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Gaplessness protected by bulk-edge correspondence

Yoshiki Fukusumi

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.

Gaplessness protected by bulk-edge correspondence

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 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.
Paper Structure (12 sections, 40 equations, 5 figures)

This paper contains 12 sections, 40 equations, 5 figures.

Table of Contents

  1. Introduction
  2. $U(1)$ part and modular property
  3. Galois shuffle and monodromy charge
  4. Fermionic simple current in Gaffnian CFT
  5. Comparison with other literature and imaginary gapping argument
  6. Zero modes in Haldane-Rezayi state and $331$ state
  7. conclusion
  8. Acknowledgement
  9. Nonuniqueness of Jordan-Wigner transformations
  10. Application of the $Z_{2}$ chiral and antichiral twists to $SU(2)_{1}$
  11. Anomalous simple charge of simple current and anomalous fermionization
  12. Modular $S$ transformation for $M(3,5)$

Figures (5)

  • Figure 1: RG flow of CFT/BQFT correspondence. The existing lierature implies a rostness of modular $S$ invariant part of the theory, which corresponds to the Lagrangian subalgebra
  • Figure 2: Galois shuffle obstraction in FQHE
  • Figure 3: Equivalence between twist and parity shift
  • Figure 4: Anomaly and (para)spin statistics obstruction in quantum-classical correspondence
  • Figure 5: Anomaly obstraction and emergence of chirality