An RG potential for the quantum Hall effects

TL;DR

This paper generalizes the RG flow analysis of quantum Hall systems to include additional symmetries and charge carrier flavors, using a single-parameter potential that fits experimental data and reveals a generalized semi-circle law.

Contribution

It introduces a simple RG potential parametrized by one variable that captures a wide range of quantum Hall scaling behaviors and symmetries.

Findings

The potential accounts for nearly all observed scaling data.

The symmetry can be enhanced to subgroups of the modular group at specific parameter values.

Experimental data suggest the parameter is real, supporting a generalized semi-circle law.

Abstract

The phenomenological analysis of fully spin-polarized quantum Hall systems, based on holomorphic modular symmetries of the renormalization group (RG) flow, is generalized to more complicated situations where the spin or other "flavors" of charge carriers are relevant, and where the symmetry is different. We make the simplest possible ansatz for a family of RG potentials that can interpolate between these symmetries. It is parametrized by a single number $a$ and we show that this suffices to account for almost all scaling data obtained to date. The potential is always symmetric under the main congruence group at level two, and when $a$ takes certain values this symmetry is enhanced to one of the maximal subgroups of the modular group. We compute the covariant RG $\beta$-function, which is a holomorphic vector field derived from the potential, and compare the geometry of this gradient flow with available temperature driven scaling data. The value of $a$ is determined from experiment by finding the location of a quantum critical point, i.e., an unstable zero of the $\beta$-function given by a saddle point of the RG potential. The data are consistent with $a \in \mathbb{R}$, which together with the symmetry leads to a generalized semi-circle law.