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

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.