Non-factorizable 4D quantum Hall state from photonic crystal defects
  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. 
Xiao Zhang, You Jian Chen, Bochen Guan, Jun Yu Lin, Nai Chao Hu, and   Ching Hua Lee
cond-mat.mes-hall cond-mat.mtrl-sci physics.optics