Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0710.0935 · Physical Review C

Self-consistent quasiparticle random-phase approximation for a multilevel pairing model

Nguyễn Quang Hưng, Nguyen Dinh Dang

Published Nov 2, 2007• 15 citations
arXivHTMLPDF

Abstract

Particle-number projection within the Lipkin-Nogami (LN) method is applied to the self-consistent quasiparticle random-phase approximation (SCQRPA), which is tested in an exactly solvable multilevel pairing model. The SCQRPA equations are numerically solved to find the energies of the ground and excited states at various numbers \ensuremath{\Omega} of doubly degenerate equidistant levels. The use of the LN method allows one to avoid the collapse of the BCS (QRPA) to obtain the energies of the ground and excited states as smooth functions of the interaction parameter $G$. The comparison between results given by different approximations such as the SCRPA, QRPA, LNQRPA, SCQRPA, and LNSCQRPA is carried out. Although the use of the LN method significantly improves the agreement with the exact results in the intermediate coupling region, we found that in the strong coupling region the SCQRPA results are closest to the exact ones.

DOI
10.1103/physrevc.76.054302

Related papers

Self-consistent quasiparticle random phase approximation for the description of superfluid Fermi systems

Aziz Rabhi, R. Bennaceur, Guy Chanfray +1 more

Self-consistent quasiparticle random phase approximation (SCQRPA) is for the first time applied to a more level pairing case. Various filling situations and values for the coupling constant are considered. Very encouraging results in comparison with the exact solution of the model are obtained. The nature of the low-lying mode in SCQRPA is identified. The strong reduction of the number fluctuation in SCQRPA vs BCS is pointed out. The transition from superfluidity to the normal fluid case is carefully investigated.

arXiv:nucl-th/0208037•Physical Review C•Dec 18, 2002•18 citations

Exact and approximate ensemble treatments of thermal pairing in a multilevel model

Nguyễn Quang Hưng, Nguyen Dinh Dang

A systematic comparison is conducted for pairing properties of finite systems at nonzero temperature as predicted by the exact solutions of the pairing problem embedded in three principal statistical ensembles, as well as the unprojected (FTBCS1$+$SCQRPA) and Lipkin$+$Nogami projected (FTLN1$+$SCQRPA) theories that include the quasiparticle number fluctuation and coupling to pair vibrations within the self-consistent quasiparticle random-phase approximation. The numerical calculations are performed for the pairing gap, total energy, heat capacity, entropy, and microcanonical temperature within the doubly folded equidistant multilevel pairing model. The FTLN1$+$SCQRPA predictions agree best with the exact grand-canonical results. In general, all approaches clearly show that the superfluid-normal phase transition is smoothed out in finite systems. A novel formula is suggested for extracting the empirical pairing gap in reasonable agreement with the exact canonical results.

arXiv:0905.2002•Physical Review C•May 28, 2009•32 citations

Canonical and microcanonical ensemble descriptions of thermal pairing within BCS and quasiparticle random-phase approximation

Nguyễn Quang Hưng, Nguyen Dinh Dang

We propose a description of pairing properties in finite systems within the canonical and microcanonical ensembles. The approach is derived by solving the BCS and self-consistent quasiparticle random-phase approximation with the Lipkin-Nogami particle-number projection at zero temperature. The obtained eigenvalues are embedded into the canonical and microcanonical ensembles. The results obtained are found in quite good agreement with the exact solutions of the doubly-folded equidistant multilevel pairing model as well as the experimental data for $^{56}\mathrm{Fe}$ nucleus. The merit of the present approach resides in its simplicity and its application to a wider range of particle number, where the exact solution is impracticable.

arXiv:1005.0758•Physical Review C•May 21, 2010•24 citations

Particle-number conservation within self-consistent random-phase approximation

Nguyen Dinh Dang

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.

arXiv:nucl-th/0411072•Physical Review C•Feb 1, 2005•8 citations

Pairing correlations studied in the two-level model

D. C. Zheng, Donald W. L. Sprung, H. Flocard

We study the pairing energy in the ground state of a system of N particles occupying a symmetric two-level model space with a level degeneracy \ensuremath{\Omega}. Assuming a pairing Hamiltonian, we calculate the energies of the projected BCS states, using the exact, the Kamlah, and the Lipkin-Nogami particle number projection methods. We find that conclusions regarding the quality of the BCS approximation as well as various approximate projection methods, drawn from studying the N=\ensuremath{\Omega} case alone, are not valid in the more general N\ensuremath{\ne}\ensuremath{\Omega} case when the pairing interaction is weak.

