When an electron gas is confined to move at the interface between two semiconductors and a magnetic field is applied perpendicular to the plane, a new state of matter (TSG1982) arises at sufficiently low temperatures. This state of matter is unique in condensed matter physics in that it has a gap to all excitations and exhibits fractional statistics. It is generally referred to as an incompressible quantum liquid or as a Laughlin liquid (L1983), in reference to the architect of this state. While the Laughlin state is mediated by the mutual repulsions among the electrons, it is the presence of the large perpendicular magnetic field that leads to the incompressible nature of this new many-body state. The precursor to this state is the integer quantum Hall state. In this state, disorder and the magnetic field conspire to limit the relevant charge transport to a narrow strip around the rim of the sample. The novel feature of this rim or edge current is that it is quantized in integer multiples of e2/h (KDP1980). The equivalent current in the Laughlin state is still quantized but rather in fractional multiples of e2/h. We present in this chapter the phenomenology and the mathematical description needed to understand the essential physics of both of these effects.

As we will see, topology is an integral part of the quantum Hall effect. Regardless of the geometry or smooth changes in the Hamiltonian, the quantization of the conductance depends solely on the existence of edge states which have a well-defined chirality.