Microscopic Definitions of Topological Data

Condensed matter physics rests its foundation on the notion of universal phases of matter. As it is not possible to understand the detailed motions of large collections of particles by tracking them individually, we must rely on the fact that, much of the time, these details are irrelevant to a comprehensive understanding of a macroscopic system. Rather, we must understand the collective behavior of materials via a small set of quantities which summarize this information. For topological phases of matter, we can understand their most basic properties from information called “topological data.” This topological data describes the phase of matter of these systems and carries with it a broad array of information about their behavior. Previously, the nature of this data was broadly understood, but in most cases was lacking a concrete interpretation in terms of the microscopic details of these systems. The goal of this thesis is to resolve this issue by giving concrete definitions of the topological data in terms of a small number of microscopic properties of these systems. These definitions serve not only as a tool to analyze these theories, but also bridge a conceptual gap between the abstract mathematical understanding of these phases of matter with the concrete physical models that physicists study. This thesis achieves this goal for two types of (2+1)D topological phases of matter: intrinsic topological phases and symmetry protected topological phases (SPTs).Intrinsically topologically ordered phases, exemplified by the fractional quantum Hall states, are characterized by long range entanglement. These systems have particle-like excitations known as anyons and their properties are summed up by information called the anyon data. The anyon data is the topological data associated with these theories, with the two most subtle pieces of data being called the F and R-symbols. In the first part of the thesis, I present microscopic definitions of all of the anyon data and then proceed to calculate the F and R-symbols in a variety of exactly solvable models. Furthermore, I show that these definitions are consistent with the known mathematical structure of the anyon data. Symmetry protected topological phases, exemplified by topological insulators, do not have long range entanglement, but are non-trivial because of their symmetries. These phases are characterized by robust modes which propagate along their boundary. The fact that the bulk ensures this non-trivial property of the boundary is a consequence of the bulk-boundary correspondence. In the second part of this thesis, I show how to make this correspondence precise by defining the bulk topological data using microscopic boundary theories. I do this for both bosonic and fermionic SPTs in 2+1 dimensions. I show how to implement these definitions in a variety of examples and, in particular, show that these definitions reproduce the known data for the (2+1)D topological insulator