  Long-lived dark states, in which an experimentally accessible qubit is not in
thermal equilibrium with a surrounding spin bath, are pervasive in solid-state
systems. We explain the ubiquity of dark states in a large class of
inhomogenous central spin models using the proximity to integrable lines with
exact dark eigenstates. At numerically accessible sizes, dark states persist as
eigenstates at large deviations from integrability, and the qubit retains
memory of its initial polarization at long times. Although the eigenstates of
the system are chaotic, exhibiting exponential sensitivity to small
perturbations, they do not satisfy the eigenstate thermalization hypothesis.
Rather, we predict long relaxation times that increase exponentially with
system size. We propose that this intermediate chaotic but non-ergodic regime
characterizes mesoscopic quantum dot and diamond defect systems, as we see no
numerical tendency towards conventional thermalization with a finite relaxation
time.
