Gaffnian holonomy through the coherent state method

TL;DR

This paper investigates the quasihole exchange statistics in the Gaffnian quantum Hall state using a coherent state approach, discussing the conditions under which adiabatic transport yields well-defined anyon statistics.

Contribution

It introduces a coherent state method to analyze Gaffnian quasihole braiding and explores conditions for valid adiabatic transport despite the state's gapless neutral excitations.

Findings

Identifies two possible unitary anyon statistics solutions for Gaffnian quasiholes.

Shows the coherent state Ansatz is valid if the Gaffnian parent Hamiltonian has a charge gap.

Finds one solution consistent with the non-Abelian spin-singlet state statistics.

Abstract

We analyze the effect of exchanging quasiholes described by Gaffnian quantum Hall trial state wave functions. This exchange is carried out via adiabatic transport using the recently developed coherent state Ansatz. We argue that our Ansatz is justified if the Gaffnian parent Hamiltonian has a charge gap, even though it is gapless to neutral excitations, and may therefore properly describe the adiabatic transport of Gaffnian quasiholes. For nonunitary states such as the Gaffnian, the result of adiabatic transport cannot agree with the monodromies of the conformal block wave functions, and may or may not lead to well-defined anyon statistics. Using the coherent state Ansatz, we find two unitary solutions for the statistics, one of which agrees with the statistics of the non-Abelian spin-singlet state by Ardonne and Schoutens.