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Duality and the Modular Group in the Quantum Hall Effect

Brian P. Dolan
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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.

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Brian P. Dolan. 1999-01-28. Duality and the Modular Group in the Quantum Hall Effect. https://doi.org/10.1088/0305-4470%2F32%2F21%2F101

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