Anyons in discrete gauge theories with Chern-Simons terms

Bais, F.A.; Driel, Peter van; Propitius, Mark de Wild

doi:10.1016/0550-3213(93)90073-x

Cited by 36 publications

(23 citation statements)

References 26 publications

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“…61 This is to be expected since both those theories and our models describe topological order classified by DW TQFTs. 52,61 In Appendix B we also discuss in detail the situation with an arbitrary number of quasiparticles (either end-A or end-B ones), describing their braiding and fusion as well as the quasi-quantum double mathematical structure.…”

Section: Braiding Matrixsupporting

confidence: 55%

“…52 Different topological orders labeled by H 3 (GG,U (1)) can be viewed as different discrete versions of the Chern-Simons terms. [60][61][62] For example, because H 3 (Z 2 ,U (1)) = Z 2 , there are two distinct topological orders described by a Z 2 gauge group. In the language of the K matrix, the two topological orders are described by K = ( 0 2 2 0 ) and K = ( 2 0 0 −2 ), respectively.…”

Section: The Classification and Connection To Previous Workmentioning

confidence: 99%

“…It has been shown 61 that certain broken gauge theories with Chern-Simons terms lead to discrete (group H ) gauge theories having such twisted algebra describing their quasiparticles. 61 In that situation, the cocycle is generated by the Chern-Simons term, and the resulting discrete gauge theory can be classified using H 3 ( H ,U(1)), 60 making connection with discrete DW TQFTs. Importantly, a nontrivial cocycle twisting of the algebra can render the resulting theory non-Abelian even though its gauge group H is Abelian.…”

Section: Local Symmetric Operators and The Twisted Extended Ribbonmentioning

confidence: 99%

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Classification of symmetry enriched topological phases with exactly solvable models

2013

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Parafermions are emergent excitations that generalize Majorana fermions and can also realize topological order. In this paper we present a non-trivial and quasi-exactly-solvable model for a chain of parafermions in a topological phase. We compute and characterize the ground-state wave-functions, which are matrix-product states and have a particularly elegant interpretation in terms of Fock parafermions, reflecting the factorized nature of the ground states. Using these wavefunc-tions, we demonstrate analytically several signatures of topological order. Our study provides a starting point for the non-approximate study of topological one-dimensional parafermionic chains with spatial-inversion and time-reversal symmetry in the absence of strong edge modes. Introduction. The study of topological order (TO) is currently one of the most active research fields in condensed-matter physics. From the AKLT model [1] to the Laughlin wavefunction [2], from the Kitaev chain [3] to the Toric code [4], this study has always benefited from the development of exactly-solvable models and of paradigmatic wavefunctions, whose detailed analysis permits the formation of a clear physical intuition, to be used in the understanding of complex experimental setups. In this letter we focus on parafermions, a generalization of Majorana fermions [5]. After the experimental clarification that two zero-energy Majorana modes can be localized at the edges of a one-dimensional fermionic wire [6, 7], the possibility of localizing parafermionic modes, and letting them interact, is currently under deep investigation. These excitations cannot appear in strictly one-dimensional spinless fermionic systems [8, 9], but may emerge at the edge of a two-dimensional fractional topological insulator coupled to alternating ferro-magnetic and superconducting materials [5, 10-15], as well as in other nanostructures or models [16-24]. In these setups, one-dimensional chains of interacting parafermions arise, which, in certain circumstances, display TO and edge Z N parafermionic modes [5, 25-34]. Such edge modes are called strong when they commute with the Hamiltonian [35] and thereby generate a N-fold degeneracy in the entire spectrum, and weak when the commutation property and associated degeneracy are restricted to the ground state manifold. TO survives weak perturbations and hosts indistinguishably weak or strong modes [36]. The importance of parafermionic zero-modes for topological quantum computation [37] motivates further investigations of these fractionalized systems. In this letter we provide a non-trivial family of parafermionic models for which the properties of the ground states can be exactly characterized. These models are gapped, display TO, have spatial-inversion and time-reversal symmetries, and feature weak edge modes; they thus belong to the same symmetry class for which weak edge modes have been discussed so far with numerical and perturbative analytical methods [28, 31, 36], with the advantage of being easy to handle. We analytically establish ...

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Section: Braiding Matrixsupporting

confidence: 55%

Section: The Classification and Connection To Previous Workmentioning

confidence: 99%

Section: Local Symmetric Operators and The Twisted Extended Ribbonmentioning

confidence: 99%

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Classification of symmetry enriched topological phases with exactly solvable models

2013

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“…Consequently, a membrane, which is the ð3 þ 1ÞD analogue of the Dirac string, is also labeled by a group element. Further, a group representation labels a particle [10].…”

Section: Membrane Operatormentioning

confidence: 99%

“…One way to study topologically ordered states is by using the exactly solvable models of discrete gauge theories introduced by Dijkgraaf and Witten [9,10]. Although these theories in ð2 þ 1ÞD do not provide an exhaustive classification of all possible topological orders [11], they describe a physically interesting set of states.…”

Section: Introductionmentioning

confidence: 99%

Generalized Modular Transformations in(3+1)DTopologically Ordered Phases and Triple Linking Invariant of Loop Braiding

2014

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In topologically ordered quantum states of matter in ð2 þ 1ÞD (spacetime dimensions), the braiding statistics of anyonic quasiparticle excitations is a fundamental characterizing property that is directly related to global transformations of the ground-state wave functions on a torus (the modular transformations). On the other hand, there are theoretical descriptions of various topologically ordered states in ð3 þ 1ÞD, which exhibit both pointlike and looplike excitations, but systematic understanding of the fundamental physical distinctions between phases, and how these distinctions are connected to quantum statistics of excitations, is still lacking. One main result of this work is that the three-dimensional generalization of modular transformations, when applied to topologically ordered ground states, is directly related to a certain braiding process of looplike excitations. This specific braiding surprisingly involves three loops simultaneously, and can distinguish different topologically ordered states. Our second main result is the identification of the three-loop braiding as a process in which the worldsheets of the three loops have a nontrivial triple linking number, which is a topological invariant characterizing closed two-dimensional surfaces in four dimensions. In this work, we consider realizations of topological order in ð3 þ 1ÞD using cohomological gauge theory in which the loops have Abelian statistics and explicitly demonstrate our results on examples with Z 2 × Z 2 topological order.

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Discrete Gauge Theories

Propitius¹,

Bais²

1999

Particles and Fields

154

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In these lecture notes, we present a self-contained treatment of planar gauge theories broken down to some finite residual gauge group H via the Higgs mechanism. The main focus is on the discrete H gauge theory describing the long distance physics of such a model. The spectrum features global H charges, magnetic vortices and dyonic combinations. Due to the Aharonov-Bohm effect, these particles exhibit topological interactions. Among other things, we review the Hopf algebra related to this discrete H gauge theory, which provides an unified description of the spin, braid and fusion properties of the particles in this model. Exotic phenomena such as flux metamorphosis, Alice fluxes, Cheshire charge, (non)abelian braid statistics, the generalized spin-statistics connection and nonabelian Aharonov-Bohm scattering are explained and illustrated by representative examples.

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Anyons in discrete gauge theories with Chern-Simons terms

Cited by 36 publications

References 26 publications

Classification of symmetry enriched topological phases with exactly solvable models

Classification of symmetry enriched topological phases with exactly solvable models

Generalized Modular Transformations in(3+1)DTopologically Ordered Phases and Triple Linking Invariant of Loop Braiding

Discrete Gauge Theories

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