Exact flat bands in a 3D photonic crystal

Photonic flat bands are hard to engineer because Maxwell's equations are vectorial: transversality obstructs the localized scalar-like bases that generate destructive-interference flat bands in tight-binding models.

We show that a three-dimensional metallic network of dipolar cavities joined by waveguide channels--a fully vectorial photonic crystal belonging to space group No.

224--hosts an exact scalar sector, carrying exact flat bands.

The twelve-band vector problem contains one self-adaptive radial dipole axis per site whose projection is exactly the scalar four-band Hamiltonian of the same network.

A microwave-scale coupled-dipole calculation confirms this scalar-vectorial duality.

The result is a symmetry-based design rule for scalar-like flat bands in reciprocal vector media.