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Bulletin of the Russian Academy of Sciences Physics

Pair transfer as a function of pairing strength in an exactly solvable model

А. К. Власников, Artem Lunev, V. M. Mikhajlov

Published Jul 1, 2013• 3 citations
Publisher

Abstract

Pair transfer matrix elements between the ground and excited states of a two-level system with monopole pairing interaction as a function of the pairing strength are calculated. The exact energies of excited states are compared with predictions of the quasiparticle random phase approximation. It is shown that at relatively low values of the pairing strength, two-particle transfer to the first and possibly second excited states must be taken into account for a more accurate description of pairing properties.

DOI
10.3103/s1062873813070277

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Pair transfer matrix elements between the ground and excited states of a two-level system with monopole pairing interaction as a function of the pairing strength are calculated. The exact energies of excited states are compared with predictions of the quasiparticle random phase approximation. It is shown that at relatively low values of the pairing strength, two-particle transfer to the first and possibly second excited states must be taken into account for a more accurate description of pairing properties.
Exact results of pair transfer probabilities for the Richardson model with equidistant or random level spacing are presented. The results are then compared either to particle-particle random-phase approximation (ppRPA) in the normal phase or quasiparticle random-phase approximation (QRPA) in the superfluid phase. We show that both ppRPA and QRPA are globally well reproducing the exact case although some differences are seen in the superfluid case. In particular, the QRPA overestimates the pair transfer probabilities to excited states in the vicinity of the normal-superfluid phase transition, which might explain the difficulty in detecting collective pairing phenomena as, for example, the giant pairing vibration. The shortcoming of QRPA can be traced back to the breaking of particle number that is used to incorporate pairing. A method based on direct diagonalization of the Hamiltonian in the space of two quasiparticles projected onto good particle number is shown to improve the description of pair transfer probabilities in superfluid systems.
Exact eigenstates of the pairing-force Hamiltonian have been calculated for 2N2N2N particles in 2N2N2N equispaced, doubly degenerate single-particle levels for N=4,8,12,and16N=4, 8, 12, \mathrm{and} 16N=4,8,12,and16. The states calculated are the ground state and the first two excited seniority-zero and seniority-two states. These states were calculated for values of the interaction strength that give pairing energies that are typical of nuclear systems. Numerical results are given for the energies and occupation probabilities of these states and are compared with the corresponding results of the Bardeen-Cooper-Schrieffer (BCS) theory. It is shown that the BCS theory is a worse approximation for the larger values of NNN than it is for the smaller values, with errors as high as 140% (20%) for the excitation energies of the seniority-zero (-two) states. The methods used in the calculation of these states are outlined.
A many-boson model is formulated and expressions for its exact eigenstates and energies are obtained for both an arbitrary finite and an infinite number of bosons. The Hamiltonian of the model contains interactions between bosons whose momenta have equal magnitudes but opposite directions. The matrix elements of this interaction are taken to be a constant over a range of momenta surrounding k = 0. The ground state of the 2N-particle system is shown to be a product of N pair-creation operators acting on the vacuum state. Each of these pair-creation operators depends upon one of N parameters which are called pair energies. The N pair energies are shown to satisfy a coupled system of nonlinear algebraic equations. The energy of the state is the sum of the pair energies and the occupation probabilities of the single-particle levels are given as simple functions of the pair energies. Similar results are derived for the excited states of the system and for the states of an odd number of particles. These results are valid for both a repulsive and an attractive interaction, since they only depend upon the form of the interaction. The equations are solved algebraically for two model systems. The first of these is one whose single-particle kinetic energy takes on only one value. The equations for this system are solved for an arbitrary interaction strength and it is shown that the pair energies are proportional to the zeros of certain Laguerre polynomials. The second system is one in which the single-particle kinetic energy can take on two values. The equations for this system are solved in the strong repulsive-interaction limit and it is shown that the pair energies are proportional to the zeros of certain Jacobi polynomials. The excitation energies of this second system are shown to be proportional to 1/n and the occupations of the two single-particle levels in the ground state are shown to be proportional to n, where n is the total number of particles. For a repulsive interaction and an arbitrary single-particle spectrum, the algebraic equations for the pair energies are converted into an approximate integral equation for the density of roots which is accurate to order 1/n. This integral equation is solved for a strong interaction which, in the context of this model, means an interaction whose strength is greater than a constant times 1/V⅔ in the limit of a large volume. From this solution, the following results are obtained: (1) the lowest two single-particle levels have occupations of order n; (2) the excitation spectrum is that of a set of noninteracting quasiparticles; and (3) the quasiparticle spectrum has two zeros corresponding to the lowest two single-particle levels. Apart from the presence of two zeros, the quasiparticle spectrum does not differ significantly from that of the noninteracting particles.
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.
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 ground state of a many body Hamiltonian considered in the quasiparticle representation is redefined by accounting for the quasiparticle quadrupole pairing interaction. The residual interaction of the newly defined quasiparticles is treated by the quasiparticle random phase approximation (QRPA). Solutions of the resulting equations exhibit specific features. In particular, there is no interaction strength where the first root is vanishing. A comparison with other renormalization methods is presented. Application to a single [Formula: see text]-shell allows for the results interpretation by comparing them with those obtained by exact calculations.