  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.
