Anyonic Chains, Topological Defects, and Conformal Field Theory

Buican, Matthew; Gromov, Andrey

doi:10.1007/s00220-017-2995-6

Cited by 84 publications

(81 citation statements)

References 59 publications

(266 reference statements)

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“…Clearly this classification forms a hierarchy, and it would be nice if we have a uniform construction that tells easily which stage of the above classification a given symmetry category C belongs to. There is a recent paper in this general direction [36], where a construction of 2d theory starting from any given symmetry category C was discussed. We hope to see more developments in the future.…”

mentioning

confidence: 99%

On finite symmetries and their gauging in two dimensions

Bhardwaj¹,

Tachikawa²

2018

J. High Energ. Phys.

236

301

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It is well-known that if we gauge a Z n symmetry in two dimensions, a dual Z n symmetry appears, such that re-gauging this dual Z n symmetry leads back to the original theory. We describe how this can be generalized to non-Abelian groups, by enlarging the concept of symmetries from those defined by groups to those defined by unitary fusion categories. We will see that this generalization is also useful when studying what happens when a non-anomalous subgroup of an anomalous finite group is gauged: for example, the gauged theory can have non-Abelian group symmetry even when the original symmetry is an Abelian group. We then discuss the axiomatization of two-dimensional topological quantum field theories whose symmetry is given by a category. We see explicitly that the gauged version is a topological quantum field theory with a new symmetry given by a dual category. 2 Recently in [5], Gaiotto, Kapustin, Komargodski and Seiberg performed an impressive study of the phase structure of thermal 4d su(2) Yang-Mills theory. One important step in the analysis is the symmetry structure of the thermal system, which is essentially three-dimensional. As a dimensional reduction from 4d, the system has a Z 2 × Z 2 0-form symmetry and a Z 2 1-form symmetry, with a mixed anomaly. Then the authors gauged the Z 2 1-form symmetry, and found that the total 0-form symmetry is now D 8 . This D 8 was then used very effectively to study the phase diagram, but that part of their paper does not directly concern us here. Their analysis of turning an anomalous Abelian symmetry by gauging a non-anomalous subgroup into a non-Abelian symmetry is a 3d analogue of what we explain in 2d. See their Sec. 4.2, Appendix B and Appendix C. Clearly an important direction to pursue is to generalize their and our constructions to arbitrary combinations of possibly-higher-form symmetries in arbitrary spacetime dimensions, but that is outside of the scope of this paper.

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confidence: 99%

On finite symmetries and their gauging in two dimensions

Bhardwaj¹,

Tachikawa²

2018

J. High Energ. Phys.

236

301

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“…This work has strong relations with the work of Petkova and Zuber [32,33], where an algebraic approach was used to construct boundary conditions in CFTs, and with the approach of Aasen, Fendley and Mong [21], where topologically invariant defects were constructed from defect commutation relations on the lattice. The generalized strange correlator construction is a Euclidean spacetime counterpart to anyonic chain models, where topological symmetries and defects have been discussed previously [34][35][36]. The class of partition functions and topological defects produced by the generalized strange correlator matches those considered in Refs.…”

mentioning

confidence: 78%

Mapping Topological to Conformal Field Theories through strange Correlators

et al. 2018

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We extend the concept of strange correlators, defined for symmetry-protected phases in [You et al., Phys. Rev. Lett. 112, 247202 (2014)], to topological phases of matter by taking the inner product between string-net ground states and product states. The resulting two-dimensional partition functions are shown to be either critical or symmetry broken, as the corresponding transfer matrices inherit all matrix product operator symmetries of the string-net states. For the case of critical systems, those non-local matrix product operator symmetries are the lattice remnants of topological conformal defects in the field theory description. Following [Aasen et al., J. Phys. A 49, 354001 (2016)], we argue that the different conformal boundary conditions can be obtained by applying the strange correlator concept to the different topological sectors of the string-net obtained from Ocneanu's tube algebra. This is demonstrated by calculating the conformal field theory spectra on the lattice in the different topological sectors for the Fibonacci (hard-hexagon) and Ising string-net. Additionally, we provide a complementary perspective on symmetry-preserving real-space renormalization by showing how known tensor network renormalization methods can be understood as the approximate truncation of an exactly coarse-grained strange correlator.

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“…The most general extension conceivable is to a set of MPOs described by a unitary fusion category 55,115,116 . While the construction of topological sectors is known in this general case 55,83,84,[115][116][117][118][119] , an ansatz which allows the symmetry to be enforced locally in the MERA remains to be found.…”

Section: Discussionmentioning

confidence: 99%

Anomalies and entanglement renormalization

Bridgeman

Williamson

2017

Phys. Rev. B

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We study 't Hooft anomalies of discrete groups in the framework of (1+1)-dimensional multiscale entanglement renormalization ansatz states on the lattice. Using matrix product operators, general topological restrictions on conformal data are derived. An ansatz class allowing for optimization of MERA with an anomalous symmetry is introduced. We utilize this class to numerically study a family of Hamiltonians with a symmetric critical line. Conformal data is obtained for all irreducible projective representations of each anomalous symmetry twist, corresponding to definite topological sectors. It is numerically demonstrated that this line is a protected gapless phase. Finally, we implement a duality transformation between a pair of critical lines using our subclass of MERA.Comment: 12+18 pages, 6+5 figures, 0+2 tables, v2 published versio

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Anyonic Chains, Topological Defects, and Conformal Field Theory

Cited by 84 publications

References 59 publications

On finite symmetries and their gauging in two dimensions

On finite symmetries and their gauging in two dimensions

Mapping Topological to Conformal Field Theories through strange Correlators

Anomalies and entanglement renormalization

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