with negative energies. It is shown that both approximations formally satisfy the Ikeda sum rule and the RPA theorem for the β_? and β₊ transition strengths also, but that the NFA requires negative strengths in the positive excitation energy region, while the NSA requires positive strengths in the negative excitation energy region, as a price of neglecting the renormalization of the divergence.     

Elliptic PDE formulation and boundary conditions of the spherical harmonics method of arbitrary order for general three-dimensional geometries

The inherent complexity of the radiative transfer equation makes the exact treatment of radiative heat transfer impossible even for idealized situations and simple boundary conditions. Therefore, a wide variety of efficient solution methods have been developed for the RTE. Among these solution methods the spherical harmonics method, the moment method, and the discrete ordinates method provide means to obtain higher-order approximate solutions to the equation of radiative transfer. Although the assembly of the governing equations for the spherical harmonics method requires tedious algebra, their final form promises great accuracy for any given order, since it is a spectral method (rather than finite difference/finite volume in the case of discrete ordinates). In this study, a new methodology outlined in a previous paper on the spherical harmonics method (P_N) is further developed. The new methodology employs successive elimination of spherical harmonic tensors, thus reducing the number of first-order partial differential equations needed to be solved simultaneously by previous P_N approximations (=(N+1)²). The result is a relatively small set (=N(N+1)/2) of second-order, elliptic partial differential equations, which can be solved with standard PDE solution packages. General boundary conditions and supplementary conditions using rotation of spherical harmonics in terms of local coordinates are formulated for the general P_N approximation for arbitrary three-dimensional geometries. Accuracy of the P_N approximation can be further improved by applying the “modified differential approximation” approach first developed for the P₁-approximation. Numerical computations are carried out with the P₃ approximation for several new two-dimensional problems with emitting, absorbing, and scattering media. Results are compared to Monte Carlo solutions and discrete ordinates simulations and a discussion of ray effects and false scattering is provided.     

Screening in a δ-doped semiconductor

The screening of an impurity in the quasi-two-dimensional (2D) electron gas in a δ-doped semiconductor structure is investigated. The screened impurity matrix elements are calculated and compared using three different approaches: the 2D random phase approximation (RPA), the corresponding 2D Thomas–Fermi theory and a quasi-three-dimensional (3D) Yukawa-like screening model. It is found that the 2D Thomas–Fermi theory differs from the RPA result, even in the limit of low q vectors, if more than one subband is occupied. This result is explained analytically by closely examining theq → 0 limit of the dielectric tensor. The 2D Thomas–Fermi theory is shown to represent a poor approximation to the RPA whereas the quasi-3D screening model agrees well with the RPA results for not too smallqvectors. Furthermore, this model reduces computing times by orders of magnitude in comparison with the RPA. Thus, our 3D screening model considerably simplifies the calculation of impurity scattering rates in the investigation of the electron mobility in a δ-doping layer.     

Dynamics of collective decoherence and thermalization

We analyze the dynamics of N interacting spins (quantum register) collectively coupled to a thermal environment. Each spin experiences the same environment interaction, consisting of an energy conserving and an energy exchange part.We find the decay rates of the reduced density matrix elements in the energy basis. We show that if the spins do not interact among each other, then the fastest decay rates of off-diagonal matrix elements induced by the energy conserving interaction is of order N², while that one induced by the energy exchange interaction is of the order N only. Moreover, the diagonal matrix elements approach their limiting values at a rate independent of N. For a general spin system the decay rates depend in a rather complicated (but explicit) way on the size N and the interaction between the spins.Our method is based on a dynamical quantum resonance theory valid for small, fixed values of the couplings. We do not make Markov-, Born- or weak coupling (van Hove) approximations.     

Effects of mode-mixing and non-Condon interaction in the vibronic spectra of multimode systems

A method of calculation of optical spectra, which allows one to take into account the mixing of an arbitrary number of modes in the transition, is developed. Thus the dependence of electronic transition matrix elements on nuclear coordinates is also taken into consideration. In this method, the calculation of the Fourier transform of the spectrum is reduced to the calculation of the determinant of the 2N×2N matrix and its reciprocal matrix; the elements of the matrix are given by time-dependent pair-correlation functions.     

