Topological aspects of the quantum Hall effect

The quantum Hall effect is a typical realization of topological effects in condensed matter physics. In this article, some of the topological aspects of the quantum Hall effect are reviewed. For the lattice fermions, the Hall conductance of the system is expressed in terms of two different topological invariants. One is the famous TKNN integer which is related to the bulk state. The other is the winding number of the edge state on the complex-energy surface which is generally a high-genus Riemann surface. We will describe them in detail. Therefore we have two topological expressions for the Hall conductance. Actually these two expressions give the same integer, although they look quite different. This means that one can explain the quantum Hall effect by using either the edge states or the bulk states, that is, sigma(xy)(edge) = sigma(xy)(bulk).