Duality and the Modular Group in the Quantum Hall Effect

TL;DR

This paper investigates the role of modular group symmetry in the quantum Hall effect, deriving universal properties, transition rules, and critical conductivities based on complex conductivity and renormalisation group assumptions.

Contribution

It introduces a modular group framework for quantum Hall transitions, providing new insights into universality and transition rules in both integer and fractional regimes.

Findings

Derivation of the selection rule |p1q2 - p2q1|=1 for transitions

Identification of critical conductivity values

Prediction of Farey sequences of transitions

Abstract

We explore the consequences of introducing a complex conductivity into the quantum Hall effect. This leads naturally to an action of the modular group on the upper-half complex conductivity plane. Assuming that the action of a certain subgroup, compatible with the law of corresponding states, commutes with the renormalisation group flow, we derive many properties of both the integer and fractional quantum Hall effects, including: universality; the selection rule $|p_1q_2 - p_2q_1|=1$ for quantum Hall transitions between filling factors $\nu_1=p_1/q_1$ and $\nu_2=p_2/q_2$; critical values for the conductivity tensor; and Farey sequences of transitions. Extra assumptions about the form of the renormalisation group flow lead to the semi-circle rule for transitions between Hall plateaus.