A review of the QHE is presented where the emphasis is placed on the role of the magnetic translation group and of the related topological properties. After a presentation of general experimental and theoretical features we briefly summarize known results of two-dimensional Conformal Field Theory relevant for the QHE. Then we show how to evaluate groundstate wave functions on the plane and on the torus by the use of CFT techniques. In the latter case it is shown how for filling ν=1/m a consistent description is achieved by means of a finite set of Coulomb Gas Vertex Operators. They describe (fractional) charged particles with the associated quantized magnetic flux (“anyons"). Furthermore, from these vertices and relative highest weight states, one does find that the g.s.w.f. for the torus should be m-fold degenerate showing, also, the role of a new kind of long-range topological order recently advocated. Then we show that the presence of m sectors of edge states for a cylinder is strictly related to such a degeneracy, and their relation with a Kac-Moody algebra is discussed.
Conductivity of a two-dimensional HgTe layer near the critical width: The role of developed edge states network and random mixture of
p
- and
n
-domains
