Fermionic symmetry fractionalization in (2+1)D

Bulmash, Daniel; Barkeshli, Maissam

doi:10.48550/arxiv.2109.10913

Cited by 3 publications

(6 citation statements)

References 29 publications

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“…Accordingly, gauge transformations of η a (g, h) and β a (g, h) shall be correlated such that (75) always holds. Combining (69), ( 70), ( 73), (74) and that…”

Section: A Basics Of Setsmentioning

confidence: 84%

“…We also become aware of the works Refs. [73][74][75] which study the general theory of fermionic SET phases. P(w 2m ) does not change under the coboundary transformation w 2m → w 2m + dc 2m−1 , or equivalently ŵ2m → ŵ2m + dc 2m−1 + 2N c 2m , where dc 2m−1 ∈ B 2m (G, Z 4N ) is the lift of dc 2m−1 ∈ B 2m (G, Z 2N ).…”

Section: Discussionmentioning

confidence: 99%

“…They are related to each other by a cohomology class [t] ∈ H 2 ρ (G, A). Once w(g, h) and β a (g, h) are given, the projective phase factor η a (g, h) is determined by ( 73) and (74).…”

Section: A Basics Of Setsmentioning

confidence: 99%

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Enforced symmetry breaking by invertible topological order

Ning¹,

Qi²,

Gu³

et al. 2021

Preprint

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It is well known that two-dimensional fermionic systems with a nonzero Chern number must break the time reversal symmetry, manifested by the appearance of chiral edge modes on an open boundary. Such an incompatibility between topology and symmetry can occur more generally. We will refer to this phenomenon as enforced symmetry breaking by topological orders. In this work, we systematically study enforced breaking of a general finite group G f by a class of topological orders, namely 0D, 1D and 2D fermionic invertible topological orders. Mathematically, the symmetry group G f is a central extension of a bosonic group G by the fermion parity group Z f 2 , characterized by a 2-cocycle λ ∈ H 2 (G, Z2). With some minor assumptions and for given G and λ, we are able to obtain a series of criteria on the existence or non-existence of enforced symmetry breaking by the fermionic invertible topological orders. Using these criteria, we discover many examples that are not known previously. For 2D systems, we define the physical quantities to describe symmetryenriched invertible topological orders and derive some obstruction functions using both fermionic and bosonic languages. In the latter case which is done via gauging the fermion parity, we find that some obstruction functions are consequences of conditional anomalies of the bosonic symmetryenriched topological states, with the conditions inherited from the original fermionic system. We also study enforced breaking of the continuous group SU f (N ) by 2D invertible topological orders through a different argument.

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“…Accordingly, gauge transformations of η a (g, h) and β a (g, h) shall be correlated such that (75) always holds. Combining (69), ( 70), ( 73), (74) and that…”

Section: A Basics Of Setsmentioning

confidence: 84%

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Enforced symmetry breaking by invertible topological order

Ning¹,

Qi²,

Gu³

et al. 2021

Preprint

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“…We believe that the q-type strings, or called Majorana-type strings, are very likely to characterize the anomaly of 3D fSPT phases with Kitaev-chain decoration 33 . Moreover, it will also be very interesting to understand the generic algebraic structure [34][35][36] of fSET phases from equivalence class of symmetric fLU transformations.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…( 13) or Eq. ( 14), and there will be another independent projective unitary condition for F -move (Similarly the Majorana numbers cancel out so that we can write down the relation for F -move): 36) where Ξ ij kln,χδ is another phase factor satisfying (Ξ ij kln,χδ ) * = Ξ ij kln,(χ×f )(δ×f ) . It depends on strings i, j, k, l, n and fusion states χ, δ.…”

Section: B the Structure Of Fixed-point Wavefunctionsmentioning

confidence: 99%

Towards a complete classification of non-chiral topological phases in 2D fermion systems

