Martin Walser and the Working World

Bullivant, Keith

doi:10.1163/9789004333789_007

Cited by 4 publications

(11 citation statements)

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“…More generally, the braiding structure of loop-like objects in the 3-disk is governed by the so-called necklace groups [21,22] when the threading flux is non-trivial and the loop-braid group [23][24][25] when the threading flux is trivial. It can be shown that the loop-like excitations of the twisted quantum triple naturally define representations of such motion groups [26]. In this way, the twisted quantum triple algebra provides a rigorous framework to describe the processes of interest in the condensed matter literature.…”

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confidence: 99%

Tube algebras, excitations statistics and compactification in gauge models of topological phases

Bullivant

Delcamp

2019

J. High Energ. Phys.

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We consider lattice Hamiltonian realizations of (d+1)-dimensional Dijkgraaf-Witten theory. In (2+1)d, it is well-known that the Hamiltonian yields point-like excitations classified by irreducible representations of the twisted quantum double. This can be confirmed using a tube algebra approach. In this paper, we propose a generalisation of this strategy that is valid in any dimensions. We then apply this generalisation to derive the algebraic structure of loop-like excitations in (3+1)d, namely the twisted quantum triple. The irreducible representations of the twisted quantum triple algebra correspond to the simple loop-like excitations of the model. Similarly to its (2+1)d counterpart, the twisted quantum triple comes equipped with a compatible comultiplication map and an R-matrix that encode the fusion and the braiding statistics of the loop-like excitations, respectively. Moreover, we explain using the language of loop-groupoids how a model defined on a manifold that is n-times compactified can be expressed in terms of another model in n-lower dimensions. This can in turn be used to recast higher-dimensional tube algebras in terms of lower dimensional analogues. arXiv:1905.08673v2 [cond-mat.str-el]

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confidence: 99%

Tube algebras, excitations statistics and compactification in gauge models of topological phases

Bullivant

Delcamp

2019

J. High Energ. Phys.

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“…Recently, there are many works [29][30][31][32][33][34][35][36] on higher gauge theories and their connection to topological phases of matter. In this paper, we present a detailed description of "lattice higher gauge theories", in a way to make their connection to non-linear σ-model explicit.…”

Section: A Backgroundmentioning

confidence: 99%

Topological nonlinear σ -model, higher gauge theory, and a systematic construction of 3+1D topological orders for boson systems

2019

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A discrete non-linear σ-model is obtained by triangulate both the space-time M d+1 and the target space K. If the path integral is given by the sum of all the complex homomorphisms φ : M d+1 → K, with an partition function that is independent of space-time triangulation, then the corresponding non-linear σ-model will be called topological non-linear σ-model which is exactly soluble. Those exactly soluble models suggest that phase transitions induced by fluctuations with no topological defects (i.e. fluctuations described by homomorphisms φ) usually produce a topologically ordered state and are topological phase transitions, while phase transitions induced by fluctuations with all the topological defects give rise to trivial product states and are not topological phase transitions. If K is a space with only non-trivial first homotopy group G which is finite, those topological nonlinear σ-models can realize all 3+1D bosonic topological orders without emergent fermions, which are described by Dijkgraaf-Witten theory with gauge group π1(K) = G. Here, we show that the 3+1D bosonic topological orders with emergent fermions can be realized by topological non-linear σ-models with π1(K) = finite groups, π2(K) = Z2, and πn>2(K) = 0. A subset of those topological non-linear σ-models corresponds to 2-gauge theories, which realize and classify bosonic topological orders with emergent fermions that have no emergent Majorana zero modes at triple string intersections. The classification of 3+1D bosonic topological orders may correspond to a classification of unitary fully dualizable fully extended topological quantum field theories in 4-dimensions. CONTENTS 7 C. Topological non-linear σ-models 8 D. Labeling simplices in a complex 8 III. Dijkgraaf-Witten gauge theories from topological non-linear σ-models 9 A. Lattice gauge theories from topological non-linear σ-models 9 B. Classification of exactly soluble 1-gauge theories 11 IV. 2-gauge theories from topological non-linear σ-models 11 A. 2-groups 11 B. 2-gauge theories 13 C. 2-group cocycles 14 D. Cohomology of 2-group 15 V. Pure 2-gauge theory of 2-gauge-group B(Π2, 2) 16 A. Pure 2-group and pure 2-gauge theory 16 B. Pure 2-gauge theory in 3+1D 17 1. n = odd case 17 2. n = even case 17 3. Properties and duality relations 17 VI. 3+1D 2-gauge theory of 2-gauge-group B(G b , Z f 2 ) 18 A. The Lagrangian and space-time path integral 18 B. The equivalence between [k0,ē2(ā),n3(ā),ν4(ā)]'s 18 C. 2-gauge transformations in the cocycle σ-model 19 D. The vanishing of the partition function 19 E. The pointlike and stringlike excitations in the 2-gauge theory 20 VII. Classify and realize 3+1D EF1 topological orders by 2-gauge theories of 2-gauge-group B(G b , Z f 2 ) 21 VIII. Realize 3+1D EF2 topological orders by topological non-linear σ-models 21 A. Construction of topological non-linear σ-models 21 B. The canonical boundary of topological non-linear σ-models 26 IX. Summary 26 A. Space-time complex, cochains, and cocycles 27 B. Lyndon-Hochschild-Serre spectral sequence 30 C. Partition f...

