A Mathematical Study of Topological Phases and Edge Modes in Mechanical Systems

Abstract

In this dissertation, we study topological wave propagation and localization in periodic media. Our main objective is to provide a rigorous mathematical theory to understand the topological phases and the edge modes in topological mechanical systems. In the first part, we study a one-dimensional periodic spring-mass system consisting of masses connected by springs with different spring constants. The band structure and the topological index, Zak phase, of the system are derived. By gluing two such semi-infinite periodic systems with distinct Zak phases, an interface is obtained. The existence of the edge mode with frequency in the common band gap of two semi-infinite systems is proved by using transfer matrices. In the second part, we study a two-dimensional mechanical spring-mass system on a honeycomb lattice. The dispersion relation is analyzed and the existence of Dirac points when the masses are identical is proved. We investigate the topological properties of the system when the inversion symmetry is broken by studying the valley Chern number. We provide a proof of the existence of edge modes that are localized at the interface of a joint system constructed from two semi-infinite systems with different valley Chern numbers. In the final part, we study a gyroscopic system obtained by replacing masses in two-dimensional honeycomb lattice with identical gyroscopes. It results in the breaking of time-reversal symmetry and creates nontrivial topological phases characterized by the Chern numbers. We numerically relate the Chern numbers with the sign of a parameter of the system, namely gravitational acceleration. A joint system is created from two semi-infinite systems with opposite sign Chern numbers. Transfer matrices and a formulation of a finite system obtained from a truncated system are developed in order to study the existence of edge modes. Numerical computations on the finite system provide an evidence of existence of edge modes with frequency in the common band gap of the gyroscopic system. The relationship between transfer matrices and the truncated system is also investigated, leading several open questions concerning the characterization of the edge states.