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

Number-operator approximation for pairing

C. Martin Vincent

Published Jan 1, 1983• 1 citation
Publisher

Abstract

Conservation of particle number in the number-operator approximation of Otsuka and Arima is shown to require a particular normalization for the pair-creation operator. Even with this normalization, two difficulties arise: (a) the average number of particles in a shell can easily violate the Pauli principle, and (b) the mean square fluctuation of the particle number can be negative. As a consequence of the tendency to Pauli violation, the approximate ground-state expectation of the single-particle energy per particle is too low and fails to increase with the number of particles.NUCLEAR STRUCTURE Number conservation and Pauli principle in number-operator approximation for pairing.

DOI
10.1103/physrevc.27.426

Related papers

Number-Conserving Approximations for the Theory of the Pairing Interaction in Nuclei. II

G. Do Dang, Abraham Klein

A previous study of the pairing interaction Hamiltonian by means of the equations of motion is extended. In the prior note, ground-state and single-quasiparticle properties were treated by an approximation method which was strictly particle conserving. According to the general theory, further improvement of the ground-state energies and single-particle occupation probabilities requires information about excited states of seniority zero and unity. A nonlinear, number-conserving random-phase approximation is developed for this purpose. Numerical calculations are carried out in a linearized version which is almost equivalent to the usual quasi-boson approximation. As part of the approximation, the conservation-of-particles condition is relaxed for the excited states. Results of the numerical study establish the consistency of the assumptions made and yield improved ground-state and single-particle properties. By comparison with the few models for which exact numerical results are available, the results of this paper are seen to be comparable with or superior to those of other improved approximation methods so far reported, especially so as the number of particles increases. Finally, it is pointed out that just as the results of the previous paper provide the input data for the present work, the results reported here can serve in like manner for more elaborate approximations, which are proposed.

Physical Review•Jul 22, 1966•23 citations

Many-particle systems VI. N-fermion systems: derivation of a lower-bound shell model

R J M Carr, H R Post

The programme of reducing N -particle problems to one-particle problems is extended to include the Pauli principle. To calculate the ground-state energy of a system of N fermions interacting by pair forces, a rigorous lower-bound shell model is derived. This shell model follows a building-up principle, and tends to improve with increasing number of particles as well as with increasing strength of interaction. For twenty particles antisymmetric in ordinary space interacting by square-well interaction of strength V 0 a 2 = 200 hslash 2 /2 m the shell model differs by less than 8% from the calculated upper bound, and hence a fortiori from the exact energy. In order to test the quality of the approximation for various interactions, it was necessary to calculate upper bounds.

Journal of Physics A Mathematical and General•Sep 1, 1968•10 citations

Normal-ordered k-body approximation in particle-number-breaking theories

Ripoche, J., A. Tichai, T. Duguet

The reach of ab initio many-body theories is rapidly extending over the nuclear chart. However, dealing fully with three-nucleon, possibly four-nucleon, interactions makes the solving of the A-body Schrodinger equation particularly cumbersome, if not impossible beyond a certain nuclear mass. Consequently, ab initio calculations of mid-mass nuclei are typically performed on the basis of the so-called normal-ordered two-body (NO2B) approximation that captures dominant effects of three-nucleon forces while effectively working with two-nucleon operators. A powerful idea currently employed to extend ab initio calculations to open-shell nuclei consists of expanding the exact solution of the A-body Schrodinger equation while authorizing the approximate solution to break symmetries of the Hamiltonian. In this context, operators are normal ordered with respect to a symmetry-breaking reference state such that proceeding to a naive truncation may lead to symmetry-breaking approximate operators. The purpose of the present work is to design a normal-ordering approximation of operators that is consistent with the symmetries of the Hamiltonian while working in the context of symmetry broken (and potentially restored) methods. Focusing on many-body formalisms in which U(1) global-gauge symmetry associated with particle-number conservation is broken (and potentially restored), a particle-number-conserving normal-orderedk-body (PNOkB) approximation of an arbitrary N-body operator is designed on the basis of Bogoliubov reference states. A numerical test based on particle-number-projected Hartree-Fock-Bogoliubov calculations permits to check the particle-number conserving/violating character of a given approximation to a particle-number conserving operator. The PNOkB approximation of an arbitrary N-body operator is formulated. Based on this systematic approach, it is demonstrated that naive extensions of the normal-ordered two-body (NO2B) approximation employed so far on the basis of symmetry-conserving reference states lead to particle non-conserving operators. Alternatively, the PNOkB procedure is now available to generate particle-number-conserving approximate operators. The formal analysis is validated numerically. Using the presently proposed PNOkB approximation, ab initio calculations based on symmetry-breaking and restored formalisms can be safely performed. The future formulation of an angular-momentum-conserving normal-ordered k-body approximation based on deformed Slater determinant or Bogoliubov reference states is envisioned.

