Exact Solution of the Isovector Neutron-Proton Pairing Hamiltonian

We present a new exactly solvable Hamiltonian with a separable pairing interaction and nondegenerate single-particle energies. It is derived from the hyperbolic family of Richardson-Gaudin models and possesses two free parameters, one related to an interaction cutoff and the other to the pairing strength. These two parameters can be adjusted to give an excellent reproduction of Gogny self-consistent mean-field calculations in the Hartree-Fock basis. Pairing is one of the most important ingredients of the effective nuclear interaction in atomic nuclei, as was recognized early on by Bohr et al. [1] in an attempt to explain the large gaps observed in even-even nuclei. They suggested that the newly proposed Bardeen-Cooper-Schriefer (BCS) [2] theory of superconductivity could be a useful tool in nuclear structure, although care should be taken with the violation of particle number in finite nuclei. Since then, BCS or the more general Hartree-Fock-Bogoliubov (HFB) theory, combined with effective or phenomenological nuclear forces, has been the standard tool to describe the low-energy properties of heavy nuclei. Improvements over BCS or HFB came through the restoration of broken symmetries, especially particle-number projection, which is still a problem not satisfactorily solved with density-dependent forces [3]. From a different perspective, Richardson found an exact solution of the constant-pairing problem with nondegenerate single-particle energies as early as 1963 [4]. Though highly schematic, the constant-pairing force has been used for decades in nuclear structure with several approximations [BCS, random-phase approximation (RPA), projected BCS (PBCS), etc.], but rarely resorting to the exact solution. Almost forgotten, the exact Richardson solution was recovered within the framework of ultrasmall superconducting grains [5], in which not only number projection but also pairing fluctuations were essential to describe the disappearance of superconductivity as a function of the grain size.By combining the Richardson exact solution with the integrable model proposed by Gaudin [6] for quantum spin systems, it was possible to derive three families of integrable models called Richardson-Gaudin (RG) models [7]. The rational family, extensively used since then, contains the Richardson model as a particular exactly solvable Hamiltonian, as well as many other exactly solvable Hamiltonians of relevance in quantum optics, cold-atom physics, quantum dots, etc. [8]. However, the other families did not find a physical realization until very recently when it was shown that the hyperbolic family could model a p-wave pairing Hamiltonian in a two-dimensional lattice [9], such that it was possible to study, with the exact solution, an exotic phase diagram having a nontrivial topological phase and a third-order quantum phase transition [10]. In this Rapid Communication, we will show that the hyperbolic family gives rise to a separable pairing Hamiltonian with two free parameters that can be adjusted to reproduce the properties of hea...

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“…[16] for well-bound nuclei. The inclusion of exact T = 1 proton-neutron pairing within this self-consistent approach is also possible [17]. …”

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Exactly solvable pairing Hamiltonian for heavy nuclei

et al. 2011

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“…When all the matrix elements of the interaction are considered of equal strength, the Hamiltonian (1) has SO(5) symmetry and its exact solutions, both for a degenerate and a non-degenerate single-particle spectrum, have been discussed extensively in the literature (e.g., see [13][14][15][16]). …”

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Quartet condensation and isovector pairing correlations inN=Znuclei

et al. 2012

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We propose a simple quartet condensation model (QCM) which describes with very high accuracy the isovector pairing correlations in self-conjugate nuclei. The quartets have an alpha-like structure and are formed by collective isovector pairs. The accuracy of the QCM is tested for N=Z nuclei for which exact shell model diagonalizations can be performed. The calculations are done with two isovector pairing forces, one extracted from standard shell model interactions and the other of seniority type, acting, respectively, upon spherical and axially-deformed single-particle states. It is shown that for all calculated nuclei the QCM gives very accurate values for the pairing correlations energies, with errors which do not exceed 1%. These results show clearly that the correlations induced by the isovector pairing in self-conjugate nuclei are of quartet type and also indicate that QCM is the proper tool to calculate the isovector proton-neutron correlations in mean field pairing models.

