Central charge from adiabatic transport of cusp singularities in the quantum Hall effect

TL;DR

This paper links the adiabatic transport of cusp singularities in quantum Hall states to the central charge, providing a geometric method to determine this key quantity through Berry curvature analysis.

Contribution

It establishes a connection between cusp singularity dimensions and the central charge in quantum Hall states, using adiabatic curvature and exact sum rules.

Findings

Berry curvature is finite and controlled by cusp singularity dimension.

Cusp singularity dimension equals the central charge for Laughlin states.

Provides a closed form for moments of mean density in integer QH states.

Abstract

We study quantum Hall (QH) states on a punctured Riemann sphere. We compute the Berry curvature under adiabatic motion in the moduli space in the large N limit. The Berry curvature is shown to be finite in the large N limit and controlled by the conformal dimension of the cusp singularity, a local property of the mean density. Utilizing exact sum rules obtained from a Ward identity, we show that for the Laughlin wave function, the dimension of a cusp singularity is given by the central charge, a robust geometric response coefficient in the QHE. Thus, adiabatic transport of curvature singularities can be used to determine the central charge of QH states. We also consider the effects of threaded fluxes and spin-deformed wave functions. Finally, we give a closed expression for all moments of the mean density in the integer QH state on a punctured disk.