Projective construction of non-Abelian quantum Hall liquids

Wen, Xiao-Gang

doi:10.1103/physrevb.60.8827

Cited by 118 publications

(203 citation statements)

References 51 publications

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“…41. This procedure works for gauge groups that are connected, while gauge groups of the form G ⋊ H, where G is connected and H is a discrete group, require further analysis.…”

Section: Summary Conclusion and Outlookmentioning

confidence: 99%

Phase transitions inZNgauge theory and twistedZNtopological phases

Barkeshli

Wen

2012

Phys. Rev. B

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We find a series of non-Abelian topological phases that are separated from the deconfined phase of ZN gauge theory by a continuous quantum phase transition. These non-Abelian states, which we refer to as the "twisted" ZN states, are described by a recently studied U (1) × U (1) ⋊ Z2 ChernSimons (CS) field theory. The U (1) × U (1) ⋊ Z2 CS theory provides a way of gauging the global Z2 electric-magnetic symmetry of the Abelian ZN phases, yielding the twisted ZN states. We introduce a parton construction to describe the Abelian ZN phases in terms of integer quantum Hall states, which then allows us to obtain the non-Abelian states from a theory of Z2 fractionalization. The non-Abelian twisted ZN states do not have topologically protected gapless edge modes and, for N > 2, break time-reversal symmetry.

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“…41. This procedure works for gauge groups that are connected, while gauge groups of the form G ⋊ H, where G is connected and H is a discrete group, require further analysis.…”

Section: Summary Conclusion and Outlookmentioning

confidence: 99%

Phase transitions inZNgauge theory and twistedZNtopological phases

Barkeshli

Wen

2012

Phys. Rev. B

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“…One approach to capture universal physics arising from topological interacting electron systems in (2+1) [20][21][22][23][24][25][26][27] and (3+1) [28][29][30] dimensions of space and time is via the parton construction. A fractional phase of electrons is obtained by constructing integer filled bands of "partons", which are then "glued" together by very strong gauge-mediated interactions so as to assemble together the physical electron.…”

Section: Introductionmentioning

confidence: 99%

Noncommutative geometry for three-dimensional topological insulators

et al. 2012

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We generalize the noncommutative relations obeyed by the guiding centers in the two-dimensional quantum Hall effect to those obeyed by the projected position operators in three-dimensional (3D) topological band insulators. The noncommutativity in 3D space is tied to the integral over the 3D Brillouin zone of a Chern-Simons invariant in momentum-space. We provide an example of a model on the cubic lattice for which the chiral symmetry guarantees a macroscopic number of zero-energy modes that form a perfectly flat band. This lattice model realizes a chiral 3D noncommutative geometry. Finally, we find conditions on the density-density structure factors that lead to a gapped 3D fractional chiral topological insulator within Feynman's single-mode approximation.

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“…[17] and [18]. One of them is [17,19] Ψ ν¼1 ðfz i gÞ ¼ ½χ 2 ðfz i gÞ 2 , where χ k ðfz i gÞ is the manyfermion wave function with k filled Landau levels. The bulk effective theory is the SUð2Þ f −2 Chern-Simons (CS) theory with three types of anyons and the edge has c ¼ 5=2 (see Supplemental Material [16]).…”

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confidence: 99%

“…The bulk effective theory is the SUð2Þ f −3 CS theory with four types of anyons [17,19]. So the state is N c ¼ 421 5 , which belongs to the same non-Abelian family as the state 226 5 in Table I (see Supplemental Material [16], which contains more examples of non-Abelian states and non-Abelian families).…”

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confidence: 99%

Hierarchy Construction and Non-Abelian Families of Generic Topological Orders

Lan

Wen

2017

Phys. Rev. Lett.

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We generalize the hierarchy construction to generic 2 þ 1D topological orders (which can be nonAbelian) by condensing Abelian anyons in one topological order to construct a new one. We show that such construction is reversible and leads to a new equivalence relation between topological orders. We refer to the corresponding equivalence class (the orbit of the hierarchy construction) as "the non-Abelian family." Each non-Abelian family has one or a few root topological orders with the smallest number of anyon types. All the Abelian topological orders belong to the trivial non-Abelian family whose root is the trivial topological order. We show that Abelian anyons in root topological orders must be bosons or fermions with trivial mutual statistics between them. The classification of topological orders is then greatly simplified, by focusing on the roots of each family: those roots are given by non-Abelian modular extensions of representation categories of Abelian groups.

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Projective construction of non-Abelian quantum Hall liquids

Cited by 118 publications

References 51 publications

Phase transitions inZNgauge theory and twistedZNtopological phases

Phase transitions inZNgauge theory and twistedZNtopological phases

Noncommutative geometry for three-dimensional topological insulators

Hierarchy Construction and Non-Abelian Families of Generic Topological Orders

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