TOPOLOGICAL PHASES OF MATTER

Topological phases of matter have been the focus of intense research for the past decades. Their intrinsic topological nature exhibits perturbation-resilient effects in macroscopic materials. These properties have applications that span sensing, electronics, spintronics, lasers, and quantum computing. They have also inspired the development of new mathematical models to study their underlying topological quantum field theory. In this work, we visit three aspects of topological phases of matter. First, we use tensor-network tools, along with the aid of Purdue's high-performance computing clusters, to simulate symmetry-protected topological phases in 1D. We update works on the realization of the Haldane phase in Cu$_2$O excitons. In addition, we related the group-cohomological classification of these phases to their order parameters. Second, we design 2D topological insulators with higher Chern numbers. This is achieved through distant physical interactions in accordance with algebraic geometry arguments. Third, we study 2D topological quantum error-correcting codes. In particular, we construct the boundaries, domain walls, and $0$D defects of Abelian twisted quantum doubles leveraging anyon condensation. We discuss these codes' advantages in correcting errors for quantum computing. We also give microscopic and macroscopic methods for computing their ground state degeneracy. These distinct but related studies illustrate the power of using physical arguments and mathematical proofs in the simulation, design, and computing aspects of topological phases.