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A two-dimensional quadrupole topological insulator on a square lattice is atypical example of a higher-order topological insulator. It hosts an edge statelocalized near each of its $90^{\circ}$ corners at an energy $E$ inside theband gap, where $E$ is set equal to zero for simplicity. Although theappearance of an edge state has been shown in simple systems with only$90^{\circ}$ corners, it is uncertain whether a similar localized state canappear at $E = 0$ near a complicated edge consisting of multiple $90^{\circ}$and $270^{\circ}$ corners. Here, we present a numerical method to determine thewavefunction of a zero-energy state localized near an arbitrary edge. Thismethod enables us to show that one localized state appears at $E = 0$ if theedge consists of an odd number of corners. In contrast, the energy of localizedstates inevitably deviates from $E = 0$ if the edge includes an even number ofcorners.

Zero-Energy State Localized near an Arbitrary Edge in Quadrupole Topological Insulators