A unified view on symmetry, anomalous symmetry and non-invertible gravitational anomaly

Ji, Wenjie; Wen, Xiao-Gang

doi:10.48550/arxiv.2106.02069

Cited by 11 publications

(11 citation statements)

References 40 publications

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“…Similar to the anomaly constraints from an ordinary symmetry, these non-invertible topological defect lines have dramatic consequences on renormalization group flows. In 1+1 dimensional quantum field theory (QFT), the topological lines have been explored extensively in [5,6,23,20,24,12,[25][26][27][28][29][30][31][32][33][34][35]22] with various dynamical applications. In particular, using a modular invariance argument, it was shown in [6] (see also [20] for generalizations) that the existence of certain non-invertible topological lines is incompatible with a trivially gapped phase.…”

Section: Introductionmentioning

confidence: 99%

Non-Invertible Duality Defects in 3+1 Dimensions

Choi,

Cordova,

Hsin

et al. 2021

Preprint

129

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For any quantum system invariant under gauging a higher-form global symmetry, we construct a non-invertible topological defect by gauging in only half of spacetime. This generalizes the Kramers-Wannier duality line in 1+1 dimensions to higher spacetime dimensions. We focus on the case of a one-form symmetry in 3+1 dimensions, and determine the fusion rule. From a direct analysis of one-form symmetry protected topological phases, we show that the existence of certain kinds of duality defects is intrinsically incompatible with a trivially gapped phase. We give an explicit realization of this duality defect in the free Maxwell theory, both in the continuum and in a modified Villain lattice model. The duality defect is realized by a Chern-Simons coupling between the gauge fields from the two sides. We further construct the duality defect in non-abelian gauge theories and the Z N lattice gauge theory.

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Section: Introductionmentioning

confidence: 99%

Non-Invertible Duality Defects in 3+1 Dimensions

Choi,

Cordova,

Hsin

et al. 2021

Preprint

129

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“…where S and T are the S and T matrices of the bulk anyon theory. We now generalize this discussion to boundary theories that are not necessarily fully chiral [38,39]. Again we assume that the boundary is described by a CFT, which could be chiral or non-chiral.…”

Section: Topological Phase With Cft Entanglement Spectrummentioning

confidence: 99%

“…In fact, M(6, 5) can be obtained from the Z 3 parafermion CFT by orbifolding the Z 2 charge conjugation symmetry. The partition function of the boundary CFT in the vacuum sector is given by [39] Z…”

Section: Exactly Solvable Limitmentioning

confidence: 99%

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Topological Disorder Parameter

Chen,

Tu,

Meng

et al. 2022

Preprint

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We introduce a many-body topological invariant, called the topological disorder parameter (TDP), to characterize gapped quantum phases with global internal symmetry in (2+1)d. TDP is defined as the constant correction that appears in the ground state expectation value of a partial symmetry transformation applied to a connected spatial region M , the absolute value of which scales generically as exp(−αl+γ) where l is the perimeter of M and γ is the TDP. Motivated by a topological quantum field theory interpretation of the operator, we show that e γ can be related to the quantum dimension of the symmetry defect, and provide a general formula for γ when the entanglement Hamiltonian of the topological phase can be described by a (1+1)d conformal field theory (CFT). A special case of TDP is equivalent to the topological Rényi entanglement entropy when the symmetry is the cyclic permutation of the replica of the gapped phase. We then investigate several examples of lattice models of topological phases, both analytically and numerically, in particular when the assumption of having a CFT edge theory is not satisfied. We also consider an example of partial translation symmetry in Wen's plaquette model and show that the result can be understood using the edge CFT. Our results establish a new tool to detect quantum topological order.

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“…The MPO representations of the algebra are characterized by the data of a fusion category, in the form of F -symbols that satisfy the well-known pentagon equations [15,16]. These MPO symmetries are referred to as topological defects or categorical symmetries, and they have become ubiquitous in the description of both topologically ordered systems in (2+1)d and their (1+1)d boundary theories [17][18][19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Classifying phases protected by matrix product operator symmetries using matrix product states

Garre-Rubio¹,

Lootens²,

Molnár³

2022

Preprint

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We classify the different ways in which matrix product states (MPS) can transform under the action of matrix product operator (MPO) symmetries. We give a local characterization of MPS tensors that generate ground spaces which remain invariant under a global MPO symmetry and generally correspond to symmetry breaking phases. This allows us to derive a set of quantities, satisfying the so-called coupled pentagon equations, invariant under continuous deformations of the MPS tensor, which provides the invariants to detect different gapped phases protected by MPO symmetries. The results obtained match with previous TQFT approaches classifying non-onsite symmetries, our techniques, however, extend the classification beyond renormalization fixed points and facilitates the numerical study of these systems. When the MPO symmetries form a global onsite representation of a group, we recover the symmetry protected topological order classification for unique and degenerate ground states. When the MPO symmetries form a representation of a group, but this representation is not onsite, we obtain restrictions on the possible ground state degeneracies depending both on the group and on the concrete form of the MPO representation. Moreover, we study the interplay between time reversal symmetry and an MPO symmetry and also provide examples of our classification together with explicit constructions of MPO and MPS. Finally, we elaborate on the connection between our setup and gapped boundaries of two-dimensional topological systems, where non-onsite symmetries also play a key role.

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A unified view on symmetry, anomalous symmetry and non-invertible gravitational anomaly

Cited by 11 publications

References 40 publications

Non-Invertible Duality Defects in 3+1 Dimensions

Non-Invertible Duality Defects in 3+1 Dimensions

Topological Disorder Parameter

Classifying phases protected by matrix product operator symmetries using matrix product states

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