•Feb 1, 2020•20 citations

Particle-number-conserving theory for nuclear pairing

Lu Jia

A new microscopic theory for nuclear pairing is proposed through the generalized density matrix formalism. Instead of the variation principle, we start from the Heisenberg equations of motion for density matrix operators ${K}_{12}\ensuremath{\equiv}{a}_{2}{a}_{1}$. The theory generalizes the well-known particle-particle random-phase approximation. Its analytical equations are as simple as that of the BCS theory and can be solved within a similar computer time if the model space is not too large. The current theory conserves the exact particle number and is valid at arbitrary pairing strength (including those below the BCS critical strength). It is also of interest to other mesoscopic systems such as ultrasmall metallic grains.

arXiv:1305.2697•Physical Review C•Oct 3, 2013•21 citations

Conservation of Particle Number in the Nuclear Pairing Model

Klaus Dietrich, Hans J. Mang, J. Pradal

The Euler-Lagrange equations corresponding to a Bardeen-Cooper-Schrieffer state that is an eigenstate of the number operator are derived and solved numerically for a $\ensuremath{\delta}$ interaction. The errors due to the nonconservation of particle number in the usual Bardeen-Cooper-Schrieffer theory are studied as a function of particle number, level density, and strength of the pairing interaction. A proof is given that for attractive pairing interactions the lowest energy solution corresponds always to real positive probability amplitudes ${v}_{\ensuremath{\nu}}$, ${u}_{\ensuremath{\nu}}$.

Physical Review•Jul 13, 1964•194 citations

Intrinsic Pair Operators and the Number-conserved Treatment of Pairing Correlations in Nuclei

Ueno, M, Hayashi, D, Miyanishi, Y

By introducing the intrinsic pair operators which commute with number fluctuation operator, a new formalism is given for the number-conserving description of the pairing correlations. The difficulty in the conventional RPA treatment for pairing rotation is removed.

