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Europhysics Letters (EPL)

Hierarchy of the Topological Orders in the Fractional Quantum Hall Effect

A. A Belov, Karén Chaltikian

Published Aug 15, 1992• 0 citations
Publisher

Abstract

Non-Abelian (NA) version of the hierarchical FQH states is proposed. The topological order (TO) in these states is shown to be described by some U ( N ) coset Chern-Simons (CS) theory. For this all the necessary information about the TO can be found in the theory of the edge excitations (EE). This theory is described in terms of the corresponding gauged Wess-Zumino-Witten model related to the bulk theory through the Witten isomorfism.

DOI
10.1209/0295-5075/19/8/008

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Non-Abelian (NA) version of the hierarchical FQH states is proposed. The topological order (TO) in these states is shown to be described by some U ( N ) coset Chern-Simons (CS) theory. For this all the necessary information about the TO can be found in the theory of the edge excitations (EE). This theory is described in terms of the corresponding gauged Wess-Zumino-Witten model related to the bulk theory through the Witten isomorfism.
Master's thesis in Theoretical Physics. The official university title is "Fractional Quantum Hall Effect: Current State and Perspectives"; this record uses the author's preferred, more descriptive title. This thesis develops a unified description of the fractional quantum Hall effect (FQHE) and topological order through three complementary frameworks: microscopic trial many-body wavefunctions, the algebraic theory of anyons and unitary modular tensor categories, and Chern-Simons effective field theory. The first part covers the physical foundations—Landau quantization, the TKNN invariant, Laughlin's wavefunctions, the hierarchy and composite fermion theories, and some non-Abelian states like the Moore-Read Pfaffian state. The second part develops the algebraic theory of anyons, from the braid group to fusion rules, quantum dimensions, topological spins, and the F- and R-matrices that characterize a unitary modular tensor category. The third part constructs the Chern-Simons effective field theory, deriving fractional charge, statistics, and ground-state degeneracy, unifying Abelian states via the K-matrix formalism, and then generalizing to non-Abelian states through 𝑈(1)ₘ × 𝑆𝑈(𝑁)ₖ Chern-Simons theory. The central result is that Chern–Simons theory provides a field-theoretic realization of the topological order introduced microscopically and algebraically in the earlier chapters, closing the circle between the three descriptions and providing a unified framework for the FQHE and topological order and anyon's physics.
We study topological order and edge excitations of the \ensuremath{\nu}=1/2 fractional quantum Hall state (FQH) of spin polarized electrons in a single-layer system. We find that the 1/2 FQH state obtained in the numerical study of Greiter et al. has a non-Abelian topological order suggested by Moore and Read. The edge excitations in such a non-Abelian FQH state are found to be described by the U(1) Kac-Moody algebra plus a chiral Majorana fermion theory. The electron and the quasiparticle propagators are calculated. Some experimental consequences are also discussed.
We study the many-body ground states of SU(N)\mathrm{SU}(N)SU(N) symmetric hardcore bosons on the topological flat-band model by using controlled numerical calculations. By introducing strong intracomponent and intercomponent interactions, we demonstrate that a hierarchy of bosonic SU(N)\mathrm{SU}(N)SU(N) fractional quantum Hall (FQH) states emerges at fractional filling factors $\ensuremath{\nu}=N/(MN+1)$ (odd M=3M=3M=3). In order to characterize this series of FQH states, we figure the effective K\mathbf{K}K matrix from the inverse of the Chern number matrix. The topological characterization of the K\mathbf{K}K matrix also reveals quantized drag Hall responses and fractional charge pumping that could be detected in future experiments. In addition, we address the general one-to-one correspondence to the spinless FQH states in topological flat bands with Chern number C=NC=NC=N at fillings $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\nu}}=1/(MC+1)$.
More profound than bulk topological order of quantum materials is only its unwinding via gapless excitations along boundaries of the sample. We recast this bulk-edge correspondence -- for the experimentally relevant case of fractional quantum Hall (FQH) systems -- in terms of effective relative higher gauge theory, controlled by choices of classifying fibrations. For FQH systems, we identify the complex Hopf fibration as classifying the bulk/boundary topological effects, and find that it yields a non-Lagrangian reconstruction of Floreanini-Jackiw/Wess-Zumino-Witten chiral edge currents. Remarkably, the resulting effective FQH higher gauge theory turns out to be "geometrically engineered" on M2/M5-branes probing A-type orbi-singularities in 11D supergravity, globally completed by flux-quantization in twisted equivariant differential (TED) Cohomotopy: Here the M-string ends of M2-branes on M5-branes engineer the FQH liquid's boundary. This geometric engineering on M-branes might naturally elucidate the curious combination of W∞W_\inftyW∞​-symmetry and of super-symmetry that is known to govern the collective excitations of FQH liquids at long wavelengths.
The fractional quantum Hall effect is the paradigmatic example of topologically ordered phases. One of its most fascinating aspects is the large variety of different topological orders that may be realized, in particular non-Abelian ones. Here we analyze a class of non-Abelian fractional quantum Hall model states which are generalizations of the Abelian Haldane-Halperin hierarchy. We derive their topological properties and show that the quasiparticles obey non-Abelian fusion rules of type su(q)k\mathrm{su}{\left(q\right)}_{k}su(q)k​. For a subset of these states we are able to derive the conformal field theory description that makes the topological properties---in particular braiding---of the state manifest. The model states we study provide explicit wave functions for a large variety of interesting topological orders, which may be relevant for certain fractional quantum Hall states observed in the first excited Landau level.
We show that different classes of topological order can be distinguished by the dynamical symmetry algebra of edge excitations. A fundamental topological order is realized when this algebra is the largest possible, the algebra of quantum area-preserving diffeomorphisms, called W 1 + ∞ . We argue that this order is realized in the Jain hierarchy of fractional quantum Hall states and show that it is more robust than the standard Abelian Chern–Simons order since it has a lower entanglement entropy due to the non-Abelian character of the quasi-particle anyon excitations. These behave as SU( m ) quarks, where m is the number of components in the hierarchy. We propose the topological entanglement entropy as the experimental measure to detect the existence of these quantum Hall quarks. Non-Abelian anyons in the ν = 2/5 fractional quantum Hall states could be the primary candidates to realize qbits for topological quantum computation.