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

Tight-binding (TB) models can accurately predict crystalline systems' band structure and topology. They have been heavily used in solid-state physics due to their versatility and low computational cost.

It is quite straightforward to build an accurate TB model of any crystalline system using the crystal's maximally localized Wannier functions (WF) as a basis.

Unfortunately, in 3D photonic crystals (PhCs), the transversality condition of Maxwell's equations precludes constructing a basis of maximally localized WF via usual techniques.

In this work, we show how to overcome this problem by using topological quantum chemistry, allowing us to express the band structure of the PhC as a difference of elementary band representations (EBRs).

This can be achieved by introducing a set of auxiliary modes recently proposed by Soljačić et al., which regularize the Γ-point obstruction arising from the transversality constraint of Maxwell's equations.

The decomposition into EBRs allows us to isolate a set of orbitals that permit us to construct an accurate transversality-enforced TB model that matches the dispersion, symmetry content, and topology of the 3D PhC 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 TB model.