Thin torus perturbative analysis of elementary excitations in the Gaffnian and Haldane-Rezayi quantum Hall states

We present a systematic perturbative approach to study excitations in the thin cylinder/torus limit of the quantum Hall states. The approach is applied to the Haldane-Rezayi and Gaffnian quantum Hall states, which are both expected to have gapless excitations in the usual two-dimensional thermodynamic limit. For the Haldane-Rezayi state, we confirm that gapless excitations are present also in the "one-dimensional" thermodynamic limit of an infinite thin cylinder, in agreement with earlier considerations based on the wave functions alone. In contrast, we identify the lowest excitations of the Gaffnian state in the thin cylinder limit, and conclude that they are gapped, using a combination of perturbative and numerical means. We discuss possible scenarios for the cross-over between the two-dimensional and the one-dimensional thermodynamic limit in this case.