Photonic arbitrary Chern vectors

The hallmark feature of three-dimensional (3D) Chern insulators is the topological Chern vector, C = (C₁, C₂, C₃), which extends the celebrated Chern-type bulk-edge correspondence from 2D to 3D systems. However, topological Chern vectors have thus far been limited to the simplest form with a single nonzero component, C = (0, 0, C₃), which restricts chiral surface states to appear only on four side surfaces with nonchiral transport along the direction of the Chern vector. Here, we theoretically propose and experimentally realize a 3D photonic Chern insulator with a Chern vector C = (1, 1, -1) in a tilted gyromagnetic photonic crystal. Its multiple nonzero components enable chiral surface states on all six crystal surfaces, exhibiting genuine chiral transport in arbitrary directions and nonreciprocal topological negative or positive refraction at the hinges. Furthermore, we demonstrate that arbitrary Chern vectors can be achieved via supercell modulation, enabling multimodal transport of chiral and torus knot or link surface states within a single crystal. This work establishes a comprehensive framework for topological Chern vectors and opens possibilities for nonreciprocal photonic devices.