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Journal of Physics Conference Series

Richardson-Gaudin description of pairing in atomic nuclei

Stijn De Baerdemacker

Published May 28, 2012• 1 citation
Publisher

Abstract

The present contribution discusses a connection between the exact Bethe Ansatz eigenstates of the reduced Bardeen-Cooper-Schrieffer (BCS) Hamiltonian and the multi-phonon states of the Tamm-Dancoff Approximation (TDA). The connection is made on the algebraic level, by means of a deformed quasi-spin algebra with a bosonic Heisenberg-Weyl algebra in the contraction limit of the deformation parameter. Each exact Bethe Ansatz eigenstate is mapped on a unique TDA multi-phonon state, shedding light on the physics behind the Bethe Ansatz structure of the exact wave function. The procedure is illustrated with a model describing neutron pairing in 56 Fe.

DOI
10.1088/1742-6596/366/1/012010

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The present contribution discusses a connection between the exact Bethe Ansatz eigenstates of the reduced Bardeen-Cooper-Schrieffer (BCS) Hamiltonian and the multi-phonon states of the Tamm-Dancoff Approximation (TDA). The connection is made on the algebraic level, by means of a deformed quasi-spin algebra with a bosonic Heisenberg-Weyl algebra in the contraction limit of the deformation parameter. Each exact Bethe Ansatz eigenstate is mapped on a unique TDA multi-phonon state, shedding light on the physics behind the Bethe Ansatz structure of the exact wave function. The procedure is illustrated with a model describing neutron pairing in 56 Fe.
A connection is made between the exact eigen states of the BCS Hamiltonian and the predictions made by the Tamm-Dancoff Approximation. This connection is made by means of a parametrised algebra, which gives the exact quasi-spin algebra in one limit of the parameter and the Heisenberg-Weyl algebra in the other. Using this algebra to construct the Bethe Ansatz solution of the BCS Hamiltonian, we obtain parametrised Richardson-Gaudin equations, leading to the secular equation of the Tamm-Dancoff Approximation in the bosonic limit. An example is discussed in depth.
Background: The reduced, level-independent, Bardeen-Cooper-Schrieffer Hamiltonian is exactly diagonalizable by means of a Bethe ansatz wave function, provided the free variables in the ansatz are the solutions of the set of Richardson-Gaudin equations. On the one side, the Bethe ansatz is a simple product state of generalized pair operators. On the other hand, the Richardson-Gaudin equations are strongly coupled in a nonlinear way, making them prone to singularities. Unfortunately, it is nontrivial to give a clear physical interpretation to the Richardson-Gaudin variables because no physical operator is directly related to the individual variables.Purpose: The purpose of this paper is to shed more light on the singular behavior of the Richardson-Gaudin equations, and how this is related to the product wave structure of the Bethe ansatz.Method: A pseudodeformation of the quasispin algebra is introduced, leading towards a Heisenberg-Weyl algebra in the contraction limit of the deformation parameter. This enables an adiabatic connection of the exact Bethe ansatz eigenstates with pure bosonic multiphonon states. The physical interpretation of this approach is an adiabatic suppression of the Pauli exclusion principle.Results: The method is applied to a so-called ``picket-fence'' model for the BCS Hamiltonian, displaying a typical critical behavior in the Richardson-Gaudin variables. It was observed that the associated bosonic multiphonon states change collective nature at the critical interaction strengths of the Richardson-Gaudin equations.Conclusions: The Pauli exclusion principle is the main responsible for the critical behavior of the Richardson-Gaudin equations, which can be suppressed by means of a pseudodeformation of the quasispin algebra.
We present a derivation of a previously announced result for matrix elements between exact eigenstates of the pairing Hamiltonian. Our results, which generalize the well-known Bardeen–Cooper–Schrieffer (BCS) (Bardeen et al 1957 Phys. Rev. 108 1175; 1957 Phys. Rev. 106 162) expressions for what are known as ‘coherence factors’, are derived based on the Slavnov (1989 Theor. Math. Phys. 79 502) formula for overlaps between Bethe-ansatz states, thus making use of the known connection between the exact diagonalization of the BCS Hamiltonian, due to Richardson (1963 Phys. Lett. 3 277; 1964 Nucl. Phys. A 52 221), and the algebraic Bethe ansatz. The resulting formula has a compact form after a suitable parameterization of the energy plane. Although we apply our method here to the pairing Hamiltonian, it may be adjusted to study what is termed the ‘Sutherland limit’ (Sutherland 1995 Phys. Rev. Lett. 74 816) for exactly solvable models, namely where a macroscopic number of rapidities form a large string.
Superconducting pairing of electrons in nanoscale metallic particles with discrete energy levels and a fixed number of electrons is described by the reduced Bardeen, Cooper, and Schrieffer model Hamiltonian. We show that this model is integrable by the algebraic Bethe ansatz. The eigenstates, spectrum, conserved operators, integrals of motion, and norms of wave functions are obtained. Furthermore, the quantum inverse problem is solved, meaning that form factors and correlation functions can be explicitly evaluated. Closed form expressions are given for the form factors and correlation functions that describe superconducting pairing.
An exact boson mapping of the deformed mean-field plus equal strength pairing Hamiltonian is considered. In the mapping, fermion pair operators are mapped exactly to the corresponding bosons. The mapping occurs at the level of the Richardson-Gaudin equations. The image of the mapping results is a Bose-Hubbard model with level-dependent hopping. Although the resultant Bose-Hubbard Hamiltonian is non-Hermitian, a part of spectrum of the Bose-Hubbard Hamiltonian with U/t = 1 determined by the corresponding Richardson-Gaudin (Bethe ansatz) equations corresponds exactly to the whole spectrum of the pairing Hamiltonian. © 2009 IOP Publishing Ltd.