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Journal of Mathematical Physics

Pairing in the limit of a large number of particles

R. W. Richardson

Published Sep 1, 1977• 111 citations
Publisher

Abstract

Expressions for the ground and excited state energies of a system of nucleons interacting through pairing forces are given as a power series in inverse powers of the number of particles. The expressions are valid for systems with superfluid ground states and either J=0 or L=0 pairing. The first three terms in the expansion are given explicitly, and they exhibit excitations with both vibrational and rotational (in isospin space) character. Analytical and numerical results are given for a model system with a two-level single-particle spectrum.

DOI
10.1063/1.523493

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Journal of Mathematical Physics•Dec 1, 1969•550 citations
Expressions for the ground and excited state energies of a system of nucleons interacting through pairing forces are given as a power series in inverse powers of the number of particles. The expressions are valid for systems with superfluid ground states and either J=0 or L=0 pairing. The first three terms in the expansion are given explicitly, and they exhibit excitations with both vibrational and rotational (in isospin space) character. Analytical and numerical results are given for a model system with a two-level single-particle spectrum.
Equations for all the seniority-zero eigenstates of 2N2N2N nucleons in an arbitrary charge- and spin-independent potential well and interacting through charge- and spin-independent pairing forces are derived. These equations are solved exactly for a large number of states of this system. The interaction in this Hamiltonian is effective in the L=0L=0L=0 states of the two-nucleon system, and its strength is independent of the remaining quantum numbers of the two nucleons. We solve our equations exactly for those states whose wave functions are totally symmetric functions of the spin-isospin coordinates of the NNN L=0L=0L=0 pairs of nucleons in the state. The wave functions of these states factor into a spin-isospin-dependent part and a spatially dependent part. The spin-isospin-dependent part of one of these wave functions is an eigenvector of three tridiagonal matrices which insure that the state is a spin, isospin, and supermultiplet eigenstate, respectively. Explicit expressions are given for the eigenvalues of these three matrices in terms of the quantum numbers of the state. The spatially-dependent part of one of these wave functions is given explicitly in terms of NNN parameters which we call pair energies. These pair energies are shown to satisfy NNN coupled algebraic equations which depend parametrically upon the supermultiplet quantum numbers of the state. An expression for the occupation probabilities of the levels of the single-particle well is given. Throughout this work, an arbitrary splitting of the single-particle levels is treated exactly.
The problem of one pair of identical nucleons sitting in N{\cal N}N single particle levels of a potential well and interacting through the pairing force is treated introducing, in the Hamiltonian formalism, even Grassmann variables. The eigenvectors are analytically expressed solely in terms of these with coefficients fixed by the eigenvalues and the single particle energies. For these a specific model is needed: in the case of the harmonic oscillator well, for any strength of the pairing interaction, an accurate expression is derived for both the collective eigenvalue and for those trapped in between the single particle levels. Notably the latter are labelled through an index upon which they depend parabolically.
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.
We address the problem of two pairs of fermions living on an arbitrary number of single particle levels of a potential well (mean field) and interacting through a pairing force. The associated solutions of the Richardson’s equations are classified in terms of a number vl, which reduces to the seniority v in the limit of large values of the pairing strength G and yields the number of pairs not developing a collective behaviour, their energy remaining finite in the G → ∞ limit. We express analytically, through the moments of the single particle levels distribution, the collective mode energy and the two critical values G + cr and G − cr of the coupling which can exist on a single particle level with no pair degeneracy. Notably G + cr and G − cr merge when the number of single particle levels goes to infinity, where they coincide with the Gcr (when it exists) of a one pair system, not envisioned by the Richardson theory. In correspondence of Gcr the system undergoes a transition from a mean field to a pairing dominated regime. We finally explore the behaviour of the excitation energies, wave functions and pair transfer amplitudes finding out that the former, for G> G − cr, come close to the BCS predictions, whereas the latter display a divergence at Gcr, signaling the onset of a long range off-diagonal order in the system. PACS numbers: 21.60.-n, 21.30 Fe, 24.10.Cn 1 1
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.