The quantum Hall effect is recognized as one of the earliest examples of a topological phase of matter. Yet, thirty-five years after its initial discovery, there remain many open questions, especially surrounding states that may host fractional excitations and exotic statistics. Through the bulk-edge correspondence, many questions can be answered by studying the low-energy edge excitations. In this thesis, we investigate analytically certain aspects of the edge excitations using Chern-Simons-Landau-Ginzburg theory. The results include some surprises: our microwave absorption proposal leads to an interferometer whose read-out is first order in the tunneling amplitude; tunneling current across a quantum point contact is affected by the presence of a neutral mode; and the bulk-edge correspondence for chiral Abelian phases can be one-to-many. We now describe these investigations in more detail.
We start by proposing an experiment to measure the microwave absorption spectrum of a quantum Hall droplet. We show that the number and velocities of charged edge modes can be directly measured from a droplet of known shape. In contrast to standard transport measurements, different edge equilibration regimes can be accessed in the same device. If there is a quantum point contact, then quasiparticle properties, including braiding statistics, can be observed. Their effects are manifested as modulations of the spectrum that are, notably, first-order in the tunneling amplitude at the point contact.
We next consider transport through a quantum point contact in states with counter-propagating neutral edge modes. We show that both the noise and the average transmitted current are affected by downstream perturbations within the standard edge state model. We argue that the change in transmitted current should be observable in experiments that have observed increased noise.
Finally, we investigate the bulk-edge correspondence for chiral Abelian quantum Hall phases. We show that the same bulk two-dimensional topological phase can have multiple distinct, fully-chiral edge phases. This can happen for both integer and fractional quantum Hall states. We give a general criterion for the existence of multiple distinct chiral edge phases for the same bulk phase and discuss experimental consequences. We find that edge phases correspond to lattices while bulk phases correspond to genera of lattices. Since there are typically multiple lattices in a genus, the bulk-edge correspondence is typically one-to-many.