Generalised Heine-Stieltjes and Van Vleck polynomials associated with degenerate, integrable BCS models

Abstract

We study the Bethe Ansatz/Ordinary Differential Equation (BA/ODE) correspondence for Bethe Ansatz equations that belong to a certain class of coupled, nonlinear, algebraic equations. Through this approach we numerically obtain the generalised Heine-Stieltjes and Van Vleck polynomials in the degenerate, two-level limit for four cases of exactly solvable Bardeen-Cooper-Schrieffer (BCS) pairing models. These are the s-wave pairing model, the p+ip-wave pairing model, the p+ip pairing model coupled to a bosonic molecular pair degree of freedom, and a newly introduced extended d+id-wave pairing model with additional interactions. The zeros of the generalised Heine-Stieltjes polynomials provide solutions of the corresponding Bethe Ansatz equations. We compare the roots of the ground states with curves obtained from the solution of a singular integral equation approximation, which allows for a characterisation of ground-state phases in these systems. Our techniques also permit for the computation of the roots of the excited states. These results illustrate how the BA/ODE correspondence can be used to provide new numerical methods to study a variety of integrable systems.

24 pages, 9 figures, 3 tables

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• Eigenvalue-based determinants for scalar products and form factors in Richardson-Gaudin integrable models coupled to a bosonic mode
• Algebraic Bethe Ansätze and eigenvalue-based determinants for Dicke-Jaynes-Cummings-Gaudin quantum integrable models
• Exact solution of the p+ip Hamiltonian revisited: duality relations in the hole-pair picture
• Common framework and quadratic Bethe equations for rational Gaudin magnets in arbitrarily oriented magnetic fields
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• Ground-state energies of the open and closed $p+ip$-pairing models from the Bethe Ansatz
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• Determinant representation of the domain-wall boundary condition partition function of a Richardson-Gaudin model containing one arbitrary spin
• Bethe ansatz solutions and hidden $sl(2)$ algebraic structure for a class of quasi-exactly solvable systems
• Integrable active atom interferometry
• On uniqueness of Heine-Stieltjes polynomials for second order finite-difference equations