Quantum Coherence of Edge States in the Quantum Hall Eect { Topological Invariants and Edge State Mixing {

Edge states in the integral quantum Hall e ect 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 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 clari ed. In a macroscopic system, mixture of the edge states are exponentially small and negligible. Quantum Coherence between the two edge states gives the quatization of xy . However, when the system is mesoscopic, the mixture between the edges states plays a physical role. We also focus on this point and show some numerical results.