An iterative approach for the exact solution of the pairing Hamiltonian

doi:10.1016/j.cpc.2022.108310

Computer Physics Communications

Volume 275, June 2022, 108310

https://doi.org/10.1016/j.cpc.2022.108310 Get rights and content

Abstract

A new iterative algorithm is established for the exact solution of the standard pairing problem, based on the Richardson-Gaudin method using the polynomial approach. It provides efficient and robust solutions for both spherical and deformed systems at a large scale. The key to its success is that the initial guess for the solutions of such a large set of the non-linear equations is provided in a physically meaningful and controllable manner. Moreover, one reduces the large-dimensional problem to a one-dimensional Monte Carlo sampling procedure, which improves the algorithm's efficiency and avoids the non-solutions and numerical instabilities that persist in most existing approaches. We calculated the ground state and low-lying excited states of equally spaced systems at different pairing strengths G. We then applied the model to study the quantum phase transitional Sm isotopes and the actinide nuclei Pu isotopes, where an excellent agreement with experimental data is obtained.

Program summary

Program Title: IterV1.m

CPC Library link to program files: https://doi.org/10.17632/rjnbhgk2p6.1

Licensing provisions: GPLv3

Programming language: Mathematica

Nature of problem: The program calculates exact pairing energies based on a new iterative algorithm. The key is the procedure of determining the initial guesses for the large-set non-linear equations involved in a controllable and physically motivated manner. It provides an efficient and robust solver for both spherical and deformed systems in super large model spaces.

Solution method: The new iterative algorithm approach starts with simple systems with k nucleon pairs and $n = k$ levels, which can be solved iteratively by including one pair and one level at each step using the Newton-Raphson algorithm with a Monte Carlo sampling procedure. Then it takes the solutions of those systems as initial values and obtain the converged results for the full space by gradually adding the remaining levels. In this way, one reduces the k-dimensional Monte Carlo sampling procedure to a one-dimensional sampling, which improves the efficiency of the algorithm and avoids the non-solutions and numerical instabilities.

Introduction

The introduction of pairing correlation represents a great pillar in our understanding of nuclear structure [1]. Nuclear pairing is now commonly accepted as one of the dominant residual correlations in atomic nuclei. It plays a key role not only in reproducing the ground-state properties of nuclei, including binding energies, excitation spectra, and level densities, odd-even staggering effects, single-particle occupancies, electromagnetic transition rates, and moments of inertia [2], [3], [4] but also in understanding nuclear reactions [5], and particle decays [6]. The simple Bardeen-Cooper-Schrieffer (BCS) approximation of superconductors and the generalized Hartree-Fock-Bogoliubov (HFB) approach are well known in describing the pairing properties of open-shell nuclei. Unlike electrons in solids, drawbacks of applying BCS theory and HFB methods to nuclei can be pronounced since the number of valence nucleons under the influence of the pairing force is too few to be treated by such particle-number nonconservation (quasi-particle) approximations. There have been efforts trying to overcome those drawbacks by using exact diagonalization [7], [8], [9], [10], [11], [12], [13], [14], [15] and particle-number conserved seniority approaches [16], [17], [18]. Nearly exact solutions may also be obtained through quantum Monte Carlo approaches [19], [20], [21]. On the other hand, the exact solution of the standard pairing problem was firstly obtained by Richardson [22] and is now referred to as the Richardson-Gaudin method [22], [23], [24]. We deem the Richardson-Gaudin method as the most promising approach for a microscopic treatment of clustering in heavy nuclei, the description of which requires a huge model space and a proper treatment of pairing and shell effects. In contrary to the exact diagonalization, the Richardson-Gaudin method has essentially no dimension limitation. Moreover, it provides structural information for each correlated pair, which can strongly influence the decay process. This paper presented an efficient new iterative algorithm for solving the standard pairing problem within the Richardson-Gaudin method in both spherical and deformed systems.

