Implications of experimental probes of the RG-flow in quantum Hall systems

We review the implications of the scaling data for the emergent symmetry of the quantum Hall system. The location of the fixed points in the conductivity plane is consistent with the global, non-Abelian discrete symmetry $\Gamma _{0}(2)$, and the renormalisation group (RG) flow-lines agree closely with that found if the symmetry acts anti-holomorphically. We extend the analysis to consider the rate of the RG flow. For a specific model in which the $\Gamma _{0}(2)$ symmetry acts anti-holomorphically the scaling close to the fixed points gives a critical delocalisation exponent $\nu = 2.38\pm 0.02$, in excellent greement with direct measurements and with numerical simulations. Both the predicted flow-lines and the flow rate also agree with the experimental measurements far away from the critical points, suggesting an emergent topological structure capable of stabilising the symmetry predictions. We hope that this agreement will stimulate further experimental study capable of conclusively testing the symmetry and exploring its associated dynamics.