Distinguishing topological corner modes in higher-order topological insulators of finite size

Topological corner modes (TCMs) can exist in two-dimensional square lattices with either quantized dipole moments or a quantized bulk quadrupole moment. In both cases, four TCMs, each localized at one corner, can be found degenerate at zero energy. Here, we analytically show that in finite-sized systems, adjacent TCMs can couple, with hopping strength and sign determined by the edge configuration, which is directly associated with the magnetic flux of the unit cell. Consequently, we found that the response functions of the coupled corner modes of different origins, i.e., quantized edge dipoles and a quantized bulk quadrupole, possess distinctive line shapes. The response functions can therefore be used as a hallmark to determine the corner modes' nature. We verify our findings in simulations using phononic crystals.