Integrability and quench dynamics in the spin-1 central spin XX model

TL;DR

This paper demonstrates that a spin-1 central spin XX model is integrable, deriving conserved quantities and exact eigenstates, and explores how this integrability influences quench dynamics and state classifications.

Contribution

It introduces the integrability of the spin-1 central spin XX model, providing explicit eigenstates and analyzing their impact on dynamics and state classification.

Findings

The model is integrable with an extensive set of conserved quantities.

Eigenstates divide into bright and dark classes with distinct entanglement properties.

Bright states cause oscillatory dynamics preventing thermalization.

Abstract

Central spin models provide an idealized description 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 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: bright states, in which the spin-1 is entangled with its surroundings, and dark states, in which it is not. On resonance, the bright states further break up into two classes depending on their weight on states with central spin polarization zero. These classes are probed in quench dynamics wherein they prevent the central spin from reaching thermal equilibrium. In the single spin-flip sector we explicitly construct the bright states and show that the central spin exhibits oscillatory dynamics as a consequence of the semilocalization of these eigenstates. We relate the integrability to the closely related class of integrable Richardson-Gaudin models, and conjecture that the spin-$s$ central spin XX model is integrable for any $s$.