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Physical Review C

Self-consistent random-phase approximation at finite temperature within the Richardson model

Nguyen Dinh Dang, K. Tanabe

Published Sep 25, 2006• 2 citations
Publisher

Abstract

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.

DOI
10.1103/physrevc.74.034326

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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.
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.
The self-consistent random-phase approximation (SCRPA) is reexamined within a multilevel-pairing model with double degeneracy. It is shown that the expressions for occupation numbers used in the original version of the SCRPA violate the particle number for nonsymmetric particle-hole (ph) spectra. A renormalization is introduced to restore the particle number, which leads to the expressions of occupation numbers similar to those derived by Hara et al. for the ph case. The results of calculations within the ph symmetric case show that this number-conserving SCRPA yields the energies of the ground state and the first excited state of the system with $\ensuremath{\Omega}+2$ particles relative to the ground state of the system with $\ensuremath{\Omega}$ particles in close agreement with those obtained within the original SCRPA. However, it gives a slightly larger correlation energy.
The consistency condition is tested within the particle-particle random-phase approximation (RPA), renormalized RPA (RRPA), and the self-consistent RPA (SCRPA) making use of the Richardson model of pairing. The two-particle separation energy is calculated in two ways, namely, as the energy of the first addition mode, which adds two particles to a core with NNN particles, and as the energy of the first removal mode, which removes two particles from the N+2N+2N+2 particle system to get back to the same NNN-particle core. The corresponding transitions generated by the pairing operators are also calculated. It is found that the results obtained in these two ways of calculations are close to each other only at large values of particle number NNN and/or small interaction strength. At $N\ensuremath{\le}10$ for a given value of the interaction strength, the discrepancy between the results obtained in two ways of calculations within the SCRPA is much smaller than those given by the RPA and RRPA.
A self-consistent version of the thermal random phase approximation (TSCRPA) is developed within the Matsubara Green’s function formalism. The TSCRPA is applied to the many-level pairing model, and the normal phase of the system is considered. The TSCRPA results are compared with the exact ones calculated for the grand canonical ensemble. Advantages of the TSCRPA over the thermal mean-field approximation and the standard thermal random phase approximation are demonstrated. Results for correlation functions, excitation energies, etc., as a function of temperature are presented.
The dynamical effects of ground state correlations for excitation energies and transition strengths near the superfluid phase transition are studied in the soluble two level pairing model, in the context of the particle-particle self-consistent random phase approximation (SCRPA). Exact results are well reproduced across the transition region, beyond the collapse of the standard particle-particle random phase approximation. The effects of two-body correlation in the SCRPA are displayed explicitly.
The random phase approximation (RPA), the renormalized RPA (RRPA), and the self-consistent RPA (SCRPA) methods are applied to calculate the wave functions of the ground and excited states of an exactly solvable model. The approximated wave functions are expanded in the basis of the exact solutions. It is found that, when the RPA collapses, the RPA wave functions are orthogonal to the exact solutions while the RRPA and SCRPA ones have small but finite overlaps with the exact results. In spite of the apparently good agreement between the results of the RRPA, the SCRPA, and the exact solution, for the energy of the first excited state beyond the point of collapse, it is found that these approximations do not correctly describe the exact wave functions.