The nuclear pairing problem: new perspectives

Zelevinsky, Vladimir; Volya, Alexander

doi:10.1016/j.nuclphysa.2005.02.047

Cited by 11 publications

(11 citation statements)

References 28 publications

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“…This is the most interesting model from the condensed matter point of view. However for the description of pairing in nuclei other choices of the parameters ǫ α are more natural 9,40,41,44 . These could be treated by a simple adaptation of our results.…”

Section: Discussionmentioning

confidence: 99%

Exact mesoscopic correlation functions of the Richardson pairing model

2008

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We study the static correlation functions of the Richardson pairing model (also known as the reduced or discrete-state BCS model) in the canonical ensemble. Making use of the algebraic Bethe ansatz formalism, we obtain exact expressions which are easily evaluated numerically for any value of the pairing strength up to large numbers of particles. We provide explicit results at half-filling and extensively discuss their finite-size scaling behavior

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Section: Discussionmentioning

confidence: 99%

Exact mesoscopic correlation functions of the Richardson pairing model

2008

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“…They therefore share common eigenstates so that, in principle, solving Eqs. (31) for negative values of g would give us the eigenstates of the central spin Hamiltonian. However, we follow the alternative road of inverting the spin quantization axis (ẑ → −ẑ) and solving Eqs.…”

Section: Central Spin Modelmentioning

confidence: 99%

“…for the occupied levels, but linearizing the Bethe Eqs. (31) tells us that every rapidity can be written as λ i = αi − g at weak coupling. We consequently solve Eqs.…”

Section: A Richardson Modelmentioning

confidence: 99%

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Gaudin models solver based on the correspondence between Bethe ansatz and ordinary differential equations

et al. 2011

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We present a numerical approach which allows the solving of Bethe equations whose solutions define the eigenstates of Gaudin models. By focusing on a new set of variables, the canceling divergences which occur for certain values of the coupling strength no longer appear explicitly. The problem is thus reduced to a set of quadratic algebraic equations. The required inverse transformation can then be realized using only linear operations and a standard polynomial root finding algorithm. The method is applied to Richardson's fermionic pairing model, the central spin model and generalized Dicke model.The correspondence between Bethe Ansatz / Ordinary Differential Equation (ODE) has been found some time ago 1 on the basis of similarity between T-Q system of the Bethe Ansatz and functional relations including spectral determinants on the ODE side. For extensive review we refer to 2 . The correspondence is related to the Langlands correspondence where the ODE part has to do with socalled Miura opers 3 . It has many interesting links with conformal field theory, quasi-exact solvability, supersymmetry and PT-symmetric quantum mechanics. It was also used it in the context of physics of cold atoms and quantum impurity problem 4 . Here we develop further this remarkable correspondence to facilitate the solving of otherwise difficult-to-solve Gaudin systems and implement it to several physically interesting situations.Indeed, numerous integrable models derived from a generalized Gaudin algebra have been used to describe the properties of physical systems. The Richardson fermionic pairing Hamiltonian 5 has explained most properties of superconducting nanograins 6,7 , numerous studies of the decoherence of a single electron spin trapped in a quantum dot rely on the central spin model 8 while the inhomogeneous Dicke models 9 have been used to study light-matter interaction in many cavity-based systems.The quantum integrability of such systems brings major simplifications to the structure of their eigenstates. In a system containing N degrees of freedom, they can be fully described by a number M of complex parameters we call rapidities. M is here a number of the same order as N despite the exponentially large Hilbert space. The possible values of this set of parameters can be obtained by finding solutions to a set of M coupled non-linear algebraic equation: the Bethe equations. However, analytically finding solutions to the Bethe equations remains impossible except in very specific cases. The problem is still numerically approachable but solving non-linear equations remains a challenge that only iterative methods can tackle.While previous efforts in designing algorithms have shown promising results [10][11][12][13][14], methods which are sufficiently fast and stable for systematically finding a large number of eigenstates remained elusive. As was shown by one of the authors, computing efficiently only a small fraction of the complete Hilbert space can be sufficient to access static 15 , dynamical 16 and even non-equilibrium dyna...

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“…The normalization factor χ N (1), occupation numbers n N 1 (6), pair-transfer amplitudes s N 1 (7), mean fields e N 1 (8) and g N 1 (9) are functions of the pair structures v 1 (2); their functional forms, as the "kinematics" of the system, have already been given in Eqs. (23) and (24) of Ref. [1] and are not repeated here.…”

Section: Formalismmentioning

confidence: 99%

Accuracy of the new pairing theory and its improvement

Jia¹

2013

Phys. Rev. C

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Recently I proposed [1] a new method for solving the pairing Hamiltonian with the pair-condensate wavefunction ansatz based on the Heisenberg equations of motion for the density matrix operators. In this work an improved version is given by deriving the relevant equations more carefully. I evaluate both versions in a large ensemble with random interactions, and the accuracy of the methods is given statistically in terms of root-mean-square derivations from the exact results. The widely used variational calculation is also done and the results and computing-time costs are compared.

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The nuclear pairing problem: new perspectives

Cited by 11 publications

References 28 publications

Exact mesoscopic correlation functions of the Richardson pairing model

Exact mesoscopic correlation functions of the Richardson pairing model

Gaudin models solver based on the correspondence between Bethe ansatz and ordinary differential equations

Accuracy of the new pairing theory and its improvement

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