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Physical Review C

Improved treatment of ground-state correlations: Modified random phase approximation

Nguyen Dinh Dang, Vladimir Zelevinsky

Published Nov 20, 2001• 38 citations
Publisher

Abstract

A method is proposed to improve the treatment of the ground-state correlations in a finite Fermi system compared with the standard random phase approximation (RPA) or earlier suggested renormalized RPA. The correlations lead to nonzero quasiparticle occupancies in the ground state. The method employs modified quasiparticles obtained by a canonical transformation of usual quasiparticles explicitly involving the quasiparticle occupation numbers. A set of equations is derived, which allows one to determine these occupation numbers along with the RPA modes. The formalism is illustrated with the Lipkin-Meshkov-Glick model, and a model for superconducting pairing at a finite temperature. With the new approach, the ground-state correlations are significantly reduced, the energy of the first excited state becomes closer to the exact solution around the region where the RPA collapses, and the superconducting gap monotonously decreases instead of the sharp phase transition. We discuss the effective equivalence of the interaction effects and variation of temperature for the ground-state correlations.

DOI
10.1103/physrevc.64.064319

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A method is proposed to improve the treatment of the ground-state correlations in a finite Fermi system compared with the standard random phase approximation (RPA) or earlier suggested renormalized RPA. The correlations lead to nonzero quasiparticle occupancies in the ground state. The method employs modified quasiparticles obtained by a canonical transformation of usual quasiparticles explicitly involving the quasiparticle occupation numbers. A set of equations is derived, which allows one to determine these occupation numbers along with the RPA modes. The formalism is illustrated with the Lipkin-Meshkov-Glick model, and a model for superconducting pairing at a finite temperature. With the new approach, the ground-state correlations are significantly reduced, the energy of the first excited state becomes closer to the exact solution around the region where the RPA collapses, and the superconducting gap monotonously decreases instead of the sharp phase transition. We discuss the effective equivalence of the interaction effects and variation of temperature for the ground-state correlations.
We present a method which allows us to treat correlations in finite Fermi systems in a more consistent way than the random-phase approximation (RPA). The quasiboson approximation (QBA), where expectation values in the ground state of the system are approximated by their values in the uncorrelated reference state, is avoided in our approach where the correlated ground state is always used. We derive a closed, nonlinear set of equations which determines the energies and wave functions of the excited states as well as the single-particle occupation numbers in the ground state. As an example we apply to metallic clusters a simplified version of the approach which represents, however, a significant improvement over previous attempts to go towards a self-consistent RPA. We show that our method allows one to correct for the inadequacy of standard RPA in cases where the use of QBA becomes questionable.
The self-consistent random-phase approximation (SCRPA) is extended to nonzero temperature. Within the SCRPA, the collective dynamics of nucleon pairs is built on the occupation numbers that deviate from the Fermi-Dirac distribution as a result of temperature-dependent correlations beyond the RPA. The equations for single-particle occupation numbers are derived using the method of the double-time Green's functions. The approach is applied to the Richardson model of pairing. The results of numerical calculations for energies of states and occupation numbers, as well as for the thermodynamic quantities such as the excitation energy and heat capacity of the system, are compared with those obtained within the conventional RPA and the perturbative thermal SCRPA and with the exact results. We find that, overall, the SCRPA offers a better agreement with the exact results.
A connection between the random-phase-approximation (RPA) total energy ERPA{E}_{\text{RPA}}ERPA​ and the GWGWGW quasiparticle energy ϵGW{ϵ}_{GW}ϵGW​ has been pointed out. More specifically we show that: $\ensuremath{\delta}{E}_{\text{RPA}}(N)/\ensuremath{\delta}N={ϵ}_{GW}$, where NNN is the total number of electron. The variational property of the RPA total energy is also discussed and the variational equations for the orbital wave functions are derived. We argue that the RPA formalism is a method which can provide both accurate ground-state energies and single quasiparticle energies.
The Lipkin-Meshkov-Glick model is used to examine the validity of some approximate methods in a many-body theory at finite temperatures. Namely, the thermal random phase approximation (TRPA) and the thermal renormalized random phase approximation (TRRPA) are studied. An average energy of the system, an average quasispin projection and a particle number variance are calculated within these approximation and exactly with the grand canonical ensemble partition function. On the whole the results of TRRPA are found to be in better agreement with the exact ones. The validity of the both approximation becomes better with increasing temperature as well as particle number.
We present a new extension of the random-phase approximation method: the quasiboson approximation is avoided and correlations are included in the ground state without resorting to renormalized operators or renormalized matrix elements; the configuration space is enlarged by considering also elementary excitations corresponding to the annihilation of a particle (hole) and the creation of another particle (hole) on the correlated ground state, together with the particle-hole ones. Two new and relevant advantages of this method with respect to the existing extensions of random-phase approximation are highlighted: (i) the energy weighted sum rules are completely satisfied; (ii) the problem of the existence of nonphysical states in the spectrum, related to the inclusion of particle-particle and hole-hole configurations, is solved: a way to unambiguously disentangle physical from nonphysical states in the excitation spectrum is presented. The method is applied here to a three-level Lipkin model where its quality can be judged by comparing with the exact results. Both advantages (i) and (ii) shall lead to feasible future applications of this extended RPA to several realistic cases.
Particle number fluctuations in the quasiparticle random-phase approximation (QRPA) and the renormalized quasiparticle random-phase approximation are investigated. A closed expression for the ground state wave function is used, which allows an exact evaluation of the ground state phonon number and correlation function, being valid to any order in the backward amplitudes. In realistic calculations a sudden increase in the particle number fluctuations is found in both approximations when the particle-particle interaction approaches the value at which the QRPA collapses. This behavior is strongly correlated with the ground state quasiparticle content and can be understood as a signature of the phase transition previously found in simpler exactly solvable models.