  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.
