Gaplessness protected by bulk-edge correspondence

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

Composing parafermions: a construction of ZNZ_{N} fractional quantum Hall systems and a modern understanding of confinement and duality

In this work, we propose a modern view of the integer spin simple currents which have played a central role in discrete torsion. We reintroduce them as nonanomalous composite particles constructed from $Z_{N}$ parafermionic field theories. These composite particles have an analogy with the Cooper pair in the Bardeen-Cooper-Schrieffer theory and can be interpreted as a typical example of anyon condensation. Based on these $Z_{N}$ anomaly free composite particles, we propose a systematic construction of the cylinder partition function of $Z_{N}$ fractional quantum Hall effects (FQHEs). One can expect realizations of a class of general topological ordered systems by breaking the bulk-edge correspondence of the bosonic parts of these FQH models. We also give a brief overview of various phenomena in contemporary condensed matter physics, such as $SU(N)$ Haldane conjecture, general gapless and gapped topological order with respect to the quantum anomaly of simple currents and bulk and boundary renormalization group flow. Moreover, we point out an analogy with the FQHE and the 2d quantum gravity coupled to matter, and propose a $Z_{N}$ generalization of supersymmetry known as "fractional supersymmetry" in the composite parafermionic theory and study its analogy with quark confinement. Our analysis gives a simple but general understanding of the contemporary physics of topological phases in the form of the partition functions derived from the operator formalism

Protected edge modes based on the bulk and boundary renormalization group: A relationship between duality and generalized symmetry

We propose a theoretical formulation of protected edge modes in the language of quantum field theories based on the contemporary understanding of the renormalization group. We use bulk and boundary renormalization arguments which have never captured enough attention in condensed matter physics and related fields. We revisit various exotic bulk and boundary phenomena in contemporary physics, and one can see the conciseness of our formulations. Moreover, in the systems with open boundaries in general space-time dimensions, we also analyze their implications under general duality implemented by the shift of defects corresponding to generalized symmetries, including higher-form, non-invertible symmetries, in principle. Our formulation opens up a new paradigm to explore the systems with protected edge modes in the established language of the renormalization group.Comment: Typos are corrected, references are adde

Fermionic fractional quantum Hall states: A modern approach of systems with bulk-boundary correspondence

In contemporary physics, especially in condensed matter physics, fermionic topological order and its protected edge modes are one of the most important objects. In this work, we propose a systematic construction of the cylinder partition corresponding to the fermionic or electric fractional quantum Hall effect (FQHE) and a general mechanism to obtain the candidates of protected edge modes. In our construction, when the underlying conformal field theory has the $Z_{2}$ duality defect corresponding to the fermionic $Z_{2}$ electric particle, we show that the FQH partition function has a fermionic T duality. This duality is analogous to (hopefully the same as) the dualities in the dual resonance models, typically known as supersymmetry and gives a renormalization group (RG) theoretic understanding of topological phases. We also introduce a modern understanding of the bulk topological degeneracies and the topological entanglement entropy. This understanding is based on the traditional tunnel problem and the recent conjecture of correspondence between bulk renormalization group flow and boundary conformal field theory. Our formalism gives an intuitive understanding of modern physics of topological ordered systems in the traditional language of RG and fermionization