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Physical Review C

Extended quasiparticle approximation and Brueckner theory

Hannes Köhler, Rudi Malfliet

Published Sep 1, 1993• 46 citations
Publisher

Abstract

The time-dependent Green's function method provides a basic theory for nuclear dynamics and heavy-ion collisions. The spectral function plays an important role in this theory containing information on correlations in the medium. These are usually neglected by using a quasiparticle approximation. In order to evaluate this particular approach it is important to establish in the static limit a link between this method and other traditional nuclear many-body theories such as the Brueckner approach. Using a self-consistent T-matrix approximation for the self-energy in the Green's-function approach, treating particles and holes on equal footing, the self-energy can be obtained and compared with the Brueckner expression. In the quasiparticle limit it contains terms up to and analogous to the Brueckner second order rearrangement energy. In an extended quasiparticle approximation (EQP) which does not violate the sum rule for the spectral function the Brueckner third-order rearrangement energy can also be reproduced. The latter approximation constitutes a bridge between the Brueckner and Green's-function methods. It assumes small absorption (and energy dependence) of the mean field. Some numerical estimates of the different approximation schemes that we discuss are shown. An iteration scheme for applying the EQP approximation is suggested.

DOI
10.1103/physrevc.48.1034

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The time-dependent Green's function method provides a basic theory for nuclear dynamics and heavy-ion collisions. The spectral function plays an important role in this theory containing information on correlations in the medium. These are usually neglected by using a quasiparticle approximation. In order to evaluate this particular approach it is important to establish in the static limit a link between this method and other traditional nuclear many-body theories such as the Brueckner approach. Using a self-consistent T-matrix approximation for the self-energy in the Green's-function approach, treating particles and holes on equal footing, the self-energy can be obtained and compared with the Brueckner expression. In the quasiparticle limit it contains terms up to and analogous to the Brueckner second order rearrangement energy. In an extended quasiparticle approximation (EQP) which does not violate the sum rule for the spectral function the Brueckner third-order rearrangement energy can also be reproduced. The latter approximation constitutes a bridge between the Brueckner and Green's-function methods. It assumes small absorption (and energy dependence) of the mean field. Some numerical estimates of the different approximation schemes that we discuss are shown. An iteration scheme for applying the EQP approximation is suggested.
Widths of low-lying states in nuclei are of the order of 30 MeV. These large widths are a consequence of the strong interactions leading to a strongly correlated many body system at the typical densities of nuclear matter. Nevertheless ''traditional'' Brueckner calculations treat these states as quasiparticles i.e. with spectral functions of zero widths. The width is related to the imaginary part of the selfenergy and is included selfconsistently in an extension of the Brueckner theory using T -matrix and Green's function (GF) techniques. A more general formulation applicable also to non-equilibrium systems is contained in the Kadanoff-Baym (KB) equations while still maintaining the basic many-body techniques of Brueckner theory. In the present work the two-time KB-equations are timestepped along the imaginary timeaxis to calculate the binding energy of nuclear matter as a function of density, including the spectral widths self-consistently. These zero temperature calculations are compared with quasiparticle calculations. The inclusion of the selfconsistent widths are found to add several MeV to the binding. The spectral widths are due to the long-ranged correlations. Short ranged correlations decrease rather than increase the binding. The method is easily extended to nonzero temperatures where the importance of the widths are expected to increase.
The Green’s function method in the Kadanoff-Baym version provides a basic theory for nuclear dynamics which is applicable also to nonzero temperature and to nonequilibrium systems. At the same time, it maintains the basic many-body techniques of the Brueckner theory that makes reasonable a comparison of the numerical results of the two methods for equilibrium systems. The correct approximation to the spectral function which takes into account the widths of energy levels is offered and discussed, and the comparison of the values of binding energy in the two methods is produced.
