On the exact solvability of the anisotropic central spin model: An operator approach

Using an operator approach based on a commutator scheme that has been previously applied to Richardson's reduced BCS model and the inhomogeneous Dicke model, we obtain general exact solvability requirements for an anisotropic central spin model with $XXZ$-type hyperfine coupling between the central spin and the spin bath, without any prior knowledge of integrability of the model. We outline the basic steps of the usage of the operator approach, and pedagogically summarize them into two \emph{Lemmas} and two \emph{Constraints}. Through a step-by-step construction of the eigen-problem, we show that the condition $g'^2_j-g_j^2=c$ naturally arises for the model to be exactly solvable, where $c$ is a constant independent of the bath-spin index $j$, and $\{g_j\}$ and $\{g'_j\}$ are the longitudinal and transverse hyperfine interactions, respectively. The obtained conditions and the resulting Bethe ansatz equations are consistent with that in previous literature.