Bethe Ansatz and Ordinary Differential Equation Correspondence for Degenerate Gaudin Models

TL;DR

This paper extends numerical methods for Gaudin models to degenerate systems, demonstrating that high degeneracies simplify computations and enable stable, efficient root extraction without arbitrary precision, exemplified by the Richardson model.

Contribution

We generalize numerical techniques for Gaudin models to degenerate cases, introducing a barycentric polynomial basis for improved stability and efficiency.

Findings

Degeneracies reduce relevant Hilbert space states.

Sparse matrix equations simplify computations.

Barycentric polynomial basis enhances root extraction stability.

Abstract

In this work, we generalize the numerical approach to Gaudin models developed earlier by us to degenerate systems showing that their treatment is surprisingly convenient from a numerical point of view. In fact, high degeneracies not only reduce the number of relevant states in the Hilbert space by a non negligible fraction, they also allow to write the relevant equations in the form of sparse matrix equations. Moreover, we introduce a new inversion method based on a basis of barycentric polynomials which leads to a more stable and efficient root extraction which most importantly avoids the necessity of working with arbitrary precision. As an example we show the results of our procedure applied to the Richardson model on a square lattice.