Exact nuclear pairing solution for large-scale configurations: I. The EP (v1.0) program at zero temperature

TL;DR

The paper presents EP v1.0, a robust tool that computes exact pairing solutions for large-scale nuclear configurations at zero temperature by leveraging an SU(2) quasispin formulation and binary representations of basis states. It introduces a fast, sparse-matrix construction and diagonalization workflow using ARPACK for large problems and LAPACK for smaller ones, along with a ground-state-block search to accelerate odd-system calculations and the Kahan summation for numerical stability. The method demonstrates strong agreement with established approaches (PairDiag and Richardson) while delivering substantial speedups and enabling configurations up to $\Omega \le 26$ (and theoretically up to $63$) levels on common hardware. This work enhances exact pairing calculations relevant to nuclear structure, reactions, and astrophysics, and lays groundwork for finite-temperature and parallel extensions. The code's accessibility and demonstrated efficiency position it as a practical alternative to mean-field-based methods in scenarios where exact particle-number-conserving solutions are essential.

Abstract

In this work, we present the ``EP code" (version 1.0), a user-friendly and robust computational tool. It computes the exact pairing eigenvalues and eigenvectors directly from the general nuclear pairing Hamiltonian, represented using SU(2) quasi-spin algebra with basis vectors in binary representation, at zero temperature for both odd and even deformed nucleon systems. In this initial release, the sparsity and symmetry of the pairing matrix are exploited for the first time to quickly construct the pairing matrix. The ARPACK and LAPACK packages are employed for the diagonalization of large- and small-scale sparse matrices, respectively. In addition, the calculation speed for odd nucleon systems is significantly improved by employing a novel technique to accurately identify the block containing the ground state in odd configurations. To ensure the high numerical stability, the Kahan compensation algorithm is employed. The current version of the EP code can efficiently expand the computational space to handle up to 26 doubly folded (deformed) single-particle levels and 26 nucleons on a standard desktop computer in approximately $10^2$ seconds with double precision. With sufficient computational resources, the code can process up to 63 deformed single-particle levels, which can accomodate from 1 to 63 nucleon pairs. The EP v1.0 code is also designed for future extensions, including the finite-temperature and parallel computations.

Figures (8)

• Figure 1: Sparse nature of the pairing matrix of $\hat{H}$. The left panel (a) shows the non-zero elements (solid dots) in the system of $N=\Omega=10$. The right panel (b) shows the log10 scale of the number of matrix rows ( Nrow) and elements. Wherein, the dimension of $\hat{H}$ is ${\tt Nrow} \times {\tt Nrow}$. Nele and NNZ are, respectively, the numbers of non-zero elements of the upper triangular part of $\hat{H}$ and that of the full $\hat{H}$, respectively, while NZE is the number of zeros in $\hat{H}$ versus $N=\Omega$, for $10 \le \Omega \le 26$ and $\Omega$ even.
• Figure 2: Illustration of all possible configurations of an even system with $N=\Omega =4$ (a), and an odd system with $\Omega =4$ and $N=3$ (b)-(e) in the binary representation.
• Figure 3: The first three eigenvalues of the configurations $\Omega = N = 14, \, 18,\, 22, \, 26$ obtained from the EP code for various $G$ values ranging from 0 to 1.
• Figure 4: Occupation numpers of the ground-state of the configurations $\Omega = N = 14, \, 18, \, 22, \, 26$ versus pairing strength $G$. The number in the circle indicates the number of level, i.e., $\Omega_j$, for $j = \overline{1,N}$.
• Figure 5: The pairing correlation energies (a) and gap (b) of the system obtained from the EP and PairDiag codes for the case where $\Omega = N = 26$, and $G$ are varied from 0 to 1 MeV.