Quadrupole topological phases and filling anomaly in all-dielectric Lieb lattice photonic crystals

While higher-order photonic topological corner states typically are created in systems with nontrivial bulk dipole polarization, they could also be created in systems with vanishing dipole polarization but with nontrivial quadrupole topology, which though is less explored. In this work, we show that simple all-dielectric photonic crystals in the Lieb lattice can host a topologically nontrivial quadrupole bandgap. Through a combination of symmetry analysis of the eigenmodes and explicit calculations of the Wannier bands and their polarization using the Wilson loop method, we demonstrate that the Lieb photonic crystals can have a bandgap with vanishing dipole polarization but with nontrivial quadrupole topology. The nontrivial bulk quadrupole moment could result in edge-localized polarization and topological corner states in systems with open edges. Interestingly, the indices of the corner states show an unusual “3+1” pattern compared to previously known “2+2” pattern, and this new pattern leads to unusual filling anomaly when the corner states are filled. Our work could not only deepen our understanding about quadrupole topology in simple all-dielectric photonic crystals but could also offer new opportunities for practical applications in integrated photonic devices.