Chern, dipole, and quadrupole topological phases of a simple magneto-optical photonic crystal with a square lattice and an unconventional unit cell

For the study of topological phases in photonics, while quantum-Hall-type first-order chiral edge states are routinely realized in magneto-optical photonic crystals, higher-order topological states are mostly explored in all-dielectric photonic crystals. In this work, we study both first- and second-order topological photonic states in magneto-optical photonic crystals. In specific, we revisit a simple magneto-optical photonic crystal in a square lattice with one gyromagnetic cylinder in each unit cell. However, rather than investigating the conventional unit cell where the cylinder is at the center of the square unit cell as previous works have done, we consider a configuration where the cylinders are located at the four corners of the square unit cell and show that this configuration hosts rich topological phases, such as dual-band Chern, dipole, and quadrupole topological phases. Our detailed characterizations of these topological states are based on the Wannier bands and their polarizations via the Wilson loop and nested Wilson loop methods. We study in detail both the edge and corner states of the different topological phases and show that they exhibit a special feature of "spectrum robustness." For example, though the edge and corner states of the dipole phases living in a band gap could be pushed into the bulk bands by tuning the boundary conditions, they can pass through the bulk bands and reappear within a different band gap. For dual-band quadrupole phases, we can find a regime where both band gaps host a set of corner states simultaneously and, intriguingly, the filling anomaly of one set of corner states can leave their signatures in the filling anomaly of the other set of corner states though they are separated by an extensive number of bulk states. The rich topological physics revealed in a simple magneto-optical photonic crystal not only provides new insights on higher-order topological phases in time-reversal symmetry-broken photonic systems, the results may also find promising applications by harnessing the potentials of both edge and corner states.