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arXiv:2212.12996

Gaplessness protected by bulk-edge correspondence

Yoshiki Fukusumi

Published Dec 26, 2022• 0 citations
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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.

DOI
10.48550/arxiv.2212.12996

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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 Z2Z_{2}Z2​ 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.
In contemporary physics, especially in condensed-matter physics, fermionic topological order and its protected edge modes is one of the most important objects. In this work, we propose a systematic construction of the cylinder partition corresponding to the fermionic fractional quantum Hall effect and a general mechanism for obtaining the candidates of the protected edge modes. In our construction, when the underlying conformal field theory has Z2{Z}_{2}Z2​ duality defects corresponding to the fermionic Z2{Z}_{2}Z2​ electric particle, we show that the fractional quantum Hall 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 it provides a renormalization-group (RG) theoretical understanding of the topological phases. We also introduce a modern understanding of bulk topological degeneracies and topological entanglement entropy. This understanding is based on the traditional tunnel problem and the recent conjecture of correspondence between bulk RG flow and boundary conformal field theory. Our formalism provides an intuitive and general understanding of the modern physics of topologically ordered systems in the traditional language of RG and fermionization, and it may serve as a complement to more mathematical physical frameworks such as fermionic category theories.
Some fractional quantum Hall states observed in experiments may be described by first-quantized wavefunctions with special clustering properties such as the Moore-Read Pfaffian for filling factor $\ensuremath{\nu}=5/2$. This wavefunction has been constructed by constructing correlation functions of a two-dimensional conformal field theory (CFT) involving a free boson and a Majorana fermion. By considering other CFTs many other clustered states have been proposed as candidate fractional quantum Hall states under appropriate circumstances. It is believed that the underlying CFT should be unitary if one wants to describe an incompressible, i.e., gapped liquid state. We show that by changing the way one derives the wavefunction from its parent CFT it is possible to obtain an incompressible candidate state when starting from a nonunitary parent. The construction mimics a global change in parameters in the phase space of the electron system. We explicit our construction in the case of the so-called Gaffnian state (a state for filling factor 2/5) and also for the Haldane-Rezayi state (a spin-singlet state at filling 1/2). We note that there are obstructions along this path in the case of the permanent spin-singlet state of Read and Rezayi which can be characterized as a robust gapless state.
Gapless quantum phases can become distinct when internal symmetries are enforced, in analogy with gapped symmetry-protected topological (SPT) phases. However, this distinction does not always lead to protected edge modes, raising the question of how the bulk-boundary correspondence is generalized to gapless cases. We propose that the spatial interface between gapless phases -- rather than their boundaries -- provides a more robust fingerprint. We show that whenever two 1+1d conformal field theories (CFTs) differ in symmetry charge assignments of local operators or twisted sectors, any symmetry-preserving spatial interface between the theories must flow to a non-invertible defect. We illustrate this general result for different versions of the Ising CFT with Z2×Z2T\mathbb{Z}_2 \times \mathbb{Z}_2^TZ2​×Z2T​ symmetry, obtaining a complete classification of allowed conformal interfaces. When the Ising CFTs differ by nonlocal operator charges, the interface hosts 0+1d symmetry-breaking phases with finite-size splittings scaling as 1/L31/L^31/L3, as well as continuous phase transitions between them. For general gapless phases differing by an SPT entangler, the interfaces between them can be mapped to conformal defects with a certain defect 't Hooft anomaly. This classification also gives implications for higher-dimensional examples, including symmetry-enriched variants of the 2+1d Ising CFT. Our results establish a physical indicator for symmetry-enriched criticality through symmetry-protected interfaces, giving a new handle on the interplay between topology and gapless phases.
Using the newly developed matrix product state formalism for non-Abelian fractional quantum Hall (FQH) states, we address the question of whether a FQH trial wave function written as a correlation function in a nonunitary conformal field theory (CFT) can describe the bulk of a gapped FQH phase. We show that the nonunitary Gaffnian state exhibits clear signatures of a pathological behavior. As a benchmark we compute the correlation length of a Moore-Read state and find it to be finite in the thermodynamic limit. By contrast, the Gaffnian state has an infinite correlation length in (at least) the non-Abelian sector, and is therefore gapless. We also compute the topological entanglement entropy of several non-Abelian states with and without quasiholes. For the first time in the FQH effect the results are in excellent agreement in all topological sectors with the CFT prediction for unitary states. For the nonunitary Gaffnian state in finite size systems, the topological entanglement entropy seems to behave like that of the composite fermion Jain state at equal filling.
We study zero modes bound to charge-conjugation defects in fractional quantum Hall-superconductor (FQH-SC) heterostructures built from Abelian and non-Abelian FQH states. Although the associated defects are well described by GGG-crossed braided tensor categories, a local operator description of their zero modes is not generally known. We generalize the bulk-to-boundary construction of localized zero mode operators to FQH-SCs with non-Abelian parent topological orders, mapping bulk Wilson operators in the folded theory to localized zero mode operators in the one-dimensional FQH-SC theory. This construction distinguishes zero mode anyons from the local zero mode operators that act on the defect Hilbert space and yields a generalized zero mode algebra. It also determines the fusion-channel dependence of tunneling: tunneling of zero mode operators probes bulk monodromy, while direct zero mode anyon transfer probes GGG-crossed associativity data. Applying this framework, we recover the Z2m\mathbb{Z}_{2m}Z2m​ parafermion zero mode structure of fermionic Laughlin FQH-SCs at filling ν=1/mν=1/mν=1/m, obtain a Z2(n)×ZN\mathbb{Z}_2^{(n)}\times\mathbb{Z}_NZ2(n)​×ZN​ structure for generalized Moore-Read states at filling ν=1/mν=1/mν=1/m, with N=2mN=2mN=2m for fermionic states and N=mN=mN=m for bosonic states, and find a Fib×ZN\mathrm{Fib}\times\mathbb{Z}_NFib×ZN​ structure for the k=3k=3k=3 Read-Rezayi state at filling ν=3/(3M+2)ν=3/(3M+2)ν=3/(3M+2), with N=2(3M+2)N=2(3M+2)N=2(3M+2) for fermionic states and N=(3M+2)/2N=(3M+2)/2N=(3M+2)/2 for bosonic states. In the Read-Rezayi case, zero mode operator tunneling and direct zero mode anyon transfer yield distinct universal amplitude ratios between the neutral fusion channels, providing a characteristic signature of the underlying Fibonacci defect structure.
The quantum Hall effect is one of the most challenging problems in condensed matter physics in which many rare physical phenomena like fractional charges and generally the fractional statistics are involved. In the integer quantum Hall effect the role of the disorder is more than others. The important point in describing these systems is that their bulk states should be gapful and in this sense the system should be incompressible. For describing the electron condensates in an arbitrary filling fraction with the odd denominator one needs the topological U(1) Chern-Simons effective field theory. Our motivation for using these field theories is the emergence of quantized conductivity. This effective field theory also has the ability of constructing the hierarchical fractional condensates with various filling factors. The excitations of these theories have fractional statistics which are investigated in this review. The non-Abelian Chern-Simons for quantum Hall systems will be also investigated. The interesting point is that the edge states of these systems are gapless and their excitations are equipped with the Kac-Moody algebra. The connection to the conformal field theories can be established using this point. The investigation of this correspondence is one of the aims of the present thesis. The connection to the logarithmic conformal field theories (e.g. for the Haldane-Rezaei wave function) is also reported in some works which will be addressed in this thesis.