$\mathbb{Z}_{2}$ Topological Index for Continuous Photonic Materials

Electronic topological insulators are one of the breakthroughs of the 21st century condensed matter physics. So far, the search for a light counterpart of an electronic topological insulator has remained elusive. This is due to the fundamentally different natures of light and matter and the different spins of photons and electrons. Here, it is shown that the theory of electronic topological insulators has a genuine analogue in the context of light wave propagation in time-reversal invariant continuous materials. We introduce a Gauge invariant $Z_2$ index that depends on the global properties of the photonic band structure and is robust to any continuous weak variation of the material parameters that preserves the time-reversal invariance. A nontrivial $Z_2$ index has two possible causes: (i) the lack of smoothness of the pseudo-Hamiltonian in the ${\bf{k}} \to \infty$ limit, and (ii) the entanglement between positive and negative frequency eigenmode branches. In particular, it is proven that electric-type plasmas and magnetic-type plasmas are topologically inequivalent for a fixed wave polarization. We propose a bulk-edge correspondence that links the number of edge modes with the topological invariants of two continuous bulk materials, and present detailed numerical examples that illustrate the application of the theory.