Elastic Trapped States at Dislocation Defects in Scaled Coupling and Hofstadter Models
Elastic topological dislocations provide a pathway for trapping elastic wave energy at internal defects, rather than being confined solely to external boundaries or corners, which are typically associated with topological insulators (TIs). However, two practical constraints persist. First, highly confined dislocation states based on conventional Su-Schrieffer-Heeger (SSH) dimerization usually require a large coupling contrast and a correspondingly enlarged bandgap, which may be challenging to realize. Second, some Hamiltonians with richer topological physics often contain complex hopping terms, synthetic gauge fields or nonlocal couplings, which substantially increase the geometric complexity of experimental samples. Here, dislocation-induced trapped states are demonstrated in both a scaled coupling (SC) model and a Hofstadter model (HM) within an elastic platform. In the SC model, the trapped mode is treated as a higher localized state in the continuum rather than an in-gap mode in the SSH model. Consequently, the SC-induced dislocation can trap an enhanced mode without the requirement of an enlarged bandgap. For the HM, Householder tridiagonalization is used to map the original tight-binding Hamiltonian with complex hopping terms onto a tridiagonal matrix with only positive-real-valued nearest-neighbour (NN) hopping terms. Truncation at a weak-hopping position preserves the topological phenomena and allows a dislocation defect to be constructed from the shortened aperiodic chain. The results establish a practical route for designing highly localized modes without relying solely on bandgap enlargement or complex couplings, which advance the topological physics of elastic wave systems and promise enhanced possibilities for elastic functional devices.