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Gaplessness protected by bulk-edge correspondence

Yoshiki Fukusumi
Dec 2022
After almost half a century of Laughlin's celebrated study of thewavefunctions of integer and fractional quantum Hall effects, there have stillexisted difficulties to prove whether the given wavefunction can describegapped phase or not in general. In this work, we show the FQH statesconstructed from nonunitary conformal field theories (CFTs), such as Gaffiinianand Haldane-Rezayi states have a difficulty gapping out under preservingbulk-edge correspondence in the cylinder geometry. Contrary to the commonunderstandings of the condensed matter communities, the gaplessness for thesesystems seems not to come from the negative conformal dimensions of nonunitaryCFTs in this setting at least directly. We propose the difficulty is comingfrom the mismatch of monodromy charge and simple charge of underlying CFTs,known as Galois shuffle. In the Haldane-Rezayi state, this corresponds to theconjugate operation of the Neveu-Schwartz and Ramond sectors for unitary Weylfermion and symplectic fermion. In the Gaffinian state, besides Galois shufflestructure, the anomalous conformal dimension of the $Z_{2}$ simple currentresults in the cylinder partition functions outside of the existing localquantum field theory. This indicates the existing gapless fractional quantumHall states have similar nonlocal structures, similar to deconfined quantumcriticality. Our work opens up a new paradigm which gives a criterion topredict whether the candidate of topological ordered states are gapped or not,and local or nonlocal, by revisiting the problem of anomaly and the duality ofsymplectic and Dirac fermion.