•Nov 7, 2001•0 citations
Conservation of particle number in the number-operator approximation of Otsuka and Arima is shown to require a particular normalization for the pair-creation operator. Even with this normalization, two difficulties arise: (a) the average number of particles in a shell can easily violate the Pauli principle, and (b) the mean square fluctuation of the particle number can be negative. As a consequence of the tendency to Pauli violation, the approximate ground-state expectation of the single-particle energy per particle is too low and fails to increase with the number of particles.NUCLEAR STRUCTURE Number conservation and Pauli principle in number-operator approximation for pairing.
A previous study of the pairing interaction Hamiltonian by means of the equations of motion is extended. In the prior note, ground-state and single-quasiparticle properties were treated by an approximation method which was strictly particle conserving. According to the general theory, further improvement of the ground-state energies and single-particle occupation probabilities requires information about excited states of seniority zero and unity. A nonlinear, number-conserving random-phase approximation is developed for this purpose. Numerical calculations are carried out in a linearized version which is almost equivalent to the usual quasi-boson approximation. As part of the approximation, the conservation-of-particles condition is relaxed for the excited states. Results of the numerical study establish the consistency of the assumptions made and yield improved ground-state and single-particle properties. By comparison with the few models for which exact numerical results are available, the results of this paper are seen to be comparable with or superior to those of other improved approximation methods so far reported, especially so as the number of particles increases. Finally, it is pointed out that just as the results of the previous paper provide the input data for the present work, the results reported here can serve in like manner for more elaborate approximations, which are proposed.
The programme of reducing N -particle problems to one-particle problems is extended to include the Pauli principle. To calculate the ground-state energy of a system of N fermions interacting by pair forces, a rigorous lower-bound shell model is derived. This shell model follows a building-up principle, and tends to improve with increasing number of particles as well as with increasing strength of interaction. For twenty particles antisymmetric in ordinary space interacting by square-well interaction of strength V 0 a 2 = 200 hslash 2 /2 m the shell model differs by less than 8% from the calculated upper bound, and hence a fortiori from the exact energy. In order to test the quality of the approximation for various interactions, it was necessary to calculate upper bounds.
The reach of ab initio many-body theories is rapidly extending over the nuclear chart. However, dealing fully with three-nucleon, possibly four-nucleon, interactions makes the solving of the A-body Schrodinger equation particularly cumbersome, if not impossible beyond a certain nuclear mass. Consequently, ab initio calculations of mid-mass nuclei are typically performed on the basis of the so-called normal-ordered two-body (NO2B) approximation that captures dominant effects of three-nucleon forces while effectively working with two-nucleon operators. A powerful idea currently employed to extend ab initio calculations to open-shell nuclei consists of expanding the exact solution of the A-body Schrodinger equation while authorizing the approximate solution to break symmetries of the Hamiltonian. In this context, operators are normal ordered with respect to a symmetry-breaking reference state such that proceeding to a naive truncation may lead to symmetry-breaking approximate operators. The purpose of the present work is to design a normal-ordering approximation of operators that is consistent with the symmetries of the Hamiltonian while working in the context of symmetry broken (and potentially restored) methods. Focusing on many-body formalisms in which U(1) global-gauge symmetry associated with particle-number conservation is broken (and potentially restored), a particle-number-conserving normal-orderedk-body (PNOkB) approximation of an arbitrary N-body operator is designed on the basis of Bogoliubov reference states. A numerical test based on particle-number-projected Hartree-Fock-Bogoliubov calculations permits to check the particle-number conserving/violating character of a given approximation to a particle-number conserving operator. The PNOkB approximation of an arbitrary N-body operator is formulated. Based on this systematic approach, it is demonstrated that naive extensions of the normal-ordered two-body (NO2B) approximation employed so far on the basis of symmetry-conserving reference states lead to particle non-conserving operators. Alternatively, the PNOkB procedure is now available to generate particle-number-conserving approximate operators. The formal analysis is validated numerically. Using the presently proposed PNOkB approximation, ab initio calculations based on symmetry-breaking and restored formalisms can be safely performed. The future formulation of an angular-momentum-conserving normal-ordered k-body approximation based on deformed Slater determinant or Bogoliubov reference states is envisioned.
A new microscopic theory for nuclear pairing is proposed through the generalized density matrix formalism. Instead of the variation principle, we start from the Heisenberg equations of motion for density matrix operators ${K}_{12}\ensuremath{\equiv}{a}_{2}{a}_{1}$. The theory generalizes the well-known particle-particle random-phase approximation. Its analytical equations are as simple as that of the BCS theory and can be solved within a similar computer time if the model space is not too large. The current theory conserves the exact particle number and is valid at arbitrary pairing strength (including those below the BCS critical strength). It is also of interest to other mesoscopic systems such as ultrasmall metallic grains.
The Euler-Lagrange equations corresponding to a Bardeen-Cooper-Schrieffer state that is an eigenstate of the number operator are derived and solved numerically for a $\ensuremath{\delta}$ interaction. The errors due to the nonconservation of particle number in the usual Bardeen-Cooper-Schrieffer theory are studied as a function of particle number, level density, and strength of the pairing interaction. A proof is given that for attractive pairing interactions the lowest energy solution corresponds always to real positive probability amplitudes ${v}_{\ensuremath{\nu}}$, ${u}_{\ensuremath{\nu}}$.