  A chiral-coupled atomic chain of two-level quantum emitters allows strong
resonant dipole-dipole interactions, which enables significant collective
couplings between every other emitters. We numerically obtain the steady-state
phase diagram of such system under weak excitations, where interaction-driven
states of crystalline orders, edge or hole excitations, and dichotomy of chiral
flow are identified. We distinguish these phases by participation ratios and
structure factors, and find two critical points which relate to
decoherence-free subradiant sectors of the system. We further investigate the
transport of excitations and emergence of crystalline orders under
spatially-varying excitation detunings, and present non-ergodic butterfly-like
system dynamics in the phase of extended hole excitations with a signature of
persistent subharmonic oscillations. Our results demonstrate the
interaction-induced quantum phases of matter with chiral couplings, and pave
the way toward simulations of many-body states in nonreciprocal quantum optical
systems.
