We study the unconventional topological phases of polaritons inside a cavity waveguide, demonstrating how strong light-matter coupling leads to a breakdown of the bulk-edge correspondence. Namely, we observe an ostensibly topologically nontrivial phase, which unexpectedly does not exhibit edge states. Our findings are in direct contrast to topological tight-binding models with electrons, such as the celebrated Su-Schrieffer-Heeger (SSH) model. We present a theory of collective polaritonic excitations in a dimerized chain of oscillating dipoles embedded inside a photonic cavity. The added degree of freedom from the cavity photons upgrades the system from a typical SSH $mathrm{SU}(2)$ model into a largely unexplored $mathrm{SU}(3)$ model. Tuning the light-matter coupling strength by changing the cavity size reveals three critical points in parameter space: when the polariton band gap closes, when the Zak phase changes from being trivial to nontrivial, and when the edge state is lost. Remarkably, these three critical points do not coincide, and thus the Zak phase is no longer an indicator of the presence of edge states. Our discoveries demonstrate some remarkable properties of topological matter when strongly coupled to light, and could be important for the growing field of topological nanophotonics.
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