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

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 Hamiltonian. 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 nonunitary 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.