The Weakly Coupled Pfaffian as a Type I Quantum Hall Liquid

The Pfaffian phase of electrons in the proximity of a half-filled Landau level is understood to be a $p+ip$ superconductor of composite fermions. We consider the properties of this paired quantum Hall phase when the pairing scale is small, {\it i.e.} in the weak-coupling, BCS, limit, where the coherence length is much larger than the charge screening length. We find that, as in a Type I superconductor, the vortices attract so that, upon varying the magnetic field from its magic value at $\nu=5/2$, the system exhibits Coulomb frustrated phase separation. We propose that the weakly and strongly coupled Pfaffian states exemplify a general dichotomy between Type I and Type II quantum Hall fluids.