Gapless excitations in the Haldane-Rezayi state: The thin torus limit

TL;DR

This paper investigates the thin torus limit of the Haldane-Rezayi state, revealing gapless excitations and explicit zero mode counting, thus providing insights into the state's quantum Hall properties and related Hamiltonians.

Contribution

It derives the thin torus limits of the Haldane-Rezayi state, identifies gapless excitations, and provides explicit zero mode counting formulas for related quantum Hall states.

Findings

Eight ground states assume a simple product form in the thin torus limit.

Two states have delocalized defect pairs forming singlets in the thin torus limit.

Gapless excitations are present in the thin torus limit, consistent with non-unitary conformal field theories.

Abstract

We study the thin torus limit of the Haldane-Rezayi state. Eight of the ten ground states are found to assume a simple product form in this limit, as is known to be the case for many other quantum Hall trial wave functions. The two remaining states have a somewhat unusual thin torus limit, where a "broken" pair of defects forming a singlet is completely delocalized. We derive these limits from the wave functions on the cylinder, and deduce the dominant matrix elements of the thin torus hollow-core Hamiltonians. We find that there are gapless excitations in the thin torus limit. This is in agreement with the expectation that local Hamiltonians stabilizing wave functions associated with non-unitary conformal field theories are gapless. We also use the thin torus analysis to obtain explicit counting formulas for the zero modes of the hollow-core Hamiltonian on the torus, as well as for the parent Hamiltonians of several other paired and related quantum Hall states.