  We investigate polariton-phonon hybrid excitations, which describe the
collective excitations of emitter-photon polaritons and vibrational phonons, in
a periodic array of vibrating two-level emitters interacting with waveguide
photons. We demonstrate the emergence of an interaction-induced quasiperiodic
structure caused by the interplay between phonon scatterings and
waveguide-mediated long-range couplings. This quasiperiodicity fundamentally
changes the excitation characteristics in the subradiant regime, which feature
an appearance of topological edge states and a transition between ergodic and
multifractal excitations. A possible realization consisting of an array of
laser-cooled atoms trapped near an optical nanofiber is also proposed. Our
results demonstrate the possibility of utilizing vibrations as a novel degree
of freedom in the exploration of many-body physics with waveguide quantum
electrodynamics systems.
