ar X iv : h ep - t h / 98 11 21 8 v 1 2 5 N ov 1 99 8 The Modular and Renormalisation Groups in the Quantum Hall Effect ∗ Brian

An analytic form for the crossover of the conductivity tensor between two Hall plateaux, as a function of the external magnetic field, is proposed. The form of the crossover is obtained from the action of a symmetry group, a particular subgroup of the modular group, on the upper-half complex conductivity plane, by assuming that the β-function describing the crossover is a holomorphic function of the conductivity. The group action also leads to a selection rule, |p1q2−p2q1| = 1, for allowed transitions between Hall plateaux with filling factors ν1 = p1/q1 and ν2 = p2/q2, where q1 and q2 are odd. 1 The Quantum Hall Effect The classical Hall effect occurs when an electric current is passed through a two dimensional slab of conducting or semi-conducting material in a perpendicular magnetic field B. If the slab lies in the x − y plane, with its longer axis aligned with the x-direction and the magnetic field in the z-direction, then a transverse voltage, the Hall voltage Vxy, is generated across the short axis of the sample, when a current I is passed through the sample by applying a longitudinal voltage, Vxx, across the long axis. Thus we can define two resistances, a longitudinal resistance Rxx and a transverse resistance Rxy using Ohm’s law, Vxx = I Rxx, Vxy = I Rxy. (1) In two dimensions resistivity and resistance have the same dimensions and the classical analysis of the Hall effect gives the transverse resistivity ρxy ∗Talk presented at the Workshop on the Exact Renormalisation Group, Faro, Portugal, 10-12th September, 1998