  We study the topological optical states in one-dimensional (1D) dimerized
ultracold atomic chains, as an extension of the Su-Schrieffer-Heeger (SSH)
model. By taking the fully retarded near-field and far-field dipole-dipole
interactions into account, we describe the system by an effective non-Hermitian
Hamiltonian, vastly different from the Hermitian Hamiltonian of the
conventional SSH model. We analytically calculate the complex bandstructures
for infinitely long chains, and show that the topological invariant, i.e., the
complex Zak phase, is still quantized and becomes nontrivial when the
dimerization parameter $\beta>0.5$, despite the broken chiral symmetry and
non-Hermiticity. We have verified the validity of the bulk-boundary
correspondence for this non-Hermitian system by further analyzing the
eigenstate distributions along with their inverse participation ratios (IPRs)
for finite chains, where topologically protected edge states are unambiguously
identified. We also reveal that such topological edge states are robust under
symmetry-breaking disorders. For transverse eigenstates, we further discover
the increase of localization length of topological edge states with the
increase of lattice period due to the presence of strong far-field
dipole-dipole interactions. Moreover, the ultra-strong scattering cross section
and ultra-narrow linewidth of a single cold atom allow us to observe in more
detail about topological states than in conventional systems, such as the
frequency shift with respect to the single-atom resonance and the largely
tunable bandgap. We envisage these topological photonic states can provide an
efficient interface between light and matter.
