  In the recent years, there has been a drive towards the realization of
topological phases beyond conventional electronic materials, including phases
defined in more than three dimensions. We propose a way to realize 2nd Chern
number topological phases with photonic crystals simply made up of defect
resonators embedded within a regular lattice of resonators. In particular,
through a novel quasiperiodic spatial modulations in the defect radii, a defect
lattice possessing topologically nontrivial Chern bands with non-abelian berry
curvature living in four-dimensional synthetic space is proposed. This system
cannot be factorized by a direct product of two 1st Chern number models,
distinguishing itself from the Hofstadter model. Such photonic systems can be
easily experimentally realized with regular photonic crystals consisting of
dielectric rods in air.
