Geometric Defects in Quantum Hall States

TL;DR

This paper introduces a geometric analogue of Laughlin quasiholes in fractional quantum Hall states, exploring their properties, statistics, and relation to topological defects called genons, with explicit calculations for Laughlin states.

Contribution

It presents a novel geometric construction of defects in quantum Hall states, linking them to topological genons and providing a method to compute their braiding statistics.

Findings

Defects can be constructed via vertex operators in conformal blocks.

Defects have assigned electric charge and spin.

Explicit braiding matrices are calculated for Laughlin states.

Abstract

We describe a geometric (or gravitational) analogue of the Laughlin quasiholes in the fractional quantum Hall states. Analogously to the quasiholes these defects can be constructed by an insertion of an appropriate vertex operator into the conformal block representation of a trial wavefunction, however, unlike the quasiholes these defects are extrinsic and do not correspond to true excitations of the quantum fluid. We construct a wavefunction in the presence of such defects and explain how to assign an electric charge and a spin to each defect, and calculate the adiabatic, non-abelian statistics of the defects. The defects turn out to be equivalent to the genons in that their adiabatic exchange statistics can be described in terms of representations of the mapping class group of an appropriate higher genus Riemann surface. We present a general construction that, in principle, allows to calculate the statistics of $\mathbb Z_n$ genons for any "parent" topological phase. We illustrate the construction on the example of the Laughlin state and perform an explicit calculation of the braiding matrices. In addition to non-abelian statistics geometric defects possess a universal abelian overall phase, determined by the gravitational anomaly.