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:1710.07570

Comparison between the Lipkin-Nogami and Richardson solutions with complex single particle energy in the Lipkin's model

R. Id Betán

Published Oct 20, 2017• 0 citations
arXivHTMLPDF

Abstract

The pairing interaction is one of the most important contribution of the residual interaction and then, it is of major importance for the study of many-body systems. One can get solutions of the pairing Hamiltonian throught the Bardeen-Cooper-Schieffer (BCS) or the Lipkin-Nogami (LN) approximations but, the pairing Hamiltonian admit exact solution worked out by Richardson. Nuclei far away from the stability line have important correlations with the continuum part of the energy spectrum, due that the Fermi level is very close to the contiuum thershold. The correlations with the continuum can be included in the many-body description through the complex energy states, called Gamow states. In this work we compare the approximates and exact solutions of the pairing Hamiltonian in real and complex-energy representations. In the application of this formulation to the symmetric Lipkin model, we found that the LN solution is in a good agreement with the exact one; besides, the extension of the BCS solution to the complex energy plane gives solution even for strength below the critical one, which is purely imaginary.

DOI
10.48550/arxiv.1710.07570

Related papers

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

A Microcanonical Comparison of Richardsons Equations With The BCS Mean Field Approximation

David From

The exact solution of the reduced pairing Hamiltonian is compared to the BCS mean field approximation for low particle numbers in finite systems. A combination of two different methods for solving the Richardson equations is presented. One of the methods originally intended to be solved analytically is instead adapted to a numerical scheme. The comparison includes the pairing gap parameter, as approximated by the three-point formula in the exact solution, and a microcanonical investigation of systems with levels of varying degeneracies. The gaps as calculated within the two approaches is found to deviate more and more from each other with increasing pairing strength. However testing the three-point-formula with the BCS approximation suggests that this might be due to the formula itself. Furthermore the BCS approximation, when re-evaluating the gap for each excitation, does a very good job at reproducing the energy spectra as obtained for the Richardson solution of the highly excited states where the pairing effects are small. However for the low lying excitations the assessment is not so good. When keeping the gap constant for all excitations the excitation spectra of the BCS approximation broaden a lot more than that of the exact method. This procedure gives unreasonable results for the systems and low particle numbers considered.

•Jan 1, 2016•0 citations

Particle-number conservation in thermodynamic systems within the Richardson model

Naziha Benhamouda, N. H. Allal, M. Fellah +1 more

To describe thermal pairing systems, an approach that combines the modified BCS method and the Lipkin-Nogami particle-number projection method is proposed. The expressions of the thermal pairing gap and internal energy are derived. The latter are numerically studied as a function of the temperature within the Richardson model.

Journal of Physics Conference Series•Aug 1, 2008•0 citations

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

Nguyễn Quang Hưng, Nguyen Dinh Dang

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.

arXiv:0710.0935•Physical Review C•Nov 2, 2007•15 citations

Solution of a pairing problem in the continuum

Alexis Mercenne, N. Michel, J. Dukelsky +1 more

We present a generalized Richardson solution for fermions interacting with the pairing interaction in both discrete and continuum parts of the single-particle (s.p.) spectrum. The pairing Hamiltonian is based on the rational Gaudin model which is formulated in the Berggren ensemble. We show that solutions of the generalized Richardson equations are exact in the two limiting situations: (i) in the pole approximation and (ii) in the s.p. continuum. If the s.p. spectrum contains both discrete and continuum parts, then the generalized Richardson equations provide accurate solutions for the Gamow shell model.

arXiv:1612.00308•Physical Review C•Feb 27, 2017•6 citations

Analytical and numerical analysis of the complete Lipkin–Meshkov–Glick Hamiltonian