A nonperturbative derivation of the Brueckner approximation to the ground-state energy of nuclear matter (with standard choice of hole and particle energies) is given within the framework of the method of correlated basis functions. The structure of the factor-cluster (FIY) expansion of the expectation value of the Hamiltonian with respect to a trial function incorporating short-range correlations in a general fashion, is analyzed for a uniform extended medium of fermions. With a special choice of correlated wave function, the Brueckner approximation is extracted from the FIY expansion by selective summation of two-body combination terms to all orders in the smallness parameter $\ensuremath{\kappa}$ associated, in this case, with both Brueckner-Bethe-Goldstone and FIY energy expansions. A reexamination of the numerical comparison of conventional Brueckner and Jastrow methods carried out earlier for two simple models of nuclear matter leads to the conclusion that it is inadvisable, at least in a Jastrow calculation satisfying the average Pauli condition, to incorporate the $O(\ensuremath{\kappa})$ contribution of the analog of the dispersion correction of reaction-matrix theory without at the same time including the $O(\ensuremath{\kappa})$ contribution of the analog of the Bethe-Faddeev term. For the uniform extended medium for the Jastrow choice of wave function, there is a formal cancellation of these two three-body contributions, the scale of this cancellation being governed by the size of $\ensuremath{\kappa}$. In the numerical example considered, $\ensuremath{\kappa}$ is about 0.22 and the cancellation goes something like +14 MeV-15 MeV=-1 MeV. These findings, when juxtaposed with the standard Brueckner results for the simple models in question, suggest that a departure from the standard prescription of hole and particle energies in reaction-matrix theory may be necessary when $\ensuremath{\kappa}$ is sizeable-as in liquid 3He^{3}\mathrm{He}3He and the high-density neutron gas, as well as equilibrium nuclear matter described with some potentials.
The self-energy method for quantum impurity models expresses the correlation part of the self-energy in terms of the ratio of two Green's functions and allows for a more accurate calculation of equilibrium spectral functions than is possible directly from the one-particle Green's function [Bulla et al., J. Phys.: Condens. Matter 10, 8365 (1998)], for example, within the numerical renormalization group method. In addition, the self-energy itself is a central quantity required in the dynamical mean field theory of strongly correlated lattice models. Here, we show how to generalize the self-energy method to the time-dependent situation for the prototype model of strong correlations, the Anderson impurity model. We use the equation-of-motion method to obtain closed expressions for the local Green's function in terms of a time-dependent correlation self-energy, with the latter being given as a ratio of a one-particle time-dependent Green's function and a higher-order correlation function. We benchmark this self-energy approach to time-dependent spectral functions against the direct approach within the time-dependent numerical renormalization group method. The self-energy approach improves the accuracy of time-dependent spectral function calculations, and the closed-form expressions for the Green's function allow for a clear picture of the time evolution of spectral features at the different characteristic time scales. The self-energy approach is of potential interest also for other quantum impurity solvers for real-time evolution, including time-dependent density matrix renormalization group and continuous-time quantum Monte Carlo techniques.
In this lecture we present many-body perturbation theory as a method to determine quasiparticle excitations in solids, especially electronic band structures, accurately from first principles.The main ingredient is the electronic self-energy that, in principle, contains all many-body exchange and correlation effects beyond the Hartree potential.As its exact mathematical expression is unknown, approximations must be used in practical calculations.The approximation is obtained using a systematic algebraic approach on the basis of Green function techniques.It constitutes an expansion of the self-energy up to linear order in the screened Coulomb potential, which describes the interaction between the quasiparticles and includes dynamic screening through the creation of exchange-correlation holes around the bare particles.The implementation of the approximation relies on a perturbative treatment starting from density functional theory.Besides a detailed mathematical discussion we focus on the underlying physical concepts and show some illustrative applications.
The Kadanoff-Baym (KB) equations are solved numerically for infinite nuclear matter. In particular we calculate correlation energies and correlation times. Approximating the Green's functions in the KB collision kernel by the free Green's functions the Levinson equation is obtained. This approximation is valid for weak interactions and/or low densities. It relates to the extended quasiclassical approximation for the spectral function. Comparing the Levinson, Born, and KB calculations allows for an estimate of higher order spectral corrections to the correlations. A decrease in binding energy is reported due to spectral correlations and off-shell parts in the reduced density matrix.