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Canadian Journal of Physics

Number-Conserving Algebraic Method for Pairing Theory

Sing Chin Pang, Abraham Klein

Published Apr 1, 1972• 12 citations
Publisher

Abstract

Several powerful approaches to the study of pairing interactions are based on the relation of seniority conservation to the number-nonconserving algebra of quasi spins. In this note, we develop an algebraic number-conserving treatment utilizing the simplest number-conserving algebra, nonlinear under commutation, which follows from the quasi-spin algebra. The development is carried out in detail for a two-level model and yields encouraging numerical results in the simplest viable approximation consistent with the limit of strong pairing interaction.

DOI
10.1139/p72-090

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Several powerful approaches to the study of pairing interactions are based on the relation of seniority conservation to the number-nonconserving algebra of quasi spins. In this note, we develop an algebraic number-conserving treatment utilizing the simplest number-conserving algebra, nonlinear under commutation, which follows from the quasi-spin algebra. The development is carried out in detail for a two-level model and yields encouraging numerical results in the simplest viable approximation consistent with the limit of strong pairing interaction.
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.
Two advances are reviewed in the application of pairing-force theory in the nuclear shell model. The first exploits a discovery that a wide range of two-nucleon interactions conserve seniority as a good quantum number. Asa consequence, the eigenstates of a hamiltonian with such an interaction belong to irreducible representations of a compact unitary-symplectic group. This makes it possible to extend the simply-solvable models with J = 0 pairing forces to a much richer set of models and still obtain states uniquely classified by their seniority and angular momentum quantum numbers. Moreover, many of the low-lying energy levels of such models can be obtained algebraically; in technical terms, the models are in some cases completely solvable and in other cases partially solvable by algebraic methods. The second advance exploits the discovery that, in a coherent state representation, states of good nucleon number can be projected analytically from BCS vacuum and excited quasiparticle states. This makes it possible to perform calculations in a number-projected BCS basis without losing much of the advantage of working in a quasiparticle scheme.
The solvable two-level pairing model is used to test two new number-conserving approximations. These approximations are compared with the exact results, with perturbation theory, with the BCS method, and with the projected BCS method of Kerman, Lawson, and Macfarlane. It is found that second-order perturbation theory and one of the tested approximations work rather well for not too high values of the pairing-force strength, while for higher values the other approximation turns out to be quite good.
A practical number-conserving method of treating pairing correlations is proposed. The theory is formulated in a form having clear correspondence with quasiparticle theory and, moreover, it works well even in the weak pairing limit. Various matrix elements are approximately expressed in terms of the occupation probabilities containing the blocking effects within the number-conserving framework. The exact occupation probabilities can be used in the case of a constant pairing force. The present method is able to take the place of previous fixed-number BCS methods and is suitable for treating residual correlations further. Application to the cranking model is discussed.