Transversality-Enforced Tight-Binding Models for 3D Photonic Crystals aided by Topological Quantum Chemistry

Tight-binding models can accurately replicate the band structure and topology of crystalline systems. They have been widely used in solid-state physics due to their versatility and low computational cost. It is straightforward to build an accurate tight-binding model of any crystalline system using the crystal's maximally localized Wannier functions as a basis. Unfortunately, in 3D photonic crystals, the transversality condition of Maxwell's equations precludes the construction of a basis of maximally localized Wannier functions via usual techniques. As a result, building reliable tight-binding models of 3D photonic crystals has not been straightforward up to now. In this work, we show how to overcome this problem using topological quantum chemistry, allowing us to express the band structure of the photonic crystal as a difference of band representations. This can be achieved by introducing a set of auxiliary modes, as recently proposed in Christensen et al., Phys. Rev. X 12, 021066 (2022), which regularizes the Gamma-point obstruction arising from the transversality constraint of Maxwell's equations. The decomposition into elementary band representations allows us to isolate a set of pseudo-orbitals that permit us to construct an accurate transversality-enforced tight-binding model that matches the dispersion, symmetry content, and topology of the 3D photonic crystal under study. Moreover, we show how to introduce the effects of a gyrotropic bias in the framework, modeled via non-minimal coupling to a static magnetic field. Our work provides the first systematic method to analytically model the photonic bands of the lowest transverse modes over the entire Brillouin zone via a transversality-enforced tight-binding model.