Topological order in the fractional quantum Hall states

Abstract

This thesis is focused on the theoretical characterization of topological order in non-Abelian fractional quantum Hall (FQH) states. The first part of the thesis is concerned with the ideal wave function approach to FQH states, where the idea is to try to obtain model wave functions and model Hamiltonians for all possible FQH states and to have a physical way of characterizing their topological order. I will explain recent attempts to do this through the so-called pattern of zeros framework and its relation to conformal field theory. The first chapter about the pattern of zeros introduces the basic concepts for single-component FQH states, how it relates to the conformal field theory approach to FQH wave functions, and how it can be used to derive various topological properties of FQH states. The second chapter extends the pattern of zeros framework to multi-component non-Abelian FQH states; this is an attempt at a full classification of possible topological orders in FQH states. Aside from the ideal wave function methods. the other known general method of constructing non-Abelian FQH states is through the parton construction. Here the idea is to break apart the electron into other fermions, called partons. and assume that they form integer quantum Hall states. This method allows us to describe all known FQH states. After reviewing the parton construction, I will demonstrate how it can be used to derive the low energy effective field theories for some of the most well-known non-Abelian FQH states, the Zk parafermion (Laughlin/Moore-Read/Read-Rezayi) states. The parton construction will motivate yet another topological field theory, the U(1) x U(1) x Z2 Chern-Simons (CS) theory. I will demonstrate how to calculate many highly non-trivial topological properties of the U(1) x U(1) x Z2 CS theory, such as ground state degeneracy on genus g surfaces and various fusion properties of the quasiparticles. Using the U(1) x U(1) x Z2 CS theory, we will study phase transitions between bilayer Abelian states and non-Abelian states. The non-Abelian ones contain a series of new states, which we call the orbifold FQH states. These orbifold FQH states turn out to be important for the conceptual foundations of the pattern of zeros/vertex algebra approach to ideal FQH wave functions. We also find a series of non-Abelian topological phases - which are not FQH states and do not have protected gapless edge modes - that are separated from the deconfined phase of ZN gauge theories by a continuous phase transition. We give a preliminary analysis of these Z2 "twisted" ZN topological phases.