Exceptional light propagation via generalized bulk-edge correspondence

In topological photonics, the bulk-edge correspondence is conventionally imported by direct analogy with electronic systems, overlooking the fundamentally distinct spacetime symmetries of Maxwell's and Schr\"{o}dinger's equations. In this work, we challenge this prevailing paradigm by demonstrating that non-trivial bulk topology alone is insufficient to guarantee localized edge states in photonic platforms. Using a Su-Schrieffer-Heeger-inspired photonic crystal, we unveil a generalized bulk-edge correspondence intrinsically shaped by the relativistic nature of electromagnetic waves. This constraint imposes a strict frequency cutoff, a feature fundamentally absent in electronic topological insulators, which enables a new regime of frequency-controlled spatial localization near the cutoff. Furthermore, we demonstrate that this generalized correspondence is polarization-dependent: transverse electric (TE) and transverse magnetic (TM) edge modes exist in different parameter regimes and exhibit distinct dispersion relations, including distinct zero-dispersion points. Our framework redefines the theoretical boundaries of topological photonics, unlocking new opportunities for polarization-selective dispersion engineering and robust pulse propagation in topological photonic platforms.