Realization of an acoustic square root topological insulator with high winding number boundary states

Square root topological insulators provide a unique framework for constructing unconventional topological phases through algebraic operations on parent Hamiltonians. Here, we experimentally realize an acoustic square root topological insulator based on an extended Su–Schrieffer–Heeger lattice incorporating long-range couplings. The square root transformation of the parent Hamiltonian leads to doubled bulk bands and symmetrically distributed in-gap boundary states. Combined full-wave simulations and acoustic measurements demonstrate that winding numbers of W = 1 and W = 2 correspond to twofold and fourfold degenerate edge modes, respectively. This work experimentally demonstrates the existence of such high degeneracy boundary states in an acoustic system through the combined effects of the square root operation and long-range coupling. The observed quadruple-degenerate states exhibit strong spatial localization and excellent agreement with theoretical predictions. Our results establish a direct bridge between algebraic topology and acoustic metamaterials, offering new opportunities for multi-channel topological waveguiding and robust acoustic energy localization.