The Modular and Renormalisation Groups in the Quantum Hall Effect

TL;DR

This paper proposes an analytic model for the transition of the conductivity tensor between quantum Hall plateaux, based on a symmetry group acting on the complex conductivity plane, and derives a selection rule for allowed transitions.

Contribution

It introduces a novel symmetry-based approach using a subgroup of the modular group to describe quantum Hall transitions and predicts a specific selection rule for plateau transitions.

Findings

Derived an explicit crossover function for conductivity tensor.

Established a selection rule |p_1q_2 - p_2q_1|=1 for transitions.

Linked the symmetry group action to physical transitions in quantum Hall effect.

Abstract

An analytic form for the crossover of the conductivity tensor between two Hall plateaux, as a function of the external magnetic field, is proposed. The form of the crossover is obtained from the action of a symmetry group, a particular subgroup of the modular group, on the upper-half complex conductivity plane, by assuming that the beta-function describing the crossover is a holomorphic function of the conductivity. The group action also leads to a selection rule, |p_1q_2-p_2q_1|=1, for allowed transitions between Hall plateaux with filling factors p_1/q_1 and p_2/q_2, where q_1 and q_2 are odd.