On the implications of discrete symmetries for the β-function of quantum Hall systems

We argue that the large discrete-symmetry group of quantum Hall systems is insufficient in itself to determine the complete β-function for the scaling of the conductivities, σxx and σxy. We illustrate this point by showing that a recent ansatz for this function is one of a many-parameter family. A clean prediction for the delocalization exponents for these systems therefore requires the specification of more information, such as past proposals that the β-function is either holomorphic or quasi-holomorphic in the variable z=(ℏ/e2)(σxy+iσxx).