Gaudin models solver based on the Bethe ansatz/ordinary differential equations correspondence

TL;DR

This paper introduces a numerical method for solving Bethe equations in Gaudin models, transforming the problem into quadratic algebraic equations that are easier to solve, demonstrated on several physical models.

Contribution

It develops a new approach that avoids divergences in Bethe equation solutions by using a novel variable set, simplifying the computational process.

Findings

Successfully applied to Richardson's fermionic pairing model

Effective for the central spin model and generalized Dicke model

Reduces complex equations to quadratic algebraic form

Abstract

We present a numerical approach which allows the solving of Bethe equations whose solutions define the eigenstates of Gaudin models. By focusing on a new set of variables, the canceling divergences which occur for certain values of the coupling strength no longer appear explicitly. The problem is thus reduced to a set of quadratic algebraic equations. The required inverse transformation can then be realized using only linear operations and a standard polynomial root finding algorithm. The method is applied to Richardson's fermionic pairing model, the central spin model and generalized Dicke model.