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.     

Boson expansions and broken symmetry

A new boson mapping for the group SU(2) × SU(2) is used to generate a perturbation expansion in a small parameter, valid in the deformed regime of the two-dimensional Moszkowski model, without violating angular-momentum conservation at any stage. The corrections to the rotational and vibrational energies and transition matrix elements are obtained to an accuracy one order beyond the RPA. These results are used as a criterion for testing the validity of two methods employing conventional boson expansions about a symmetry-broken minimum and correcting for the symmetry violation a posteriori. It is shown that both methods—that of Marshalek and Weneser, and that of Bes et al., adapted here to boson expansions—reproduce the results of the criterion.     

Systematics of vibrational intensities for “diatomic” molecules: Approximate formulas

Several approximations are discussed which enable one to estimate electric dipole moment transition matrix elements for diatomic molecules based on a very limited amount of experimental data. For strongly polar molecules, a simplistic point-charge dipole model in which there is no vibrational charge flux predicts a simple relation between the permanent moment, M₀, and its derivative with respect to internuclear separation, M₁. This is found to be approximately valid for a surprisingly large number of polar molecules. Next, approximate matrix elements for the fundamental and overtone bands are presented for a linear dipole moment function. Finally, systematic trends for molecules having similar electronic structures are investigated and combined with the approximations discussed above in order to estimate the transition moments for several molecules of astrophysical interest. An extension to the ν₃ fundamental and overtone bands of CH₄ is suggestive that some of these ideas may be applicable to certain polyatomic molecules as well.     

An extended Lipkin-type model with residual proton-neutron interaction

The Lipkin-Meshkov-Glick (LMG) model is extended to explicitly take into account the proton and neutron degrees of freedom. The proton and neutron Hamiltonians are taken to be of the LMG form, and, in addition, a residual proton-neutron interaction is include. Exact solutions in an SU(2) ⊗ SU(2) basis as well as the RPA solutions for the energy spectrum of the model Hamiltonian are obtained. The spectrum of the exact solutions is degenerate in the limit of no proton-neutron residual interaction, but this degeneracy is totally removed when this type of residual interaction is turned on. The spectrum obtained with RPA is compressed or expanded, as compared to the LMG model with the same total number of nucleons (N), depending of the relative magnitudes of the like- and unlike-particle residual interactions. Furthermore, the behavior of the first collective excited state is studied as function of the interaction parameters of the model using the exact and RPA methods. At a given N, it was found that the RPA result is closer to the exact one when the like- and unlike-particle residual interactions are both taken into account in the model Hamiltonian, as compared to the case when residual interaction of only one type (e.g. either like- or unlike-particle type) is involved.     

Time-dependent Gutzwiller approximation for the Hubbard model

We develop a time-dependent Gutzwiller approximation (GA) for the Hubbard model analogous to the time-dependent Hartree-Fock (HF) method. This new formalism incorporates ground state correlations of the random phase approximation (RPA) type beyond the GA. Static quantities like ground state energy and double occupancy are in excellent agreement with exact results in one dimension up to moderate coupling and in two dimensions for all couplings. We find a substantial improvement over traditional GA and HF+RPA treatments. Dynamical correlation functions can be computed and are also substantially better than HF+RPA ones and obey well behaved sum rules.     

Local forces and the 16O reaction matrix

It is demonstrated that the G-matrix elements obtained from a solution of the Bethe-Goldstone equation for finite nuclei and a given NN interaction can be very well approximated by an effective local interaction. The local approximation is determined from the reaction matrix in nuclear matter using the same realistic NN interaction. The comparison is performed on the level of RPA calculations for the excited states in ¹⁶O. Very good agreement is found between the results for both interactions except for scalar-isoscalar states. It is shown that this is due to the energy dependence of the reaction matrix and can be cured rather easily. A comparison with experiment for isovector states is very satisfactory.     

