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Tight-binding models can accurately predict the band structure and topologyof crystalline systems and they have been heavily used in solid-state physicsdue to their versatility and low computational cost. It is quitestraightforward to build an accurate tight-binding model of any crystallinesystem using the maximally localized Wannier functions of the crystal as abasis. In 1D and 2D photonic crystals, it is possible to express the waveequation as two decoupled scalar eigenproblems where finding a basis ofmaximally localized Wannier functions is feasible using standard Wannierizationmethods. Unfortunately, in 3D photonic crystals, the vectorial nature of theelectromagnetic solutions cannot be avoided. This precludes the construction ofa basis of maximally localized Wannier functions via usual techniques. In thiswork, we show how to overcome this problem by using topological quantumchemistry which will allow us to express the band structure of the photoniccrystal as a difference of elementary band representations. This can beachieved by the introduction of a set of auxiliary modes, as recently proposedby Solja\v{c}i\'c et. al., which regularize the $\Gamma$-point obstructionarising from transversality constraint of the Maxwell equations. Thedecomposition into elementary band representations allows us to isolate a setof pseudo-orbitals that permit us to construct an accuratetransversality-enforced tight-binding model (TETB) 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 theframework, modeled via non-minimal coupling to a static magnetic field. Ourwork provides the first systematic method to analytically model the photonicbands of the lowest transverse modes over the entire BZ via a TETB model.

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