Dolan, Brian P.
(1999)
Duality and the modular group in the quantum Hall effect.
Journal of Physics A: Mathematical and General, 32 (21).
L243-L248.
ISSN 0305-4470
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 renormalization group flow, we derive many properties of both the integer and fractional quantum Hall effects including: universality; the selection rule |p1q2-p2q1| = 1 for transitions between quantum Hall states characterized by filling factors nu1 = p1/q1 and nu2 = p2/q2; critical values of the conductivity tensor; and Farey sequences of transitions. Extra assumptions about the form of the renormalization group flow lead to the semicircle rule for transitions between Hall plateaux.
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