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Numerical algorithm for the standard pairing problem based on the Heine-Stieltjes correspondence and the polynomial approach. (English) Zbl 1360.82005

Summary: We present a detailed study of the computational complexity of a numerical algorithm based on the Heine-Stieltjes correspondence following the new approach we proposed recently for solving the Bethe ansatz (Gaudin-Richardson) equations of the standard pairing problem. For \(k\) pairs of valence nucleons in \(n\) non-degenerate single-particle energy levels, the approach utilizes that solutions of the Bethe ansatz equations can be obtained from two matrices of dimensions \((k + 1) \times(k + 1)\) and \((n - 1) \times(k + 1)\), 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 provides an efficient and systematic way to solve the problem. The method reduces to solving a system of \(k\) polynomial equations, which can be efficiently implemented by the fast Newton-Raphson algorithm with a Monte Carlo sampling procedure for the initial guesses. By extension, the present algorithm can also be used to solve a large class of Gaudin-type quantum many-body problems, including an efficient angular momentum projection method for multi-particle systems.

MSC:

82-04 Software, source code, etc. for problems pertaining to statistical mechanics
82B23 Exactly solvable models; Bethe ansatz
82B80 Numerical methods in equilibrium statistical mechanics (MSC2010)
65C05 Monte Carlo methods

Software:

exactPairingHS
PDFBibTeX XMLCite
Full Text: DOI

References:

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