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

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