Gaplessness protected by bulk-edge correspondence

TL;DR

This paper investigates why certain fractional quantum Hall states constructed from nonunitary conformal field theories remain gapless, revealing that the mismatch in monodromy and simple charge, rather than negative conformal dimensions, causes the gaplessness.

Contribution

It introduces a new criterion based on monodromy charge mismatch and Galois shuffle to predict the gapped or gapless nature of topological states, challenging previous assumptions.

Findings

Gaffinian and Haldane-Rezayi states are gapless due to monodromy-charge mismatch.

The gaplessness is linked to nonlocal structures similar to deconfined quantum criticality.

Proposes a new paradigm for predicting topological order gappedness based on anomaly and duality analysis.

Abstract

After almost half a century of Laughlin's celebrated study of the wavefunctions of integer and fractional quantum Hall effects, there have still existed difficulties to prove whether the given wavefunction can describe gapped phase or not in general. In this work, we show the FQH states constructed from nonunitary conformal field theories (CFTs), such as Gaffiinian and Haldane-Rezayi states have a difficulty gapping out under preserving bulk-edge correspondence in the cylinder geometry. Contrary to the common understandings of the condensed matter communities, the gaplessness for these systems seems not to come from the negative conformal dimensions of nonunitary CFTs in this setting at least directly. We propose the difficulty is coming from the mismatch of monodromy charge and simple charge of underlying CFTs, known as Galois shuffle. In the Haldane-Rezayi state, this corresponds to the conjugate operation of the Neveu-Schwartz and Ramond sectors for unitary Weyl fermion and symplectic fermion. In the Gaffinian state, besides Galois shuffle structure, the anomalous conformal dimension of the $Z_{2}$ simple current results in the cylinder partition functions outside of the existing local quantum field theory. This indicates the existing gapless fractional quantum Hall states have similar nonlocal structures, similar to deconfined quantum criticality. Our work opens up a new paradigm which gives a criterion to predict whether the candidate of topological ordered states are gapped or not, and local or nonlocal, by revisiting the problem of anomaly and the duality of symplectic and Dirac fermion.