Giampaolo Co’, Stefano De Leo

The Lipkin–Meshkov–Glick is a simple, but not trivial, model of a quantum many-body system which allows us to solve the many-body Schrödinger equation without making any approximation. The model, which in its unperturbed case is composed only by two energy levels, includes two interacting terms. A first one, the [Formula: see text] interaction, which promotes or degrades pairs of particles, and a second one, the [Formula: see text] interaction, which scatters one particle in the upper and another in the lower energy level. In comparing this model with other approximation methods, the [Formula: see text] term interaction is often set to zero. In this paper, we show how the presence of this interaction changes the global structure of the system, generates degeneracies between the various eigenstates and modifies the energy eigenvalues structure. We present analytical solutions for systems of two and three particles and, for some specific cases, also for four, six and eight particles. The solutions for systems with more than eight particles are only numerical but their behavior can be well understood by considering the extrapolations of the analytical results. Of particular interest is the study of how the [Formula: see text] interaction affects the energy gap between the ground state and the first-excited state.

arXiv:1805.12442•International Journal of Modern Physics E•Apr 24, 2018•12 citations
The pairing interaction is one of the most important contribution of the residual interaction and then, it is of major importance for the study of many-body systems. One can get solutions of the pairing Hamiltonian throught the Bardeen-Cooper-Schieffer (BCS) or the Lipkin-Nogami (LN) approximations but, the pairing Hamiltonian admit exact solution worked out by Richardson. Nuclei far away from the stability line have important correlations with the continuum part of the energy spectrum, due that the Fermi level is very close to the contiuum thershold. The correlations with the continuum can be included in the many-body description through the complex energy states, called Gamow states. In this work we compare the approximates and exact solutions of the pairing Hamiltonian in real and complex-energy representations. In the application of this formulation to the symmetric Lipkin model, we found that the LN solution is in a good agreement with the exact one; besides, the extension of the BCS solution to the complex energy plane gives solution even for strength below the critical one, which is purely imaginary.
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 exact solution of the reduced pairing Hamiltonian is compared to the BCS mean field approximation for low particle numbers in finite systems. A combination of two different methods for solving the Richardson equations is presented. One of the methods originally intended to be solved analytically is instead adapted to a numerical scheme. The comparison includes the pairing gap parameter, as approximated by the three-point formula in the exact solution, and a microcanonical investigation of systems with levels of varying degeneracies. The gaps as calculated within the two approaches is found to deviate more and more from each other with increasing pairing strength. However testing the three-point-formula with the BCS approximation suggests that this might be due to the formula itself. Furthermore the BCS approximation, when re-evaluating the gap for each excitation, does a very good job at reproducing the energy spectra as obtained for the Richardson solution of the highly excited states where the pairing effects are small. However for the low lying excitations the assessment is not so good. When keeping the gap constant for all excitations the excitation spectra of the BCS approximation broaden a lot more than that of the exact method. This procedure gives unreasonable results for the systems and low particle numbers considered.
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.
We present a generalized Richardson solution for fermions interacting with the pairing interaction in both discrete and continuum parts of the single-particle (s.p.) spectrum. The pairing Hamiltonian is based on the rational Gaudin model which is formulated in the Berggren ensemble. We show that solutions of the generalized Richardson equations are exact in the two limiting situations: (i) in the pole approximation and (ii) in the s.p. continuum. If the s.p. spectrum contains both discrete and continuum parts, then the generalized Richardson equations provide accurate solutions for the Gamow shell model.
The Lipkin–Meshkov–Glick is a simple, but not trivial, model of a quantum many-body system which allows us to solve the many-body Schrödinger equation without making any approximation. The model, which in its unperturbed case is composed only by two energy levels, includes two interacting terms. A first one, the [Formula: see text] interaction, which promotes or degrades pairs of particles, and a second one, the [Formula: see text] interaction, which scatters one particle in the upper and another in the lower energy level. In comparing this model with other approximation methods, the [Formula: see text] term interaction is often set to zero. In this paper, we show how the presence of this interaction changes the global structure of the system, generates degeneracies between the various eigenstates and modifies the energy eigenvalues structure. We present analytical solutions for systems of two and three particles and, for some specific cases, also for four, six and eight particles. The solutions for systems with more than eight particles are only numerical but their behavior can be well understood by considering the extrapolations of the analytical results. Of particular interest is the study of how the [Formula: see text] interaction affects the energy gap between the ground state and the first-excited state.