Nonperturbative Renormalization Group Function for Quantum Hall Plateau Transitions Imposed by Global Symmetries

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As a unified theory of integer and fractional quantum Hall plateau transitions, a nonperturbative theory of the two-parameter scaling renormalization group function is presented. By imposing global symmetries known as ``the law of corresponding states'', we seek a possible form of renormalization group flows. Asking for consistency with the result from weak-localization perturbation theory, such restriction is so intense that we can analytically determine its concrete form. Accordingly, the critical exponent ν and the irrelevant scaling index y are obtained analytically and turn out to be irrational. Their values (ν≈ 2.1, y≈ 0.3) agree favorably well with experiments and numerics.