Topological Band Transition and Corresponding Topologically Protected Interface States in Origami-based Chains

Abstract

In this thesis, a tunable one-dimensional chain is designed based on an origami structure with two degrees of freedom. We begin this analysis by constructing the Hamiltonian matrix for the origami unit cell. This is followed by obtaining the equations-of-motion matrix for the origami finite chain using the spring-mass model. An eigenvalue analysis is conducted on the matrix obtained by both methods, which provides the dispersion curve of the origami chain. We then polarize the dispersion curve for the purpose of understanding the effects of a coupled two degrees of freedom system on the bands and bandgaps. We use the polarization to distinguish the domination of the eigenmode, i.e., translational domination, or rotational domination, for each band. We calculate the Zak phase and the sign of the impendence for each band and bandgaps so that we can study the topological inversion before and after the transition. With the determination of the band’s Zak phase and the sign of impendence of bandgaps, we discuss how the interface occurs inside the bandgaps. Based on this conceptual study, we find the topologically protected interface state in certain configurations of this highly tunable origami chain. With these results, we can use this origami chain as an exceptional energy harvesting system and transfer the rotation motion to the translational motion or the other way around.