Crystallographic splitting theorem for band representations and fragile topological photonic crystals

Two complementary themes in foundational band theory have recently emerged in full force: (i) A particle traversing a loop in momentum space does not always return to itself, but may acquire a Berry phase that reflects the geometric curvature of wave functions. (ii) In the dual real-space perspective, a basic question regards whether electrons can form exponentially localized wave packets, known as Wannier functions. Here, the authors rigorously derive the correspondence between the Berry phase (in momentum space) and the symmetry constraint of Wannier functions for nearly all crystallographic space groups.