The Renormalization Group and Quantum Hall Edge States

TL;DR

This paper explores how boundary conditions in quantum systems lead to edge states relevant for the quantum Hall effect, linking microscopic boundary choices to macroscopic quantized conductance and edge anomalies.

Contribution

It demonstrates how specific boundary conditions induce edge states and anomalies in quantum Hall systems through renormalization group analysis and second quantization.

Findings

Edge states arise from boundary conditions in Schrödinger systems.

Quantized Hall conductivity linked to edge anomalies.

Boundary conditions influence the nature of excitations in quantum Hall systems.

Abstract

The role of edge states in phenomena like the quantum Hall effect is well known. In this paper we show how the choice of boundary conditions for a one-particle Schr\"odinger equation can give rise to states localized at the edge of the system. We consider both the example of a free particle and the more involved example of a particle in a magnetic field. In each case, edge states arise from a non-trivial scaling limit involving the boundary conditions. Second quantization of these quantum mechanical systems leads to a multi-particle ground state carrying a persistent current at the edge. We show that the theory quantized with this vacuum displays an ``anomaly'' at the edge which is the mark of a quantized Hall conductivity in the presence of an external magnetic field. We also offer interpretations for the physics of such boundary conditions which may have a bearing on the nature of the excitations of these systems.