Zhou¹,

Gu²

2021

Preprint

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In recent years, fermionic topological phases of quantum matter has attracted a lot of attention. In a pioneer work by Gu, Wang and Wen, the concept of equivalence classes of fermionic local unitary(FLU) transformations was proposed to systematically understand non-chiral topological phases in 2D fermion systems and an incomplete classification was obtained. On the other hand, the physical picture of fermion condensation and its corresponding super pivotal categories give rise to a generic mathematical framework to describe fermionic topological phases of quantum matter. In particular, it has been pointed out that in certain fermionic topological phases, there exists the so-called q-type anyon excitations, which have no analogues in bosonic theories. In this paper, we generalize the Gu, Wang and Wen construction to include those fermionic topological phases with qtype anyon excitations. We argue that all non-chiral fermionic topological phases in 2+1D are characterized by a set of tensorsijm,αβ kln,χδ , ni, di), which satisfy a set of nonlinear algebraic equations parameterized by phase factors Ξ ijm,αβ kl , Ξ ij kln,χδ , Ω kim,αβ jland Ω ki jln,χδ . Moreover, consistency conditions among algebraic equations give rise to additional constraints on these phase factors which allow us to construct a topological invariant partition for an arbitrary triangulation of 3D spin manifold. Finally, several examples with q-type anyon excitations are discussed, including the Fermionic topological phase from Tambara-Yamagami category for Z2N , which can be regarded as the Z2N parafermion generalization of Ising fermionic topological phase. CONTENTS Z 2N b. Fermionic topological order TY t,κ Z 2N /ψ N (N odd) C. Equivalence relation for diagonal fusion states

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Factorization and Global Symmetries in Holography

Benini¹,

Copetti²,

Pietro³

2022

Preprint

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We consider toy models of holography arising from 3d Chern-Simons theory. In this context a duality to an ensemble average over 2d CFTs has been recently proposed. We put forward an alternative approach in which, rather than summing over bulk geometries, one gauges a one-form global symmetry of the bulk theory. This accomplishes two tasks: it ensures that the bulk theory has no global symmetries, as expected for a theory of quantum gravity, and it makes the partition function on spacetimes with boundaries coincide with that of a modular-invariant 2d CFT on the boundary. In particular, on wormhole geometries one finds a factorized answer for the partition function. In the case of non-Abelian Chern-Simons theories, the relevant one-form symmetry is non-invertible, and its "gauging" corresponds to the condensation of a Lagrangian anyon.the lack of factorization and lead to well-defined quantum mechanical systems with discrete spectrum.3 Besides, in both cases there usually are 0-form symmetries as well.gauging for non-invertible symmetries. In both cases, after gauging, the bulk Chern-Simons theory acquires the following pleasant properties:1) The theory becomes trivial in the bulk, in the sense that the Euclidean partition function on any (oriented) closed 3-manifold equals 1. After all, this is what we expect from a holographic theory: the degrees of freedom only live at the boundary.2) Given a (possibly disconnected) 2d boundary and a boundary condition, the Euclidean partition function on any 3-manifold with that boundary gives the same result (this follows from point 1). Chern-Simons theory is a generally-covariant theory [37,38], in the sense that its partition function on closed 3-manifolds does not depend on a choice of metric -but it does depend on the topology. After gauging, the theory becomes independent from the bulk topology as well.3) The partition function of the gravitational theory is defined as the one on an arbitrary 3-manifold with the given boundary conditions (not as a sum over all possible geometries, or some subset thereof), because of point 2). Factorization in the case of disconnected boundaries immediately follows.4) The partition function with boundary conditions equals the partition function of a single and well-defined boundary CFT. The details of such a CFT are encoded in the bulk gauge group and in the specific chosen gauging of the 1-form symmetry.5) What we described extends to correlation functions of the boundary theory, and hinges on the fact that all lines are transparent in the bulk (because of point 1).

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Fermionic symmetry fractionalization in (2+1)D

Cited by 3 publications

References 29 publications

Enforced symmetry breaking by invertible topological order

Enforced symmetry breaking by invertible topological order

Towards a complete classification of non-chiral topological phases in 2D fermion systems

Factorization and Global Symmetries in Holography

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