  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
non-trivial 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.
