Heine-Stieltjes correspondence and the polynomial approach to the standard pairing problem

Abstract

A new approach for solving the Bethe ansatz (Gaudin-Richardson) equations of the standard pairing problem is established based on the Heine-Stieltjes correspondence. For k pairs of valence nucleons on n different single-particle levels, it is found that solutions of the Bethe ansatz equations can be obtained from one (k + 1) × (k + 1) and one (n − 1) × (k + 1) matrices, which are associated with the extended Heine-Stieltjes and Van Vleck polynomials, respectively. Since the coefficients in these polynomials are free from divergence with variations in contrast to the original Bethe ansatz equations, the approach thus provides with a new efficient and systematic way to solve the problem, which, by extension, can also be used to solve a large class of Gaudin-type quantum many-body problems and to establish a new efficient angular momentum projection method for multi-particle systems.PACS numbers: 21.60. Cs, 21.60.Fw, 03.65.Fd, 71.10.Li, 74.20.Fg, 02.60.Cb It is well known that the pairing force, similar to that in the Bardeen-Cooper-Schrieffer (BCS) theory of superconductors [1], as one of main residual interactions introduced to the nuclear shell model, is key to manifest ground state properties and low energy spectroscopy of nuclei, such as binding energies, odd-even effects, singleparticle occupancies, excitation spectra, electromagnetic transition rates, beta-decay probabilities, transfer reaction amplitudes, low-lying collective modes, level densities, and moments of inertia, and so on [2]- [4]. Unlike electrons in solids, the drawbacks of the application of the BCS theory and its extensions to nuclei are noticeable due to the fact that the number of valence nucleons under the influence of the pairing force is too few to be treated by such particle-number nonconservation (quasiparticle) approximations

Eigenvalue-based method and form-factor determinant representations for integrable XXZ Richardson-Gaudin models

“…Once the coefficients P N−m are known, the roots of this polynomial can be determined to find the variables. This method arises naturally for different problems in the theory of integrable systems, such as the Heine-Stieltjes connection [37,38], the numerical methods by Guan et al [18] and Rombouts et al [15], and the weak-coupling limit in RG models [2]. The definition of P (z) can now be used to consider…”

“…Unfortunately, the Bethe ansatz or Richardson-Gaudin equations [6,11,12] that need to be solved are highly nonlinear and give rise to singularities, making a straightforward numerical solution challenging [13,14]. Several methods have been introduced as a way to resolve this difficulty, such as a change in variables [14,15], a (pseudo)deformation of the algebra [16,17], or a Heine-Stieltjes connection, reducing the problem to a differential equation [18]. The ground-state energy in the thermodynamic limit has also been obtained by treating the interaction as an effective temperature [19].…”

Exact solution of the mean-field plus separable pairing model reexamined

“…Therefore, solutions of (20) or (21) can not be obtained easily as those in the standard pairing case shown in [7,8].…”

“…Since {a i (x (ζ) µ )} (i = 1, 2, · · · , q) and {x (ζ) µ,q=p−1 } are expressed in terms of {β(x (ζ) µ )} and {x (ζ) µ,i } (i = 1, 2, · · · , q − 1) according to (7), q equations given by (22) provide expressions of x (ζ) µ,i (i = 1, 2, · · · , q − 1) and the final expres-…”

“…It has been shown that either spherical or deformed mean-field plus the standard (orbit-independent) pairing interaction can be solved exactly by using the Gaudin-Richardson method [3][4][5]. The Gaudin-Richardson equations in this case can be solved more easily by using the extended Heine-Stieltjes polynomial approach [6][7][8][9]. The deformed and spherical mean-field plus the extended pairing models have also been proposed, which can be solved more easily than the standard pairing model, especially when both the number of valence nucleon pairs and the number of single-particle orbits are large [10,11].…”