Theoretical Research on Hopf Insulators and Model Construction in One-Dimensional Phononic Crystals

Abstract

In this Letter, a theoretical model of the Hopf insulator is constructed to realize topologically protected boundary states and higher-order corner states via parameter-space modulation. The nontrivial topological order is characterized by the Hopf mapping; by tuning the Hamiltonian coupling parameters, phase transitions are induced, and the localized stability of the boundary and corner states is theoretically verified. Subsequently, this framework is mapped onto a phononic crystal platform, leveraging the tunability of artificial bands to construct a one-dimensional equivalent physical model of the Hopf insulator. Through the precise design of resonator geometries and coupling parameters, an effective correspondence of the topological bands is achieved. Analytical calculations and finite-element simulations exhibit excellent agreement in both band dispersions and eigenstate distributions, confirming the feasibility of this model in acoustic systems and the robustness of its topological states.