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

TL;DR

This work proves that Hall conductance κ in an infinite-plane quantum Hall setting is rational whenever the low-energy excitation content forms a finite braided C*-tensor category 𝓜 of anyons. By constructing a U(1)–symmetric framework and a current observable J, the authors connect κ to the braiding statistics via θ(ρ,ρ) = e^{-{i} (2π)^2 κ} for a simple object ρ in 𝓜. The paper then shows that if 𝓜 has finitely many simple objects, there exists p such that 2π κ ∈ ℤ/p, i.e., κ is rational; this extends prior finite-volume and invertible-state results to the infinite-plane setting without Local Topological Quantum Order (LTQO) assumptions. The approach relies on the Doplicher–Haag–Roberts (DHR) framework for superselection sectors adapted to lattice systems and yields a uniform, category-theoretic perspective on the interplay between macroscopic transport and microscopic anyon statistics, with potential implications for understanding fractional quantum Hall signals in realistic models.

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.

Figures (5)

• Figure 1: The various cones used in the construction of the category ${\mathcal{M}}$. Forbidden directions are represented by the arc in the lower half plane.
• Figure 2: The four quadrants used to define the Hall conductance
• Figure 3: The half plane $\Gamma_\Lambda$ associated with the cone $\Lambda$
• Figure 4: The region $\Gamma$ corresponding to the disjoint cones $\Lambda,\Lambda'$. The unitary $V_{\Lambda,\Lambda'}$ is almost localized along the thicker grey line.
• Figure 5: The cones used in the proof of Lemma \ref{['thetaRhoSigmaIsHomRhoRho']}

Theorems & Definitions (56)

• Definition 1
• Theorem : MTC, Theorems 5.2 & 6.1
• Theorem 1: Existence of Anyons
• Theorem 2: Quantization of Hall conductance
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof