Duality and the Modular Group in the Quantum Hall Effect

Details Duality and the Modular Group in the Quantum Hall Effect Duality and the Modular Group in the Quantum Hall Effect

1998-05-14

arxiv.org/abs/cond-mat/9805171v2

J.Phys. A32 (1999) L243

Physics

Condensed Matter

Mesoscale and Nanoscale Physics

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

Scientific paper

10.1088/0305-4470/32/21/101

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 $\nu_1=p_1/q_1$ 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.

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