On the Implications of Discrete Symmetries for the Beta Function of Quantum Hall Systems

TL;DR

This paper discusses how the large discrete symmetry group in quantum Hall systems alone cannot fully determine the beta function for conductivity scaling, emphasizing the need for additional constraints like holomorphicity assumptions.

Contribution

It demonstrates that the beta function in quantum Hall systems is not uniquely determined by symmetry alone and highlights the necessity of extra conditions for precise predictions.

Findings

The symmetry group does not fix the beta function uniquely.

A family of possible beta functions exists consistent with the symmetry.

Additional assumptions like holomorphicity are needed for definitive predictions.

Abstract

We argue that the large discrete symmetry group of quantum Hall systems is insufficient in itself to determine the complete beta function for the scaling of the conductivities, $\sigma_{xx}$ and $\sigma_{xy}$. 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 beta function is either holomorphic or quasi-holomorphic in the variable $z = (\hbar/e^2)(\sigma_{xy} + i\sigma_{xx})$.