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29 March 2024
 
  » arxiv » cond-mat/9805171

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Duality and the Modular Group in the Quantum Hall Effect
Brian P. Dolan ;
Date 14 May 1998
Journal J.Phys. A32 (1999) L243
Subject Mesoscopic Systems and Quantum Hall Effect | cond-mat.mes-hall hep-th
AbstractWe 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 $ u_1=p_1/q_1$ and $ u_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.
Source arXiv, cond-mat/9805171
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