Manipulating Topological Boundary States: From Corner-Bound to Arbitrarily Positioned Extended States.

Topological insulators with bulk-boundary correspondence have been widely explored in condensed-matter, photonic, and phononic systems. Two-dimensional second-order topological insulators are typically characterized by zero-dimensional boundary states localized at sharp corners. While these corner bound states have been widely explored across multiple platforms as robust signatures of second-order topology, their strong confinement to sharp corners severely limits practical applications. Here, we experimentally demonstrate control over topological boundary states in a honeycomb-lattice acoustic second-order topological insulator. Specifically, we reveal that the boundary Dirac mass exhibits a sensitive and diverse dependence on the edge types, enabled by sublattice degrees of freedom. By engineering the edges, we achieve not only arbitrary spatial relocation of topological boundary states from their conventional corner positions, but also precise, continuous tuning of their wave-function profiles from bound to extended as desired. Our Letter establishes a universal approach for manipulating topological boundary states, offering possibilities for designing advanced acoustic devices.