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 $|p_1{q}_2-p_2{q}_1|=1$ for quantum Hall transitions between filling factors ${\nu }_1=p_1/q_1$ and ${\nu }_2=p_2/q_2$; 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.

3 pages, 1 table and 2 figures. Typeset in RevTeX. Some minor modifications and typos corrected

References in corpus (2)

• Symmetry in the insulator - quantum Hall - insulator transitions observed in a Ge/SiGe quantum well
• Modular Groups, Visibility Diagram and Quantum Hall Effect

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