Duality and the Modular Group in the Quantum Hall Effect

Abstract

We explore the consequences of introducing a complex conductivity into the quantum Hall effect. This leads naturally to an action of the modular group on the upper-half complex conductivity plane. Assuming that the action of a certain subgroup, compatible with the law of corresponding states, commutes with the renormalisation group flow, we derive many properties of both the integer and fractional quantum Hall effects, including: universality; the selection rule $|p_1q_2 - p_2q_1|=1$ for quantum Hall transitions between filling factors $ν_1=p_1/q_1$ and $ν_2=p_2/q_2$; critical values for the conductivity tensor; and Farey sequences of transitions. Extra assumptions about the form of the renormalisation group flow lead to the semi-circle rule for transitions between Hall plateaus.