Gaplessness of the Gaffnian

TL;DR

This paper investigates the Gaffnian wavefunction in fractional quantum Hall systems, providing evidence for its gapless nature through numerical and theoretical analysis, and exploring its conformal field theory structure.

Contribution

It offers a detailed analysis of the Gaffnian wavefunction's gapless behavior and its underlying non-unitary conformal field theory, using interpolating Hamiltonians and geometry-based studies.

Findings

Evidence for the absence of an energy gap in the Gaffnian state

Identification of heterotic conformal towers in the edge spectrum

Connection of the wavefunction to non-unitary conformal field theory

Abstract

We study the Gaffnian trial wavefunction proposed to describe fractional quantum Hall correlations at Bose filling factor $\nu=2/3$ and Fermi filling $\nu=2/5$. A family of Hamiltonians interpolating between a hard-core interaction for which the physics is known and a projector whose ground state is the Gaffnian is studied in detail. We give evidence for the absence of a gap by using large-scale exact diagonalizations in the spherical geometry. This is in agreement with recent arguments based on the fact that this wavefunction is constructed from a non-unitary conformal field theory. By using the cylinder geometry, we discuss in detail the nature of the underlying minimal model and we show the appearance of heterotic conformal towers in the edge energy spectrum where left and right movers are generated by distinct primary operators.