Quantum Coherence of Edge States in the Quantum Hall Effect -- Topological Invariants and Edge State Mixing --

TL;DR

This paper reviews the topological nature of edge states in the quantum Hall effect, emphasizing the role of quantum coherence and edge state mixing in determining conductance, especially in mesoscopic systems.

Contribution

It clarifies the relation between bulk and edge topological invariants and explores the physical effects of edge state mixing in mesoscopic quantum Hall systems.

Findings

Hall conductance is given by a winding number of edge states.

Edge state mixture is negligible in macroscopic systems.

Quantum coherence influences conductance quantization in mesoscopic systems.

Abstract

Edge states in the integral quantum Hall effect on a lattice are reviewed from a topological point of view. For a system with edges which is realized inevitably in an experimental situation, the Hall conductance $\sigma_{xy}$ is given by a winding number of the edge state on a complex energy surface. A relation between two topological invariants (bulk and edge) is also clarified. In a macroscopic system, mixture of the edge states are exponentially small and negligible. Quantum Coherence between the two edge states gives the quantization of $\sigma_{xy}$. However, when the system is mesoscopic, the mixture between the edges states plays a physical role. We focus on this point and show numerical results.