One-Dimensional Flows in the Quantum Hall System

TL;DR

This paper constructs a c-function to analyze the renormalization group flow of conductivities in quantum Hall systems, assuming symmetry invariance and quasi-holomorphic flow, thereby determining the phase diagram and flow characteristics.

Contribution

It introduces a novel c-function framework for quantum Hall systems based on symmetry and quasi-holomorphic assumptions, advancing understanding of their RG flow and phase structure.

Findings

The c-function and phase diagram are fully determined under the assumptions.

The RG flow is characterized by a specific metric compatible with the c-function.

The approach extends to other systems with similar symmetries, like supersymmetric QED.

Abstract

We construct the c-function whose gradient determines the RG flow of the conductivities (sigma_xy and sigma_xx) for a quantum Hall system, subject to two assumptions. (1) We take the flow to be invariant with respect to the infinite discrete symmetry group, recently proposed by several workers to explain the `superuniversality' of the delocalization exponents in these systems. (2) We also suppose the flow to be `quasi-holomorphic' (which we make precise) in the sense that it is as close as possible to a one-dimensional flow in the complex parameter sigma_xy +i sigma_xx. These assumptions together with the known asymptotic behaviour for large sigma_xx, completely determine the c-function, and so the phase diagram, for these systems. A complete description of the RG flow also requires a metric in addition to the c-function, and we identify the features which are required for this by the RG. A similar construction produces the c-function for other systems enjoying an infinite discrete symmetry, such as for supersymmetric QED.