Exactly solvable pairing Hamiltonian for heavy nuclei

Abstract

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...

Structure of the number-projected BCS wave function

“…Likewise, more general pairing Hamiltonians are exactly solvable if they can be expressed as a linear combination of the set of integrals that define the Richardson-Gaudin models [8]. This exact solvability has enabled the test of approximate methods of treating pairing for a wide variety of systems, like small superconducting grains [9], realistic atomic nuclei [10][11][12][13], and more recently in quantum chemistry [14]. Such tests have illustrated that for a large enough number of active orbits and weak pairing, the PBCS approximation misses important pairing correlations, making its use in large-scale energy density functional treatments of finite nuclei suspect.…”

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

“…One class of these systems is the class of Richardson-Gaudin (RG) integrable systems, which can be derived from a generalized Gaudin algebra [1,2]. The pairing model in the reduced BCS approximation, used to describe superconductivity, has been shown to be RG integrable [3], as has the p x + ip y pairing Hamiltonian [4,5], the central spin model [6], factorizable pairing models in heavy nuclei [7], an extended d + id pairing Hamiltonian [8], and several atom-molecule Hamiltonians such as the inhomogeneous Dicke model [9,10]. For these models, diagonalizing the Hamiltonian in an exponentially scaling Hilbert space can be reduced to solving a set of nonlinear equations scaling linearly with system size.…”

Solution of a pairing problem in the continuum

“…states. In general, pairing models with the state-dependent pairing interaction are not integrable with the exception of the hyperbolic model [6,19] where the Gaussian weights w q should be a linear function of the s. p. energies q in order for the system to be exactly solvable. One has then to look for reliable approximations to the Hamiltonian (13) or to the commutation relations (10) for the non-resonant scattering states, which break the SU(2) commutator algebra, that could lead to an ansatz for an exact eigenstate.…”