Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1810.09469 · Journal of Mathematical Physics

Fusing binary interface defects in topological phases: The Z/pZ case

Jacob C. Bridgeman, Daniel Barter, Corey Jones

Published Dec 1, 2019• 19 citations
arXivHTMLPDF

Abstract

A binary interface defect is any interface between two (not necessarily invertible) domain walls. We compute all possible binary interface defects in Kitaev’s Z/pZ model and all possible fusions between them. Our methods can be applied to any Levin-Wen model. We also give physical interpretations for each of the defects in the Z/pZ model. These physical interpretations provide a new graphical calculus, which can be used to compute defect fusion.

DOI
10.1063/1.5095941

Related papers

Computing defects associated to bounded domain wall structures: the Z / p Z case

Jacob C. Bridgeman, Daniel Barter

Abstract We discuss domain walls and defects in topological phases occurring as the Drinfeld center of some fusion category. Domain walls between such phases correspond to bimodules between the fusion categories. Point defects correspond to functors between the bimodules. A domain wall structure consists of a planar graph with faces labeled by fusion categories. Edges are labeled by bimodules. When the vertices are labeled by point defects we get a compound defect. We present an algorithm, called the domain wall structure algorithm, for computing the compound defect. We apply this algorithm to show that the bimodule associator , related to the O 3 obstruction of Etingof et al (2010 Quantum Topol . 1 209), is trivial for all domain walls of Vec Z / p Z . In the language of this paper, the ground states of the Levin–Wen model are compound defects. We use this to define a generalization of the Levin–Wen model with domain walls and point defects. The domain wall structure algorithm can be used to compute the ground states of these generalized Levin–Wen type models.

arXiv:1901.08069•Journal of Physics A Mathematical and Theoretical•Mar 6, 2020•11 citations

Computing data for Levin-Wen with defects

Jacob C. Bridgeman, Daniel Barter

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.

arXiv:1907.06692•Quantum•Jun 4, 2020•33 citations

2 Models for gapped boundaries and domain walls

Alexei Kitaev, Liang Kong

We define a class of lattice models for two-dimensional topological phases with boundary such that both the bulk and the boundary excitations are gapped. The bulk part is constructed using a unitary tensor category $\calC$ as in the Levin-Wen model, whereas the boundary is associated with a module category over $\calC$. We also consider domain walls (or defect lines) between different bulk phases. A domain wall is transparent to bulk excitations if the corresponding unitary tensor categories are Morita equivalent. Defects of higher codimension will also be studied. In summary, we give a dictionary between physical ingredients of lattice models and tensor-categorical notions.

•Aug 17, 2016•431 citations

Fermionic defects of topological phases and logical gates

Ryohei Kobayashi

We discuss the codimension-1 defects of (2+1)D bosonic topological phases, where the defects can support fermionic degrees of freedom. We refer to such defects as fermionic defects, and introduce a certain subclass of invertible fermionic defects called “gauged Gu-Wen SPT defects” that can shift self-statistics of anyons. We derive a canonical form of a general fermionic invertible defect, in terms of the fusion of a gauged Gu-Wen SPT defect and a bosonic invertible defect decoupled from fermions on the defect. We then derive the fusion rule of generic invertible fermionic defects. The gauged Gu-Wen SPT defects give rise to interesting logical gates of stabilizer codes in the presence of additional ancilla fermions. For example, we find a realization of the CZ logical gate on the (2+1)D \mathbb{Z}_2 ℤ 2 toric code stacked with a (2+1)D ancilla trivial atomic insulator. We also investigate a gapped fermionic interface between (2+1)D bosonic topological phases realized on the boundary of the (3+1)D Walker-Wang model. In that case, the gapped interface can shift the chiral central charge of the (2+1)D phase. Among these fermionic interfaces, we study an interesting example where the (3+1)D phase has a spatial reflection symmetry, and the fermionic interface is supported on a reflection plane that interpolates a (2+1)D surface topological order and its orientation-reversal. We construct a (3+1)D exactly solvable Hamiltonian realizing this setup, and find that the model generates the \mathbb{Z}_8 ℤ 8 classification of the (3+1)D invertible phase with spatial reflection symmetry and fermion parity on the reflection plane. We make contact with an effective field theory, known in literature as the exotic invertible phase with spacetime higher-group symmetry.

arXiv:2211.12394•SciPost Physics•Jul 25, 2023•2 citations

Extend the Levin-Wen model to two-dimensional topological orders with gapped boundary junctions

Hongyu Wang, Yuting Hu, Yidun Wan

Abstract A realistic material may possess defects, which often bring the material new properties that have practical applications. The boundary defects of a two-dimensional topologically ordered system are thought of as an alternative way of realizing topological quantum computation. To facilitate the study of such boundary defects, in this paper, we construct an exactly solvable Hamiltonian model of topological orders with gapped boundary junctions, where the boundary defects reside, by placing the Levin-Wen model on a disk, whose gapped boundary is separated into multiple segments by junctions. We derive a formula of the ground state degeneracy and an explicit ground-state basis of our model. We propose the notion of mobile and immobile charges on the boundary and find that they are quantum observables and label the ground-state basis. Our model is computation friendly.

