Square-root higher-order topological insulator on a decorated honeycomb lattice

Square-root topological insulators are recently proposed intriguing topological insulators, where the topologically nontrivial nature of Bloch wave functions is inherited from the square of the Hamiltonian. In this paper, we propose that higher-order topological insulators can also have their square-root descendants, which we term square-root higher-order topological insulators. There, emergence of in-gap corner states is inherited from the squared Hamiltonian which hosts higher-order topology. As an example of such systems, we investigate the tight-binding model on a decorated honeycomb lattice, whose squared Hamiltonian includes a breathing kagome-lattice model, a well-known example of higher-order topological insulators. We show that the in-gap corner states appear at finite energies, which coincides with the nontrivial bulk polarization. We further show that the existence of in-gap corner states results in characteristic single-particle dynamics; namely, setting the initial state to be localized at the corner, the particle stays at the corner even after a long time. Such characteristic dynamics may experimentally be detectable in photonic crystals.