Integrability and quench dynamics of the spin-1 Central Spin Model

Central spin models provide idealized models of interactions between a central degree of freedom and a mesoscopic environment of surrounding spins. We show that the family of models with a spin-1 degree of freedom at the center and XX interactions of arbitrary strength with surrounding spins is integrable. Specifically, we derive an extensive set of conserved quantities, and obtain the exact eigenstates using the Bethe ansatz. As in the homogenous limit, the states divide into two exponentially large classes: emph{bright} states, in which the spin-1 is entangled with its surroundings, and emph{dark} states, in which it is not. The bright states further break up into two classes on resonance; these classes prevent the central spin from equilibrating on quenching to resonance. In the single spin-flip sector, analytical computations of real-time dynamics are feasible using the explicit form of the Bethe states.

We discuss how the structure of the integrability differs from that of the spin-1/2 model and the closely related class of Richardson-Gaudin models, and conjecture that the spin-$S$ central spin XX model is integrable for any $S$.