Numerical algorithm for the standard pairing problem based on the Heine–Stieltjes correspondence and the polynomial approach

Guan, Xi‐Wen; Launey, Kristina D.; Xie, Mei; Bao, Lina; Pan, Feng; Draayer, J. P.

doi:10.1016/j.cpc.2014.05.023

Cited by 29 publications

(33 citation statements)

References 25 publications

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“…This limits the applicability of the methodology to relatively small systems. However, it has been shown recently that the set of Gaudin-Richardson equations for the standard pairing case can be solved relatively easily by using the extended Heine-Stieltjes polynomial approach [229,249]. Since solutions of the standard pairing model can be obtained from zeros of the associated extended HeineStieltjes polynomials, the approach can be applied to study the model with more pairs over a larger number of single-particle levels.…”

Section: Seniority Scheme and Exact Pairing Theorymentioning

confidence: 99%

Symmetry-guided large-scale shell-model theory

Launey

Dytrych

Draayer

2016

Progress in Particle and Nuclear Physics

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127

173

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In this review, we present a symmetry-guided strategy that utilizes exact as well as partial symmetries for enabling a deeper understanding of and advancing ab initio studies for determining the microscopic structure of atomic nuclei. These symmetries expose physically relevant degrees of freedom that, for large-scale calculations with QCD-inspired interactions, allow the model space size to be reduced through a very structured selection of the basis states to physically relevant subspaces. This can guide explorations of simple patterns in nuclei and how they emerge from first principles, as well as extensions of the theory beyond current limitations toward heavier nuclei and larger model spaces. This is illustrated for the ab initio symmetry-adapted no-core shell model (SA-NCSM) and two significant underlying symmetries, the symplectic Sp(3, R) group and its deformation-related SU(3) subgroup. We review the broad scope of nuclei, where these symmetries have been found to play a key role -from the light p-shell systems, such as 6 Li, 8 B, 8 Be, 12 C, and 16 O, and sd-shell nuclei exemplified by 20 Ne, based on first-principle explorations; through the Hoyle state in 12 C and enhanced collectivity in intermediate-mass nuclei, within a no-core shell-model perspective; up to strongly deformed species of the rare-earth and actinide regions, as investigated in earlier studies. A complementary picture, driven by symmetries dual to Sp(3, R), is also discussed. We briefly review symmetry-guided techniques that prove useful in various nuclear-theory models, such as Elliott model, ab initio SA-NCSM, symplectic model, pseudo-SU(3) and pseudo-symplectic models, ab initio hyperspherical harmonics method, ab initio lattice effective field theory, exact pairing -plusshell model approaches, and cluster models, including the resonating-group method. Important implications of these approaches that have deepened our understanding of emergent phenomena in nuclei, such as enhanced collectivity, giant resonances, pairing, halo, and clustering, are discussed, with a focus on emergent patterns in the framework of the ab initio SA-NCSM with no a priori assumptions.

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Section: Seniority Scheme and Exact Pairing Theorymentioning

confidence: 99%

Symmetry-guided large-scale shell-model theory

Launey

Dytrych

Draayer

2016

Progress in Particle and Nuclear Physics

Self Cite

127

173

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“…(36) also suggests an alternative way of solving the equations. From the RG equations it is possible to obtain a Heine-Stieltjes equation for P (z), which can be used to numerically determine the polynomial in the monomial basis [18,39]. In this method it would not be necessary to start from the weak-coupling limit when numerically solving the equations, and only two coupled matrix equations need to be solved.…”

Section: Inverting the Transformationmentioning

confidence: 99%

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

et al. 2015

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We propose an extension of the numerical approach for integrable Richardson-Gaudin models based on a new set of eigenvalue-based variables [A. Faribault et al., Phys. Rev. B 83, 235124 (2011); O. El Araby et al., ibid. 85, 115130 (2012)]. Starting solely from the Gaudin algebra, the approach is generalized towards the full class of XXZ Richardson-Gaudin models. This allows for a fast and robust numerical determination of the spectral properties of these models, avoiding the singularities usually arising at the so-called singular points. We also provide different determinant expressions for the normalization of the Bethe ansatz states and form factors of local spin operators, opening up possibilities for the study of larger systems, both integrable and nonintegrable. Remarkably, these results are independent of the explicit parametrization of the Gaudin algebra, exposing a universality in the properties of Richardson-Gaudin integrable systems deeply linked to the underlying algebraic structure.

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“…where ζ is an additional quantum number introduced to label the ζ-th excitation state, the explicit operator form of S + (x (ζ) µ ) is still given by (8), which was also used in [20][21][22]. Using (6), (9), and (11), we can directly check that…”

Section: The Model and Its General Solutionmentioning

confidence: 99%

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

Section: The Model and Its General Solutionmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2017

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Exact solution of the nuclear mean-field plus separable pairing model is reexamined. New auxiliary constraints for solving the Bethe ansatz equations of the model are proposed. By using these auxiliary constraints, the Bethe ansatz form of eigenvectors of the mean-field plus separable pairing Hamiltonian with non-degenerate single-particle energies and non-degenerate separable pairing strengths purposed previously is verified. Since the solutions of the model with one-and two-orbit cases are known, verification of the solutions for these two special cases is made. In order to demonstrate structure and features of the solution, the model with three orbits in the ds-shell is taken as a nontrivial example, of which two-pair results and the ground state of the three-pair case are provided explicitly. Since the number of equations involved increases with the number of orbits and pairs, to solve these equations for large number of orbits and pairs seems still difficult.

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Numerical algorithm for the standard pairing problem based on the Heine–Stieltjes correspondence and the polynomial approach

Cited by 29 publications

References 25 publications

Symmetry-guided large-scale shell-model theory

Symmetry-guided large-scale shell-model theory

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

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

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