Physical Review C•Oct 1, 1992•40 citations

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

Nguyen Dinh Dang, K. Tanabe

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.

Physical Review C•Sep 25, 2006•2 citations
Particle-number projection within the Lipkin-Nogami (LN) method is applied to the self-consistent quasiparticle random-phase approximation (SCQRPA), which is tested in an exactly solvable multilevel pairing model. The SCQRPA equations are numerically solved to find the energies of the ground and excited states at various numbers \ensuremath{\Omega} of doubly degenerate equidistant levels. The use of the LN method allows one to avoid the collapse of the BCS (QRPA) to obtain the energies of the ground and excited states as smooth functions of the interaction parameter GGG. The comparison between results given by different approximations such as the SCRPA, QRPA, LNQRPA, SCQRPA, and LNSCQRPA is carried out. Although the use of the LN method significantly improves the agreement with the exact results in the intermediate coupling region, we found that in the strong coupling region the SCQRPA results are closest to the exact ones.
Self-consistent quasiparticle random phase approximation (SCQRPA) is for the first time applied to a more level pairing case. Various filling situations and values for the coupling constant are considered. Very encouraging results in comparison with the exact solution of the model are obtained. The nature of the low-lying mode in SCQRPA is identified. The strong reduction of the number fluctuation in SCQRPA vs BCS is pointed out. The transition from superfluidity to the normal fluid case is carefully investigated.
A systematic comparison is conducted for pairing properties of finite systems at nonzero temperature as predicted by the exact solutions of the pairing problem embedded in three principal statistical ensembles, as well as the unprojected (FTBCS1+++SCQRPA) and Lipkin+++Nogami projected (FTLN1+++SCQRPA) theories that include the quasiparticle number fluctuation and coupling to pair vibrations within the self-consistent quasiparticle random-phase approximation. The numerical calculations are performed for the pairing gap, total energy, heat capacity, entropy, and microcanonical temperature within the doubly folded equidistant multilevel pairing model. The FTLN1+++SCQRPA predictions agree best with the exact grand-canonical results. In general, all approaches clearly show that the superfluid-normal phase transition is smoothed out in finite systems. A novel formula is suggested for extracting the empirical pairing gap in reasonable agreement with the exact canonical results.
We propose a description of pairing properties in finite systems within the canonical and microcanonical ensembles. The approach is derived by solving the BCS and self-consistent quasiparticle random-phase approximation with the Lipkin-Nogami particle-number projection at zero temperature. The obtained eigenvalues are embedded into the canonical and microcanonical ensembles. The results obtained are found in quite good agreement with the exact solutions of the doubly-folded equidistant multilevel pairing model as well as the experimental data for 56Fe^{56}\mathrm{Fe}56Fe nucleus. The merit of the present approach resides in its simplicity and its application to a wider range of particle number, where the exact solution is impracticable.
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.
We study the pairing energy in the ground state of a system of N particles occupying a symmetric two-level model space with a level degeneracy \ensuremath{\Omega}. Assuming a pairing Hamiltonian, we calculate the energies of the projected BCS states, using the exact, the Kamlah, and the Lipkin-Nogami particle number projection methods. We find that conclusions regarding the quality of the BCS approximation as well as various approximate projection methods, drawn from studying the N=\ensuremath{\Omega} case alone, are not valid in the more general N\ensuremath{\ne}\ensuremath{\Omega} case when the pairing interaction is weak.
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.