New tools to shape topological boundary states: Hidden symmetry and nonlinearity

Topological insulators have generated renewed interest in robust wave propagation across various physical systems. In this talk, we introduce two powerful tools—hidden symmetry and nonlinearity—that can shape novel topological boundary states in mechanical lattices. First, we explore spinner dimer lattices, where hidden chiral symmetry protects topological edge states with distinct localization profiles at opposite ends. Then, we investigate how introducing nonlinearity into a Kagome lattice allows for the tuning of higher-order topological states' frequency and stability, unveiling new families of states absent in linear systems. Together, these tools offer fresh insight into controlling the shape and stability of topological boundary states.