arXiv:2201.04072•Journal of High Energy Physics•Jul 1, 2022•10 citations

Fusion of one-dimensional gapped phases and their domain walls

David T. Stephen, Xie Chen

Finite depth quantum circuits provide an equivalence relation between gapped phases. Moreover, there can be nontrivial domain walls either within the same gapped phase or between different gapped phases, whose equivalence relations are given by finite depth quantum circuits in one lower dimension. In this paper, we use such unitary equivalence relations to study the fusion of one-dimensional gapped phases. In particular, we use finite depth circuits to fuse two gapped phases, local unitaries to fuse two domain walls, and a combination of both to fuse gapped phases with domain walls. This provides a concrete illustration of some simple aspects of the higher-category structure of gapped defects in a higher-dimensional trivial gapped bulk state.

arXiv:2403.19068•Physical review. B./Physical review. B•Oct 18, 2024•5 citations
A binary interface defect is any interface between two (not necessarily invertible) domain walls. We compute all possible binary interface defects in Kitaev’s Z/pZ model and all possible fusions between them. Our methods can be applied to any Levin-Wen model. We also give physical interpretations for each of the defects in the Z/pZ model. These physical interpretations provide a new graphical calculus, which can be used to compute defect fusion.
Abstract We discuss domain walls and defects in topological phases occurring as the Drinfeld center of some fusion category. Domain walls between such phases correspond to bimodules between the fusion categories. Point defects correspond to functors between the bimodules. A domain wall structure consists of a planar graph with faces labeled by fusion categories. Edges are labeled by bimodules. When the vertices are labeled by point defects we get a compound defect. We present an algorithm, called the domain wall structure algorithm, for computing the compound defect. We apply this algorithm to show that the bimodule associator , related to the O 3 obstruction of Etingof et al (2010 Quantum Topol . 1 209), is trivial for all domain walls of Vec Z / p Z . In the language of this paper, the ground states of the Levin–Wen model are compound defects. We use this to define a generalization of the Levin–Wen model with domain walls and point defects. The domain wall structure algorithm can be used to compute the ground states of these generalized Levin–Wen type models.
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.
We define a class of lattice models for two-dimensional topological phases with boundary such that both the bulk and the boundary excitations are gapped. The bulk part is constructed using a unitary tensor category $\calC$ as in the Levin-Wen model, whereas the boundary is associated with a module category over $\calC$. We also consider domain walls (or defect lines) between different bulk phases. A domain wall is transparent to bulk excitations if the corresponding unitary tensor categories are Morita equivalent. Defects of higher codimension will also be studied. In summary, we give a dictionary between physical ingredients of lattice models and tensor-categorical notions.
We discuss the codimension-1 defects of (2+1)D bosonic topological phases, where the defects can support fermionic degrees of freedom. We refer to such defects as fermionic defects, and introduce a certain subclass of invertible fermionic defects called “gauged Gu-Wen SPT defects” that can shift self-statistics of anyons. We derive a canonical form of a general fermionic invertible defect, in terms of the fusion of a gauged Gu-Wen SPT defect and a bosonic invertible defect decoupled from fermions on the defect. We then derive the fusion rule of generic invertible fermionic defects. The gauged Gu-Wen SPT defects give rise to interesting logical gates of stabilizer codes in the presence of additional ancilla fermions. For example, we find a realization of the CZ logical gate on the (2+1)D \mathbb{Z}_2 ℤ 2 toric code stacked with a (2+1)D ancilla trivial atomic insulator. We also investigate a gapped fermionic interface between (2+1)D bosonic topological phases realized on the boundary of the (3+1)D Walker-Wang model. In that case, the gapped interface can shift the chiral central charge of the (2+1)D phase. Among these fermionic interfaces, we study an interesting example where the (3+1)D phase has a spatial reflection symmetry, and the fermionic interface is supported on a reflection plane that interpolates a (2+1)D surface topological order and its orientation-reversal. We construct a (3+1)D exactly solvable Hamiltonian realizing this setup, and find that the model generates the \mathbb{Z}_8 ℤ 8 classification of the (3+1)D invertible phase with spatial reflection symmetry and fermion parity on the reflection plane. We make contact with an effective field theory, known in literature as the exotic invertible phase with spacetime higher-group symmetry.
Abstract A realistic material may possess defects, which often bring the material new properties that have practical applications. The boundary defects of a two-dimensional topologically ordered system are thought of as an alternative way of realizing topological quantum computation. To facilitate the study of such boundary defects, in this paper, we construct an exactly solvable Hamiltonian model of topological orders with gapped boundary junctions, where the boundary defects reside, by placing the Levin-Wen model on a disk, whose gapped boundary is separated into multiple segments by junctions. We derive a formula of the ground state degeneracy and an explicit ground-state basis of our model. We propose the notion of mobile and immobile charges on the boundary and find that they are quantum observables and label the ground-state basis. Our model is computation friendly.
Finite depth quantum circuits provide an equivalence relation between gapped phases. Moreover, there can be nontrivial domain walls either within the same gapped phase or between different gapped phases, whose equivalence relations are given by finite depth quantum circuits in one lower dimension. In this paper, we use such unitary equivalence relations to study the fusion of one-dimensional gapped phases. In particular, we use finite depth circuits to fuse two gapped phases, local unitaries to fuse two domain walls, and a combination of both to fuse gapped phases with domain walls. This provides a concrete illustration of some simple aspects of the higher-category structure of gapped defects in a higher-dimensional trivial gapped bulk state.