Entangled four-dimensional multicomponent topological states from photonic crystal defects

Recently, 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 versatile and experimentally realistic approach of realizing a large variety of multicomponent topological phases in two-dimensional (2D) photonic crystals with quasiperiodically modulated defects. With a length scale introduced by a background resonator lattice, the defects are found to host various effective orbitals of $s$-, $p$-, and $d$-type symmetries, thus providing a monolithic platform for realizing multicomponent topological states without requiring separate internal degrees of freedom in the physical setup. Notably, by coupling the defect modulations diagonally, we report the realization of ``entangled'' 4D quantum Hall (QH) phases which cannot be factorized into two copies of 2D QH phases, each described by the first Chern number. The structure of this nonfactorizability can be quantified by a classical entanglement entropy inspired by quantum information theory. In another embodiment, we present 4D $p$-orbital nodal lines in a nonsymmorphic photonic lattice, hosting boundary states with an exotic manifold. Our simple and versatile approach holds the promise of topological optoelectronic and photonic applications such as one-way optical fibers.