$Γ(2)$ modular symmetry, renormalization, group flow and the quantum Hall effect

Details $Γ(2)$ modular symmetry, renormalization, group flow and the quantum Hall effect $Γ(2)$ modular symmetry, renormalization, group flow and the quantum Hall effect

1999-06-14

arxiv.org/abs/cond-mat/9906193v1

Physics

Condensed Matter

Mesoscale and Nanoscale Physics

14 pages plain TeX, 7 figures in a postscript file

Scientific paper

10.1088/0305-4470/33/1/303

We construct a family of holomorphic $\beta$-functions whose RG flow preserves the $\Gamma(2)$ modular symmetry and reproduces the observed stability of the Hall plateaus. The semi-circle law relating the longitudinal and Hall conductivities that has been observed experimentally is obtained from the integration of the RG equations for any permitted transition which can be identified from the selection rules encoded in the flow diagram. The generic scale dependance of the conductivities is found to agree qualitatively with the present experimental data. The existence of a crossing point occuring in the crossover of the permitted transitions is discussed.

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