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

Physical Review C

Exact results for the particle-number-projected BCS approach with isovector proton-neutron pairing

Apolodor Aristotel Raduta, M. I. Krivoruchenko, Amand Faessler

Published May 17, 2012• 21 citations
Publisher

Abstract

The mean values of a many-body Hamiltonian including a proton-neutron pairing term and matrix elements of one-, two-, and four-body operators within a basis of particle-number-projected BCS states are analytically expressed in terms of a single function $Q(N)$ depending on the number of particles, $N$. The function $Q(N)$ is calculated using a recursion in $N$ in which the shells and the BCS angles are kept the same for any iteration step. An illustrative example is numerically considered in a restricted single-particle space. Some specific features of the standard BCS, the projection after variation approach, and the variation after projection formalism are pointed out.

DOI
10.1103/physrevc.85.054314

Related papers

Some exact results for the particle number projected BCS approach of the isovector proton-neutron pairing

A. A. Raduta, M. I. Krivoruchenko, Amand Faessler

The mean values of a many-body Hamiltonian including a proton-neutron pairing term and matrix elements of one-, two- and four-body operators within a basis of particle number projected BCS states, are analytically expressed in terms of a single function Q(N) depending on the number of particles, $N$. The function Q(N) is calculated using a recursion in $N$ in which the shells and the BCS angles are kept the same for any step of iteration. An illustrative example is numerically considered in a restricted single particle space. Some specific features for the standard BCS, the projection after variation approach as well as for the variation after projection formalism, are pointed out.

arXiv:1201.6297•Jan 30, 2012•0 citations

Exact Solution of the Isovector Neutron-Proton Pairing Hamiltonian

J. Dukelsky, V. G. Gueorguiev, P. Van Isacker +3 more

The complete exact solution of the T = 1 neutron-proton pairing Hamiltonian is presented in the context of the SO(5) Richardson-Gaudin model with nondegenerate single-particle levels and including isospin symmetry-breaking terms. The power of the method is illustrated with a numerical calculation for for 64Ge for a pf + g9/2 model space which is out of reach of modern shell-model codes.

arXiv:nucl-th/0601082•Physical Review Letters•Feb 23, 2006•57 citations

Analytical approach for the quartet condensation model

V. Baran, Doru Sabin Delion

Background: Within the quartet condensation model (QCM), the isovector pairing correlations for $N=Z$ nuclei are described with a very high accuracy by a ``condensate'' of $\ensuremath{\alpha}$-like quartets. The usual approach involves cumbersome recurrence relations in order to compute numerically the relevant quantities of the model: the norm of the quartet states and the mean value of the isovector pairing Hamiltonian as functions of the pair mixing amplitudes.Purpose: We present the final analytical expressions for the above-mentioned quantities, for all cases up to four quartets in the valence shell.Method: The analytical QCM expressions were obtained by a straightforward implementation of the SO(5) algebra in the symbolic computer algebra system cadabra2.Results: The norm of the quartet states and the mean value of the Hamiltonian are polynomial functions of the mixing amplitudes. The numerical implementation of the QCM model is thus made trivial as a matter of copying and pasting the presented formulas.Conclusions: We introduce in this work a method of computer-aided analytical calculus for a many-body setting. In particular, we provide precise and easy-to-use tools for the description of isovector pairing correlations.

arXiv:1901.04759•Physical Review C•Mar 13, 2019•10 citations

DENSITY FUNCTIONAL FOR PAIRING WITH PARTICLE NUMBER CONSERVATION

Denis Lacroix A, Guillaume Hupin

In this work, a new functional is introduced to treat pairing correlations in finite many-body systems. Guided by the projected BCS framework, the energy is written as a functional of occupation numbers. It is shown to generalize the BCS approach and to provide an alternative to Variation After Projection framework. Illustrations of the new approach are given for the pairing Hamiltonian for various particle numbers and coupling strengths. In all case, a very good agreement with the exact solution is found.

arXiv:1005.0300•Modern Physics Letters A•Jul 19, 2010•0 citations

Improvement to the Projected BCS Approximation

Stuart Pittel, J. Dukelsky, Carlos Esebbag

We consider the structure of the number projected BCS wave function in the particle-hole basis, and use it to study several approximate treatments of pairing. The analysis is carried out for the exactly solvable Richardson model involving a pure pairing hamiltonian acting in a space of equally spaced doubly degenerate levels at half filling.

