Identifying topological markers in gapless photonic systems with topological phase transitions at an interface

Photonic topological insulators exhibit bulk-boundary correspondence, which requires that boundary-localized states appear at the interface between topologically distinct insulating materials. However, in many topological photonic systems, this raises a subtle problem, as free-space is gapless for photons above the light-line. Although experiments have observed localized photonic edge states in this regime, their topology has been defined using theories which effectively approximate free-space to be insulating. Here, we show that even at the interface between a gapless material and a topological insulator these systems still exhibit both a topological phase transition and bulk-boundary correspondence. To do so, we employ the system's spectral localizer, which uses a system’s real-space description to calculate local topological markers and local topological protection independent of the material’s bulk band gap (or lack thereof). Ultimately, we show that topological photonic crystals can still demonstrate topological behavior and associated edge localizer resonances while in contact with free-space or any other gapless media.