The Neutral Excitations in the Gaffnian State

TL;DR

This paper investigates the neutral excitations of the Gaffnian state, a gapless fractional quantum Hall wavefunction, using extended single-mode approximation and numerical methods to understand its exotic properties.

Contribution

It introduces a representation highlighting the pairing structure of the Gaffnian state and extends the single-mode approximation to three-body interactions.

Findings

Clarifies the physical origin of gaplessness in the Gaffnian state

Provides second quantized expressions for Haldane pseudopotentials in multiple geometries

Demonstrates systematic extraction of physics in large-electron-number limits

Abstract

We study a model fractional quantum Hall (FQH) wavefunction called the Gaffnian state, which is believed to represent a gapless, strongly correlated state that is very different from conventional metals. To understand this exotic gapless state better, we provide a representation based on work of Halperin in which the pairing structure of the Gaffnian state becomes more explicit. We employ the single-mode approximation introduced by Girvin, MacDonald, and Platzman (GMP), here extended to three-body interactions, in order to treat a neutral collective exitation mode in order to clarify the physical origin of the gaplessness of the Gaffnian state. We discuss approaches to extract systematically the relevant physics in the long-distance, large-electron-number limit of FQH states using numerical calculations with relatively few electrons. In an appendix, we provide second quantized expressions for many-body Haldane pseudopotentials in various geometries including the plane, sphere, cylinder, and the torus based on the proper definition of the relative angular momentum.