Folding approach to topological order enriched by mirror symmetry

Qi, Yang; Jian, Chao-Ming; Wang, Chenjie

doi:10.1103/physrevb.99.085128

Cited by 15 publications

(41 citation statements)

References 97 publications

(202 reference statements)

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“…In Theorem B below, we show that if this anomaly vanishes for the action on C, then it must also vanish for the action on C loc A constructed in Theorem A. However, if the SETO C has an anomalous symmetry fractionalization class o 4 ∈ H 4 (G, U (1)), we have not yet determined whether the phase C loc A after anyon condensation is anomalous or not [QJW17]. We leave this question for future study.…”

Section: Introductionmentioning

confidence: 87%

Spontaneous symmetry breaking from anyon condensation

Bischoff

Jones

Lü

et al. 2019

J. High Energ. Phys.

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In a physical system undergoing a continuous quantum phase transition, spontaneous symmetry breaking occurs when certain symmetries of the Hamiltonian fail to be preserved in the ground state. In the traditional Landau theory, a symmetry group can break down to any subgroup. However, this no longer holds across a continuous phase transition driven by anyon condensation in symmetry enriched topological orders (SETOs). For a SETO described by a Gcrossed braided extension C ⊆ C × G , we show that physical considerations require that a connected etale algebra A ∈ C admit a G-equivariant algebra structure for symmetry to be preserved under condensation of A. Given any categorical action G → Aut br ⊗ (C) such that g(A) ∼ = A for all g ∈ G, we show there is a short exact sequence whose splittings correspond to G-equivariant algebra structures. The non-splitting of this sequence forces spontaneous symmetry breaking under condensation of A. Furthermore, we show that if symmetry is preserved, there is a canonically associated SETO of C loc A , and gauging this symmetry commutes with anyon condensation.

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Section: Introductionmentioning

confidence: 87%

Spontaneous symmetry breaking from anyon condensation

Bischoff

Jones

Lü

et al. 2019

J. High Energ. Phys.

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“…Instead of attaching the WW model to the boundary of the 4+1d bulk, we can insert it into the middle of the bulk (at a fixed w). The folding approach [11] implies that a WW domain wall in the bulk of the 4+1d toric code is equivalent to having a WW gapped boundary for two copies of the 4+1d toric code. We can arrange the DOFs from the two models to be aligned in a similar way (edges in the WW dual to plaquettes in the 3+1d submanifold with fixed w inside the 4+1d bulk) and coupled in a similar way (with a Z ⊗ τ Z…”

Section: B Lattice: Decorated Smooth Boundarymentioning

confidence: 99%

“…We remark that the factor of i implies that U (1,1) , U (1,1) transform differently under timereversal symmetry. 11 While Q permutes m ↔ ψ, Q 2 does not permute the types of excitations,…”

Section: Mapping Between M Loop and ψ Loopmentioning

confidence: 99%

Loops in 4+1d Topological Phases

Xie¹,

Dua²,

Hsin³

et al. 2021

Preprint

Self Cite

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2+1d topological phases are well characterized by the fusion rules and braiding/exchange statistics of fractional point excitations. In 4+1d, some topological phases contain only fractional loop excitations. What kind of loop statistics exist?We study the 4+1d gauge theory with 2-form Z 2 gauge field (the loop only toric code) and find that while braiding statistics between two different types of loops can be nontrivial, the self 'exchange' statistics are all trivial. In particular, we show that the electric, magnetic, and dyonic loop excitations in the 4+1d toric code are not distinguished by their self-statistics. They tunnel into each other across 3+1d invertible domain walls which in turn give explicit unitary circuits that map the loop excitations into each other. The SL(2, Z 2 ) symmetry that permutes the loops, however, cannot be consistently gauged and we discuss the associated obstruction in the process. Moreover, we discuss a gapless boundary condition dubbed the 'fractional Maxwell theory' and show how it can be Higgsed into gapped boundary conditions.We also discuss the generalization of these results from the Z 2 gauge group to Z N .

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“…[39]. A number of subsequent works further studied and rederived these anomaly formulas through various other methods [40][41][42][43].…”

Section: A Relation To Prior Workmentioning

confidence: 99%

Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions

Bulmash

Barkeshli

2020

Phys. Rev. Research

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Certain patterns of symmetry fractionalization in (2+1)-dimensional [(2+1)D] topologically ordered phases of matter can be anomalous, which means that they possess an obstruction to being realized in purely (2+1)D. In this paper we demonstrate how to compute the anomaly for symmetry-enriched topological states of bosons in complete generality. We demonstrate how, given any unitary modular tensor category (UMTC) and symmetry fractionalization class for a global symmetry group G, one can define a (3+1)-dimensional [(3+1)D] topologically invariant path integral in terms of a state sum for a G-symmetry-protected topological (SPT) state. We present an exactly solvable Hamiltonian for the system and demonstrate explicitly a (2+1)D G-symmetric surface termination that hosts deconfined anyon excitations described by the given UMTC and symmetry fractionalization class. We present concrete algorithms that can be used to compute anomaly indicators in general. Our approach applies to general symmetry groups, including anyon-permuting and antiunitary symmetries. In addition to providing a general way to compute the anomaly, our result also shows, by explicit construction, that every symmetry fractionalization class for any UMTC can be realized at the surface of a (3+1)D SPT state. As a by-product, this construction also provides a way of explicitly seeing how the algebraic data that defines symmetry fractionalization in general arises in the context of exactly solvable models. In the case of unitary orientation-preserving symmetries, our results can also be viewed as providing a method to compute the H 4 (G, U (1)) obstruction that arises in the theory of G-crossed braided tensor categories, for which no general method has been presented to date.

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Folding approach to topological order enriched by mirror symmetry

Cited by 15 publications

References 97 publications

Spontaneous symmetry breaking from anyon condensation

Spontaneous symmetry breaking from anyon condensation

Loops in 4+1d Topological Phases

Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions

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