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

TL;DR

This paper develops a perturbative method to analyze elementary excitations in the thin cylinder limit of quantum Hall states, confirming gapless excitations in the Haldane-Rezayi state and identifying gapped excitations in the Gaffnian state.

Contribution

It introduces a systematic perturbative approach for studying excitations in the thin torus limit of quantum Hall states, providing new insights into their gap properties.

Findings

Haldane-Rezayi state exhibits gapless excitations in the thin cylinder limit.

Gaffnian state has gapped excitations in the same limit.

Discussion of crossover scenarios between 2D and 1D thermodynamic limits.

Abstract

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.