Non-Abelian quantum Hall states and their quasiparticles: From the pattern of zeros to vertex algebra

Lü, Yuan; Wen, Xiao-Gang; Wang, Zhenghan; Wang, Ziqiang

doi:10.1103/physrevb.81.115124

Cited by 29 publications

(32 citation statements)

References 45 publications

(91 reference statements)

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“…Fractional quantum Hall states [15,16], chiral spin liquids [17,18], Z 2 spin liquids [19][20][21], non-Abelian fractional quantum Hall states [22][23][24][25], etc., are examples of topologically ordered phases. The mathematical foundation of topological orders is closely related to tensor category theory [9,12,26,27] and simple current algebra [22,28]. Using this point of view, we have developed a systematic and quantitative theory for topological orders with gappable edge for (2+1)-dimensional [(2+1)D] interacting boson and fermion systems [9,12,27].…”

Section: A Short-and Long-range Entangled Statesmentioning

confidence: 99%

Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinearσmodels and a special group supercohomology theory

2014

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Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have T 2 = 1 time-reversal Z T 2 and fermion-number-parity Z f 2 symmetries described by a full symmetry group Z T 2 × Z f 2 ). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the K-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group Z 2 × Z f 2 . Those 2D fermionic SPT phases all have central-charge c = 1 gapless edge excitations, if the symmetry is not broken.

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Section: A Short-and Long-range Entangled Statesmentioning

confidence: 99%

Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinearσmodels and a special group supercohomology theory

2014

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“…Based on the simplest examples, it is natural to expect that the critical phenomena will generally be described by a suitable class of generalized parafermion CFTs, possibly including those described in Ref. 52,58,59.…”

Section: Critical Phenomena Between Gapped Edge Statesmentioning

confidence: 99%

Theory of defects in Abelian topological states

2013

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The structure of extrinsic defects in topologically ordered states of matter is host to a rich set of universal physics. Extrinsic defects in 2+1 dimensional topological states include line-like defects, such as boundaries between topologically distinct states, and point-like defects, such as junctions between different line defects. Gapped boundaries in particular can themselves be \it topologically \rm distinct, and the junctions between them can localize topologically protected zero modes, giving rise to topological ground state degeneracies and projective non-Abelian statistics. In this paper, we develop a general theory of point defects and gapped line defects in 2+1 dimensional Abelian topological states. We derive a classification of topologically distinct gapped boundaries in terms of certain maximal subgroups of quasiparticles with mutually bosonic statistics, called Lagrangian subgroups. The junctions between different gapped boundaries provide a general classification of point defects in topological states, including as a special case the twist defects considered in previous works. We derive a general formula for the quantum dimension of these point defects, a general understanding of their localized "parafermion" zero modes, and we define a notion of projective non-Abelian statistics for them. The critical phenomena between topologically distinct gapped boundaries can be understood in terms of a general class of quantum spin chains or, equivalently, "generalized parafermion" chains. This provides a way of realizing exotic 1+1D generalized parafermion conformal field theories in condensed matter systems.Comment: 24 pages, 11 figures. Some of the results here are also summarized in arXiv:1304.7579; v2: strengthened braiding results and mapping to genons, updated refs/acknowledgment

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“…The mathematical foundation of topological orders is closely related to tensor category theory 9,10,24,25 and simple current algebra. 20,26 Using this point of view, we have developed a systematic and quantitative theory for non-chiral topological orders in 2D interacting boson and fermion systems. 9,10,25 Also for chiral 2D topological orders with only Abelian statistics, we find that we can use integer K-matrices to describe them.…”

Section: Introductionmentioning

confidence: 99%

Quantized topological terms in weak-coupling gauge theories with a global symmetry and their connection to symmetry-enriched topological phases

2013

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We study the quantized topological terms in a weak-coupling gauge theory with gauge group Gg and a global symmetry Gs in d space-time dimensions. We show that the quantized topological terms are classified by a pair (G, ν d ), where G is an extension of Gs by Gg and ν d an element in group cohomology H d (G, R/Z). When d = 3 and/or when Gg is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriched topological (SET) phases (ie gapped long-range entangled phases with symmetry). Thus, those SET phases are classified bywhere G/Gg = Gs. We also apply our theory to a simple case Gs = Gg = Z2, which leads to 12 different SET phases in 2+1D, where quasiparticles have different patterns of fractional Gs = Z2 quantum numbers and fractional statistics. If the weak-coupling gauge theories are gapless, then the different quantized topological terms may describe different gapless phases of the gauge theories with a symmetry Gs, which may lead to different fractionalizations of Gs quantum numbers and different fractional statistics (if in 2+1D).

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Non-Abelian quantum Hall states and their quasiparticles: From the pattern of zeros to vertex algebra

Cited by 29 publications

References 45 publications

Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinearσmodels and a special group supercohomology theory

Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinearσmodels and a special group supercohomology theory

Theory of defects in Abelian topological states

Quantized topological terms in weak-coupling gauge theories with a global symmetry and their connection to symmetry-enriched topological phases

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