Hilbert Space Fragmentation and Scar Time Crystallinity in Driven Homogeneous Central Spin Models

We study the stroboscopic non-equilibrium quantum dynamics of periodically driven central spin systems with XXZ-type interaction. The system exhibits a strong fragmentation of Hilbert space into a four-dimensional Floquet Krylov subspace and discrete time-translation symmetry breaking due to oscillations between two disjoint regions within each Krylov subspace. Analytical and numerical analyses of the stability of the fragmented subspaces reveal the existence of scar states corresponding to fully polarized satellite spins, which also possess high overlap with Floquet eigenstates having atypically low bipartite entanglement entropy. We show that the period-doubling indicative of time crystalline behavior arises for highly polarized initial states and persists in the presence of magnetic field disorder, pulse errors, and other perturbative effects. Our work is relevant for experiments on semiconductor quantum dots or defect spins hyperfine-coupled to surrounding nuclei.