Nonlinear dynamics of square-root higher-order topological corner states in a decorated honeycomb lattice

Square-root higher-order topological insulators are systems whose corner states are inherited from a topological parent Hamiltonian through a square-root operation. Here, we investigate the linear and nonlinear dynamics of these states in a decorated honeycomb lattice. The squared Hamiltonian can be written as the direct sum of a honeycomb lattice and a breathing Kagome lattice, with the higher-order topology originating from the latter. In the linear regime, bulk polarization confirms the topological origin of the corner states. We show that tuning the coupling strengths gives rise to two corner-state branches in the band gaps, namely out-of-phase and in-phase states. We then demonstrate that the two branches exhibit distinct nonlinear evolution. These results are supported by linear stability analysis and noisy propagation simulations. Our work extends square-root higher-order topology into the nonlinear regime and points to new possibilities for controlling localized topological states in photonic platforms.