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In the recent years, there has been a drive towards the realization oftopological phases beyond conventional electronic materials, including phasesdefined in more than three dimensions. We propose a way to realize 2nd Chernnumber topological phases with photonic crystals simply made up of defectresonators embedded within a regular lattice of resonators. In particular,through a novel quasiperiodic spatial modulations in the defect radii, a defectlattice possessing topologically nontrivial Chern bands with non-abelian berrycurvature living in four-dimensional synthetic space is proposed. This systemcannot be factorized by a direct product of two 1st Chern number models,distinguishing itself from the Hofstadter model. Such photonic systems can beeasily experimentally realized with regular photonic crystals consisting ofdielectric rods in air.

Non-factorizable 4D quantum Hall state from photonic crystal defects