Computing defects associated to bounded domain wall structures: the $\mathbb{Z}/p\mathbb{Z}$ case

Bridgeman, Jacob C.; Barter, Daniel

doi:10.1088/1751-8121/ab7d60

Cited by 7 publications

(9 citation statements)

References 54 publications

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“…This is the essence of the 'inflation trick' developed in [BBJ18,BB19b,BB19a]. We refer the interested reader to these references for more details.…”

Section: Fusion Categoriesmentioning

confidence: 99%

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Gauging defects in quantum spin systems: A case study

et al. 2020

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The goal of this work is to build a dynamical theory of defects for quantum spin systems. A kinematic theory for an indefinite number of defects is first introduced exploiting distinguishable Fock space. Dynamics are then incorporated by allowing the defects to become mobile via a microscopic Hamiltonian. This construction is extended to topologically ordered systems by restricting to the ground state eigenspace of Hamiltonians generalizing the golden chain. We illustrate the construction with the example of a spin chain with Vec(Z/2Z) fusion rules, employing generalized tube algebra techniques to model the defects in the chain. The resulting dynamical defect model is equivalent to the critical transverse Ising model.

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“…This is the essence of the 'inflation trick' developed in [BBJ18,BB19b,BB19a]. We refer the interested reader to these references for more details.…”

Section: Fusion Categoriesmentioning

confidence: 99%

“…For fixed representation (fixed, x, y), we find a unique vector up to a multiplicative scalar for each α , so each morphism space is one dimensional. We define the basis to be The 'inflation trick' developed in [BBJ18,BB19b,BB19a] picks out the representation P 0,0 , and we work with that from here on. This means…”

Section: Computation Of the Bimodule Vertexmentioning

confidence: 99%

Gauging defects in quantum spin systems: A case study

et al. 2020

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“…This paper is a continuation of the work begun in Refs. [34][35][36]. In the current work, we generalize the methods developed there for Vec(Z/pZ) to more general input (unitary) fusion categories, and provide code to compute essentially complete data on the defect theories.…”

Section: What Is Being Computed In This Workmentioning

confidence: 99%

“…The term domain wall structure first appeared in Ref. [36]. Surfaces decorated with domain walls play a central role in the theory of defect TQFTs.…”

Section: Definition 4 (Vec(g)mentioning

confidence: 99%

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Computing data for Levin-Wen with defects

Bridgeman

Barter

2020

Quantum

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We demonstrate how to do many computations for doubled topological phases with defects. These defects may be 1-dimensional domain walls or 0-dimensional point defects.Using Vec⁡(S3) as a guiding example, we demonstrate how domain wall fusion and associators can be computed using generalized tube algebra techniques. These domain walls can be both between distinct or identical phases. Additionally, we show how to compute all possible point defects, and the fusion and associator data of these. Worked examples, tabulated data and Mathematica code are provided.

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Invertible Bimodule Categories and Generalized Schur Orthogonality

Bridgeman

Lootens

Verstraete

2023

Commun. Math. Phys.

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The Schur orthogonality relations are a cornerstone in the representation theory of groups. We utilize a generalization to weak Hopf algebras to provide a new, readily verifiable condition on the skeletal data for deciding whether a given bimodule category is invertible and therefore defines a Morita equivalence. As a first application, we provide an algorithm for the construction of the full skeletal data of the invertible bimodule category associated to a given module category, which is obtained in a unitary gauge when the underlying categories are unitary. As a second application, we show that our condition for invertibility is equivalent to the notion of MPO-injectivity, thereby closing an open question concerning tensor network representations of string-net models exhibiting topological order. We discuss applications to generalized symmetries, including a generalized Wigner-Eckart theorem.

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Computing defects associated to bounded domain wall structures: the $\mathbb{Z}/p\mathbb{Z}$ case

Abstract: Definition 3 (domain wall structure). A domain wall structure consists of a graph embedded into a disc where the edges don't have critical points. We label the faces of the graph with fusion categories, and the edges of the graph * * c + bl + n

Cited by 7 publications

References 54 publications

Gauging defects in quantum spin systems: A case study

Gauging defects in quantum spin systems: A case study

Computing data for Levin-Wen with defects

Invertible Bimodule Categories and Generalized Schur Orthogonality

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