Classification of Photonic Bands using Stable Real-Space Invariants

Topological Quantum Chemistry (TQC) has been extensively applied to study and engineer the topology of photonic bands. Despite being quite successful for 1D and 2D photonic crystals (PhCs), a complete application of TQC to the lowest bands of 3D PhCs is precluded by the polarization singularity due to the transversality of light at zero frequency and momentum. One proposal to resolve this singularity involves adding unphysical auxiliary bands to the system. We adapt this approach to show that the lowest bands in a topologically trivial photonic crystal can be expressed as differences of elementary band representations. This allows us to introduce a form of stable equivalence to classify these bands. We show that the newly-introduced stable real-space invariants (SRSIs) naturally capture this equivalence. Using this framework, we then find examples of photonic bands whose symmetry properties have no electronic counterparts. Finally, we develop a tight-binding model and perform ab-initio simulation for one of these examples, and characterize the polarization singularity using Wilson loops.