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“…etc., which are defined by the following chain rules: [23], k) now reads graphically and algebraically as…”

Section: 2)mentioning

confidence: 99%

“…(3.9) becomes f (1 ′ 1, 12 ′ , 2 ′ 2, 24) = f (1 ′ 1, 12 ′ , 2 ′ 2, 23). Since the group elements [23] and [24] are arbitrary in K. We must have f (1 ′ 1, 12 ′ , 2 ′ 2, x) = c ∀x ∈ K, where c is a constant. By the normalization condition (2.3), however, f (1 ′ 1, 12 ′ , 2 ′ 2, x = 1) = 1; hence, we must have c = 1.…”

Section: 2)mentioning

confidence: 99%

“…Systems bearing intrinsic topological orders in two [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and three [15,[17][18][19][20][21][22][23][24][25][26][27][28][29] spatial dimensions have received substantial attention recently because they not only have greatly expanded and deepened our understanding of phases of matter but also have practical applications, in particular certain 2-dimensional topologically ordered systems can support topological quantum computation [5,30]. Celebrated candidates of two-dimensional topological phases include chiral spin liquids [2,31], Z 2 spin liquids [32][33][34], Abelian quantum Hall states [35][36][37], and non-Abelian fractional quantum Hall states [38][39][40][41][42].…”

Section: Introductionmentioning

confidence: 99%

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Gapped boundary theory of the twisted gauge theory model of three-dimensional topological orders

Wang

et al. 2018

J. High Energ. Phys.

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We extend the twisted gauge theory model of topological orders in three spatial dimensions to the case where the three spaces have two dimensional boundaries. We achieve this by systematically constructing the boundary Hamiltonians that are compatible with the bulk Hamoltonian. Given the bulk Hamiltonian defined by a gauge group G and a four-cocycle ω in the fourth cohomology group of G over U (1), a boundary Hamiltonian can be defined by a subgroup K of G and a three-cochain α in the third cochain group of K over U (1). The boundary Hamiltonian to be constructed must be gapped and invariant under the topological renormalization group flow (via Pachner moves), leading to a generalized Frobenius condition. Given K, a solution to the generalized Frobenius condition specifies a gapped boundary condition. We derive a closed-form formula of the ground state degeneracy of the model on a three-cylinder, which can be naturally generalized to three-spaces with more boundaries. We also derive the explicit ground-state wavefunction of the model on a three-ball. The ground state degeneracy and ground-state wavefunction are both presented solely in terms of the input data of the model, namely, {G, ω, K, α}.

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Martin Walser and the Working World

Cited by 4 publications

References 0 publications

Tube algebras, excitations statistics and compactification in gauge models of topological phases

Tube algebras, excitations statistics and compactification in gauge models of topological phases

Topological nonlinear σ -model, higher gauge theory, and a systematic construction of 3+1D topological orders for boson systems

Gapped boundary theory of the twisted gauge theory model of three-dimensional topological orders

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