Quadratic operator relations and Bethe equations for spin-1/2 Richardson-Gaudin models

Understand

In this work we demonstrate how one can, in a generic approach, derive a set of $N$ simple quadratic Bethe equations for integrable Richardson-Gaudin (RG) models built out of $N$ spins-1/2.

• These equations depend only on the $N$ eigenvalues of the various conserved charges so that any solution of these equations defines, indirectly through the corresponding set of eigenvalues, one particular eigenstate.
• The proposed construction covers the full class of integrable RG models of the XYZ (including the subclasses of XXZ and XXX models) type realised in terms of spins-1/2, coupled with one another through $\sigma_i^x \sigma_j^x $, $\sigma_i^y \sigma_j^y $, $\sigma_i^z \sigma_j^z $ terms, including, as well, magnetic field-like terms linear in the Pauli matrices.
• The approach exclusively requires integrability, defined here only by the requirement that $N$ conserved charges $R_i$ (with $i = 1,2 \dots N$) such that $\left[R_i,R_j\right] =0 \ (\forall \ i,j)$ exist .