Section snippets

Exact solution to the pairing problem

For the standard pairing model, the Hamiltonian of is given as $\hat{H} = \sum_{j}^{n} ϵ_{j} {\hat{n}}_{j} - G \sum_{j j^{'}} S_{j}^{+} S_{j^{'}}^{-},$ where the sum runs over single-particle levels j with total number n, $G > 0$ is the overall pairing interaction strength, $ϵ_{j}$ are single-particle energies, ${\hat{n}}_{j} = \sum_{m} a_{j m}^{†} a_{j m}$ is the number operator for valence particles in the j-th level, and $S_{j}^{+} = \sum_{m} {(-)}^{j - m} a_{j m}^{†} a_{j - m}^{†}$ ( $S_{j}^{-} = {(S_{j}^{+})}^{†}$ ) are pair creation (annihilation) operators. For a simple system with doubly-degenerate levels (i.e., Nilsson levels in deformed nuclei), one

Numerical method

The new iterative approach to the exact pairing solutions can be implemented using a simple MATHEMATICA code (see Appendix A) with the following steps.

Step 0.
Input values for the number of levels n, the number of pairs k, pairing strength G and the initial values of $r o o t a_{0, \dots, k - 2} = a_{0, \dots, k - 1}$ , single-particle energies $ϵ_{k}$ , dimensionality $Ω_{1, \dots, n}$ , pairing strength G.
Step 1.
Construct symbolic polynomials $P (x) = \sum_{μ = 0}^{k} a_{μ} x^{μ}$ and it's derivative $P P (x) = {(\frac{P^{'} (x)}{P (x)})}^{'}$ in terms of the symbolic variables, x, ${a_{0}, \dots, a_{k}}$ ; Set $a_{k} = 1$ .
Step 2.

Comparison of approaches and numerical algorithms

We compare the runtimes of the Bethe-Ansatz approach (BAE), the Heine-Stieltjes polynomial approach (HS) in Refs. [29] and the present iterative approach with the single-particle energies are taken as equally spaced with $ϵ_{i} = i / 10$ MeV, $i = 0, 1, \dots, n$ for doubly-degenerate system with $Ω_{i} = 1 (j = 1 / 2)$ . Execution times are quoted for a $3.40 G H z$ CPU/32 GB RAM desktop computer with windows 10 and using MATHEMATICA v.11.2. The BAE approach is limited to only a few pairs. For example, for two pairs, it takes $0.5 s ($

Conclusion

In summary, a new iterative algorithm is developed for solving the standard pairing problem within the Richardson-Gaudin method. It is based on the polynomial approach, Eq. (8), and provides an efficient and robust solver for both spherical and deformed systems in super large model spaces. For that, we start with simple systems with k nucleon pairs and $n = k$ levels, which can be solved iteratively by including one pair and one level at each step using the Newton-Raphson algorithm with a Monte

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the Swedish Research Council (VR) under grant nos. 621-2012-3805, 621-2013-4323 and Natural Science Foundation of China (grant no. 12175097). Liaoning Provincial Universities Overseas Training Program (grant no. 2019GJWYB024).

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^☆: The review of this paper was arranged by Prof. Z. Was.

^☆☆: This paper and its associated computer program are available via the Computer Physics Communications homepage on ScienceDirect (http://www.sciencedirect.com/science/journal/00104655).

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Computer Physics Communications

An iterative approach for the exact solution of the pairing Hamiltonian☆,☆☆

Abstract

Program summary

Introduction

Section snippets

Exact solution to the pairing problem

Numerical method

Comparison of approaches and numerical algorithms

Conclusion

Declaration of Competing Interest

Acknowledgements

Prog. Part. Nucl. Phys.

Phys. Lett. B

Nucl. Phys. A

Phys. Lett. B

Comput. Phys. Commun.

Comput. Phys. Commun.

Phys. Rev. C

J. Phys.

Phys. Lett. B

Phys. Rev. Lett.

Phys. Rev. C

JPS Conf. Proc.

Phys. Rev. C

Nuclear Structure, vol. 1: Single-Particle Motion

Mat.-Fys. Medd. Danske Vid. Selsk.

The Nuclear Many-Body Problem

Rev. Mod. Phys.

JPS Conf. Proc.

Phys. At. Nucl.

Phys. Rev. C