Spatial Inversion Kramers Degeneracy

Photonic crystals are artificial structures with periodic geometries that manipulate light at the wavelength scale. Recently, they have garnered significant attention in the field of topological photonics, where advanced band manipulation is explored to achieve breakthroughs in topological device applications. Conventional photonic crystals predominantly rely on a single spatial domain, characterized by spatially distributed dielectric modulation or metallic resonance. However, the degrees of freedom offered by spatial domains are often overlooked. Here, we theoretically propose and experimentally demonstrate photonic metacrystals that manipulate electromagnetic waves via minimal surfaces in three-dimensional space. These minimal surfaces partition the three-dimensional space into two complementary yet dual subspaces, each supporting a chiral topological semimetal. This spatial duality induces double-band degeneracy, analogous to the well-known time-reversal Kramers degeneracy, which has long been sought in classical systems. Previous proposals either relied on unrealizable strict symmetries or achieved double degeneracy only along high-symmetry lines. In contrast, we demonstrate that our double degeneracy exists globally, akin to the intrinsic spin degree of freedom of electrons. Furthermore, we extend the zero-frequency sixfold degeneracy of Maxwell's equations to a nonzero frequency range, which significantly facilitates the characterization of the intrinsic topological properties. Our study provides new insights into the geometric foundations of topological physics.