Quasi-relativistic treatment of the low-lying KCs states

The potential energy curves, permanent and transition dipole moments as well as spin-orbit and angular coupling matrix elements between the KCs electronic states converging to the lowest three dissociation limits were evaluated in the basis of the spin-averaged wavefunctions corresponding to pure Hund’s coupling case (a). The quasi-relativistic matrix elements have been obtained for a wide range of internuclear distance by using of small (9-electrons) effective core pseudopotentials of both atoms. The core-valence correlation has been accounted for a large scale multi-reference configuration interaction method combined with semi-empirical core polarization potentials. The static dipole polarizabilities of the ground X¹Σ⁺ and a³Σ⁺ states were extracted from the closed-shell coupled-cluster energies by the finite-field method. Among the singlet and triplet Σ⁺ states manifold the pronounced avoided crossing effect between repulsive walls of the (2,3)³Σ⁺ states has been discovered and analyzed by finite-difference calculation of radial coupling matrix elements. The resulting transition dipole moments and potentials were used to predict radiative lifetimes and emission branching ratios of excited vibronic states while the calculated angular coupling matrix elements were transformed to Λ-doubling constants of the (1,2)¹Π states and magnetic g-factor of the ground state. The accuracies of the present results are discussed by comparing with experimental data and preceding calculations.     

Effect of ground state correlations on the imaginary part of the optical-model potential

The effect of the ground state correlations on the imaginary part of the nucleon-nucleus optical-model potential is qualitatively discussed within the random phase approximation (RPA). We also investigate the effect of terms of order higher than two in the microscopic nucleon-nucleus interaction. A rough estimate of both effects is given for the case K³⁹+p.     

The effect of weak magnetism and induced pseudoscalar coupling in neutrinoless double-beta decay

In calculating the amplitude of the majorana neutrino-mass mechanism of neutrinoless double-beta decay (0νββ-decay), several approximations of the nucleon current have been done. for example, effects from induced current such as weak magnetism and pseudoscalar coupling have been neglected. we shall show in this work that, although such terms do not contribute significantly to the 2νββ-decay amplitude, they are important in the case of 0νββ decay. performing calculations within the renormalized quasiparticle random phase approximation (pn-rqrpa) for all nuclei undergoing double-beta decay in the region a=76 to a=150, we have found that these additional contributions of the nucleon current reduce considerably the matrix elements in all cases for the light neutrino as well as for the heavy neutrino mass. in the light-neutrino mass, we find reductions up to thirty percent, while in the heavy-neutrino mass, up to almost a factor of five. these reductions make the limits on the lepton-number-violating parameters 〈m _ν〉 and η_N less stringent.     

Die Hartree-Fock-Theorie mit Zwischenzuständen: Klärung ihrer Struktur bei Zeitumkehrinvarianz und Anwendung auf das Kleinsche Rotations-Vibrationsmodell

In an earlier paper we have extended the new Tamm-Dancoff method to the “New Tamm-Dancoff method with intermediate states”. This extension makes it possible to treat the effect of nearby levels in many body systems with Green's functions. In addition to well-known approximations, such as the Hartree-Fock theory and the Hartree-Bogoliubov theory, we obtain a series of new approximations. The “Hartree-Fock theory with intermediate states”, which is the subject of the present investigation, is one of these. By using time reversal invariance we have succeeded in clarifying its structure, and we give the solution procedure. The exchange terms in theN-particle intermediate states can be represented by an additional potentialY, which (as is the case for the generalized density matrixρ) has to be determined selfconsistently. In this way we have overcome the difficulties, that Kerman and Klein met in their “generalized Hartree-Fock approximation”, which has some close similarities with our Hartree-Fock theory with intermediate states. We demonstrate our method for the exactly soluble rotational-vibrational model of Klein et al. Hereby we show how to treat conservation laws and the degeneracy of levels. The Hartree-Fock equations with three intermediate states turn out to give analytical expressions for the energies and the matrix elements. These agree excellently with the exact values in the rotational part of the spectrum.