An eigenvalue-based method and determinant representations for general integrable XXZ Richardson-Gaudin models

TL;DR

This paper introduces a new eigenvalue-based numerical method for XXZ Richardson-Gaudin models, enabling efficient spectral analysis and determinant formulas for state normalization and form factors, applicable to larger systems.

Contribution

It extends eigenvalue-based techniques to the full class of XXZ Richardson-Gaudin models, providing universal determinant expressions independent of parametrization.

Findings

Fast and robust spectral computation avoiding singularities

Determinant formulas for normalization and form factors

Applicability to larger, more complex systems

Abstract

We propose an extension of the numerical approach for integrable Richardson-Gaudin models based on a new set of eigenvalue-based variables. Starting solely from the Gaudin algebra, the approach is generalized towards the full class of XXZ Richardson-Gaudin models. This allows for a fast and robust numerical determination of the spectral properties of these models, avoiding the singularities usually arising at the so-called singular points. We also provide different determinant expressions for the normalization of the Bethe Ansatz states and form factors of local spin operators, opening up possibilities for the study of larger systems, both integrable and non-integrable. These expressions can be written in terms of the new set of variables and generalize the results previously obtained for rational Richardson-Gaudin models and Dicke-Jaynes-Cummings-Gaudin models. Remarkably, these results are independent of the explicit parametrization of the Gaudin algebra, exposing a universality in the properties of Richardson-Gaudin integrable systems deeply linked to the underlying algebraic structure.