Journal of Physics Conference Series•Sep 14, 2015•0 citations

Variational Approach to Configuration Interaction

Richard R. Chasman

A variational method is developed to determine configuration interaction wave functions. The method is straightforward and is applied to a pairing Hamiltonian with constant matrix elements, for which exact eigenvalues are available. Comparisons are made with the exact results. Calculations can be carried out to any desired degree of accuracy. The method is also applied to a Hamiltonian that has neutron-neutron, proton-proton, and neutron-proton pairing. No difficulties are found in extending the method to this Hamiltonian that has many collective modes. In practice, the method scales linearly with , where is the number of variational basis states.

Physical Review Letters•Dec 20, 2005•5 citations
The mean values of a many-body Hamiltonian including a proton-neutron pairing term and matrix elements of one-, two-, and four-body operators within a basis of particle-number-projected BCS states are analytically expressed in terms of a single function Q(N)Q(N)Q(N) depending on the number of particles, NNN. The function Q(N)Q(N)Q(N) is calculated using a recursion in NNN in which the shells and the BCS angles are kept the same for any iteration step. An illustrative example is numerically considered in a restricted single-particle space. Some specific features of the standard BCS, the projection after variation approach, and the variation after projection formalism are pointed out.
The mean values of a many-body Hamiltonian including a proton-neutron pairing term and matrix elements of one-, two- and four-body operators within a basis of particle number projected BCS states, are analytically expressed in terms of a single function Q(N) depending on the number of particles, NNN. The function Q(N) is calculated using a recursion in NNN in which the shells and the BCS angles are kept the same for any step of iteration. An illustrative example is numerically considered in a restricted single particle space. Some specific features for the standard BCS, the projection after variation approach as well as for the variation after projection formalism, are pointed out.
Background: Within the quartet condensation model (QCM), the isovector pairing correlations for N=ZN=ZN=Z nuclei are described with a very high accuracy by a ``condensate'' of $\ensuremath{\alpha}$-like quartets. The usual approach involves cumbersome recurrence relations in order to compute numerically the relevant quantities of the model: the norm of the quartet states and the mean value of the isovector pairing Hamiltonian as functions of the pair mixing amplitudes.Purpose: We present the final analytical expressions for the above-mentioned quantities, for all cases up to four quartets in the valence shell.Method: The analytical QCM expressions were obtained by a straightforward implementation of the SO(5) algebra in the symbolic computer algebra system cadabra2.Results: The norm of the quartet states and the mean value of the Hamiltonian are polynomial functions of the mixing amplitudes. The numerical implementation of the QCM model is thus made trivial as a matter of copying and pasting the presented formulas.Conclusions: We introduce in this work a method of computer-aided analytical calculus for a many-body setting. In particular, we provide precise and easy-to-use tools for the description of isovector pairing correlations.
In this work, a new functional is introduced to treat pairing correlations in finite many-body systems. Guided by the projected BCS framework, the energy is written as a functional of occupation numbers. It is shown to generalize the BCS approach and to provide an alternative to Variation After Projection framework. Illustrations of the new approach are given for the pairing Hamiltonian for various particle numbers and coupling strengths. In all case, a very good agreement with the exact solution is found.
A variational method is developed to determine configuration interaction wave functions. The method is straightforward and is applied to a pairing Hamiltonian with constant matrix elements, for which exact eigenvalues are available. Comparisons are made with the exact results. Calculations can be carried out to any desired degree of accuracy. The method is also applied to a Hamiltonian that has neutron-neutron, proton-proton, and neutron-proton pairing. No difficulties are found in extending the method to this Hamiltonian that has many collective modes. In practice, the method scales linearly with , where is the number of variational basis states.