Acoustic square-root topological states

Square-root topological states are new topological phases, whose topological property is inherited from the square of the Hamiltonian. Because of the square-root procedure, the bulk gap of the squared Hamiltonian is doubled. We realize the first-order and second-order square-root topological insulators in phononic crystals by putting additional cavities on the connecting tubes in the acoustic Su-Schrieffer-Heeger model and the honeycomb lattice, respectively. In the doubled bulk gaps, multiple acoustic localized modes, i.e., the end and corner states, are evidently confirmed by our calculations and experimental observations. We further propose a second-order square-root topological semimetal by stacking the decorated honeycomb lattice to three dimensions.