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“…From (10) and their solutions (15,16,17), it is easy to show that the operator K(λ) can be written as:…”

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“…The SU(3) version presented here is associated with an interaction between pairs forming a doublet, we call it spinorial pairing or T = 1/2 pairing. As it was done in [15] for the SO(5) algebra, from a linear combination of the rational integrals of motion (33), the following pairing Hamiltonian for spinorial pairs results:where the scalar product is σ=↑↓ P † σm P σn , and we have fixed the parameters ξ 1 = 1/g and ξ 2 = 0. In this particle-particle representation of the SU(3) algebra, the Dynkin labels of the highest weight states are associated with the number of unpaired particles in each single particle level m and their transformation properties under the subgroup SU T (2).…”

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“…Recently, some applications to physical problems of RG models based on higher rank algebras have been published. The detailed derivation of the exact solution was considered for the SO(5) algebra of proton-neutron pairing in [14] and with more generality including numerical applications in [15]. In [16] the RG model based on the non-compact SO(3,2) algebra is used in the context of the Interacting Boson Model II, which describes the interaction between two different species of bosons (π and ν) for each non-degenerated level.…”

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SU(3) Richardson–Gaudin models: three-level systems

Errea¹

2007

J. Phys. A: Math. Theor.

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Abstract. We present the exact solution of the Richardson-Gaudin models associated with the SU(3) Lie algebra. The derivation is based on a Gaudin algebra valid for any simple Lie algebra in the rational, trigonometric and hyperbolic cases. For the rational case additional cubic integrals of motion are obtained, whose number is added to that of the quadratic ones to match, as required from the integrability condition, the number of quantum degrees of freedom of the model. We discuss different SU(3) physical representations and elucidate the meaning of the parameters entering in the formalism. By considering a bosonic mapping limit of one of the SU(3) copies, we derive new integrable models for three level systems interacting with two bosons. These models include a generalized Tavis-Cummings model for three level atoms interacting with two modes of the quantized electric field.PACS numbers: 02.30. Ik, 03.65.Fd, 31.15.Hz, Submitted to: J. Phys. A: Math. Gen. IntroductionThe Richardson-Gaudin (RG) exactly solvable models [1,2] can be traced back to the exact solution of the BCS Hamiltonian given by Richardson in the early 1960's [3] and to the integrable spin model developed by Gaudin in the seventies [4]. They are diagonalizable by Bethe Ansatz techniques and are the so-called classical limit of two dimensional vertex models. We refer the reader to [5], where the connection between the RG models and the inhomogeneous XXX vertex model with twisted boundary conditions is established in detail for the case of the rank-1 SU(2) algebra. Previously, in [6] it was shown that the solution of the Gaudin models [those without linear term in the integrals of motion, cf.(19) below] associated with more general Lie algebras, can be used to get a solution of the corresponding Classical Yang-Baxter Equations (CYBE). This connection has been largely exploited to diagonalize the RG models (see [2,7] for some examples). In practice this method consists of using the already known solutions of the general-Lie-algebra Yang-Baxter Equations (obtained by using the Quantum Inverse Scattering method) to obtain the corresponding solution of the RG models by taking the classical limit of the respective Bethe equations. From a Conformal Field Theory context, Asorey, Falceto and Sierra [8] obtained the integrals SU(3) Richardson-Gaudin models 2 of motion and respective eigenvalues of the rational RG models for any simple Lie algebra as a limiting case of the Chern-Simons theory.Here we follow an alternative and more direct approach to diagonalize the RG models. Though essentially equivalent, it differs in practice. This approach, where no reference to the YBE is necessary, is similar to the presented by Ushveridze [9] for the rational case. The method is based on the introduction of an infinite dimensional algebra (the Gaudin algebra) associated with the Lie-algebra of a simple group. By taking a Casimir-like operator in the Gaudin algebra, one gets the transfer matrix of the YBE approach and, consequently, a set of independent quadratic ...

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Exact Solution of the Isovector Neutron-Proton Pairing Hamiltonian

Cited by 72 publications

References 19 publications

Exactly solvable pairing Hamiltonian for heavy nuclei

Exactly solvable pairing Hamiltonian for heavy nuclei

Quartet condensation and isovector pairing correlations inN=Znuclei

SU(3) Richardson–Gaudin models: three-level systems

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