Topological Robust Corner States of a Two-Dimensional Square Lattice with $\mathbf C_{\mathbf 4}$ Symmetry in Fully Coupled Dipolar Arrays

Higher-order topological insulators(HOTIs) is an exciting topic. We constructed a square lattice dipole arrays, it supports out-of-plane and in-plane modes by going beyond conventional scalar coupling. In-plane modes naturally break $\mathrm C_{4}$ symmetry, we only studied the out-of-plane modes that maintain $\mathrm C_{4}$ symmetry. Due to the slowly decaying long-range coupling, we consider its fully coupled interactions by using the lattice sums technique and combined with the coupled dipole method (CDM) to study its topological properties in detail. Interestingly, even when the full coupling is considered, the topological properties of the system remain similar to those of the 2D Su-Schrieffer-Heeger(SSH) model, but very differently, it supports robust zero-energy corner states (ZECSs) with $\mathrm C_{4}$ symmetry, we calculate the bulk polarization and discuss in detail the topological origin of the ZECSs. The lattice sums technique in the article can be applied to arbitrary fully coupled 2D dipole arrays. The materials we used can be able to confine light into the deep subwavelength scale, it has a great potential in enhancing light-matter interactions in the terahertz (THz) range.