Emergent Conformal Symmetry of Quantum Hall States on Singular surfaces

TL;DR

This paper demonstrates that quantum Hall states on surfaces with conical singularities exhibit conformal symmetry properties, with the gravitational anomaly dictating their behavior near the singularities, including density structure and exchange statistics.

Contribution

It reveals the conformal primary nature of quantum Hall states on singular surfaces and links gravitational anomaly to their geometric and statistical properties.

Findings

Quantum Hall states near conical singularities behave as conformal primaries.

The conformal dimension is governed by the gravitational anomaly.

The electronic fluid at the cone tip has intrinsic angular momentum equal to the conformal dimension.

Abstract

We show that quantum Hall states on surfaces with conical singularities behave as conformal primaries near the singular points, with a conformal dimension controlled by the gravitational anomaly. We show that the electronic fluid at the cone tip possesses an intrinsic angular momentum equal to the conformal dimension, in units of the Planck constant. Finally, we argue that the gravitational anomaly also controls { the fine structure of electronic density at the tip, and } the exchange statistics of cones in the Laughlin states, arising from adiabatically braiding conical singularities. Thus, the gravitational anomaly, which appears as a finite size correction on smooth surfaces, dominates geometric transport on singular surfaces.