Title: Mathematical Theories for Topological Phases and Edge Modes in Mechanical Systems

 

 

Abstract: We examine the topological phases of the spring-mass lattices when the spatial inversion symmetry of the system is broken and prove the existence of edge modes when two lattices with different topological phases are glued together. In particular, for the one-dimensional lattice consisting of an infinite array of masses connected by springs, we show that the Zak phase of the lattice is quantized, only taking the value 0 or π. We also prove the existence of an edge mode when two semi-infinite lattices with distinct Zak phases are connected. We characterize the valley Chern numbers for the two-dimensional honeycomb lattice when the masses on the lattice vertices are uneven. The existence of edge modes is proved for a joint honeycomb lattice formed by gluing two semi-infinite lattices with opposite valley Chern numbers together.