Tensor category describing anyons in the quantum Hall effect and quantization of conductance

TL;DR

This paper demonstrates that in a quantum Hall system with a finite braided tensor category of anyons, the Hall conductance is quantized as a rational number, extending understanding beyond previous assumptions.

Contribution

It establishes that the Hall conductance is rational when the underlying anyonic tensor category is finite, under minimal assumptions.

Findings

Hall conductance is well-defined in the considered system.

Hall conductance is rational if the tensor category of anyons is finite.

Quantization holds without relying on weak interactions or finite volume assumptions.

Abstract

In this study, we examine the quantization of Hall conductance in an infinite plane geometry. We consider a microscopic charge-conserving system with a pure, gapped infinite-volume ground state. While Hall conductance is well-defined in this scenario, existing proofs of its quantization have relied on assumptions of either weak interactions, or properties of finite volume ground state spaces, or invertibility. Here, we assume that the conditions necessary to construct the braided $C^*$-tensor category which describes anyonic excitations are satisfied, and we demonstrate that the Hall conductance is rational if the tensor category is finite.