Duality and the Modular Group in the Quantum Hall Effect

Dolan, Brian P. (1998) Duality and the Modular Group in the Quantum Hall Effect. (Preprint)

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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_1*q_2 - p_2*q_1| = 1 for quantum Hall transitions between filling factors ν_1 = p_1/q_1 and ν_2 = p_2/q_2; critical values of 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.

Item Type: Article
Divisions: School of Theoretical Physics > Preprints
Date Deposited: 19 Jun 2018 14:07
Last Modified: 18 Jul 2018 10:09
URI: http://dair.dias.ie/id/eprint/682

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