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Weak Hopf symmetry and tube algebra of the generalized multifusion string-net model

Zhian Jia, Sheng Tan, Dagomir Kaszlikowski

TL;DR

<3-5 sentence high-level summary> The paper develops a comprehensive framework for the generalized multifusion Levin-Wen string-net model with weak Hopf symmetry. It builds a macroscopic theory connecting unitary multifusion categories to weak Hopf algebras, and introduces a weak Hopf tube algebra whose representation category matches the bulk topological excitations; gapped boundaries and domain walls are described via boundary and domain-wall tube algebras, anyon condensation, and Morita theory. The authors provide explicit lattice constructions of the generalized string-net ground states, a local commutative projector Hamiltonian, and a detailed tube-algebra treatment of bulk, boundary, and domain-wall excitations, illustrating how defective string-nets arise as restricted multifusion string-nets. The work lays a foundation for analyzing SET phases, boundary phenomena, and defects within a unified weak Hopf symmetry framework, with potential impact on quantum codes and topological quantum computation.

Abstract

We investigate the multifusion generalization of string-net ground states and lattice Hamiltonians, delving into its associated weak Hopf symmetry. For the multifusion string-net, the gauge symmetry manifests as a general weak Hopf algebra, leading to a reducible vacuum string label; the charge symmetry, serving as a quantum double of gauge symmetry, constitutes a connected weak Hopf algebra. This implies that the associated topological phase retains its characterization by a unitary modular tensor category (UMTC). The bulk charge symmetry can also be captured by a weak Hopf tube algebra. We offer an explicit construction of the weak Hopf tube algebra structure and thoroughly discuss its properties. The gapped boundary and domain wall models are extensively discussed, with these $1d$ phases characterized by unitary multifusion categories (UMFCs). We delve into the gauge and charge symmetries of these $1d$ phases, as well as the construction of the boundary and domain wall tube algebras. Additionally, we illustrate that the domain wall tube algebra can be regarded as a cross product of two boundary tube algebras. As an application of our model, we elucidate how to interpret the defective string-net as a restricted multifusion string-net.

Weak Hopf symmetry and tube algebra of the generalized multifusion string-net model

TL;DR

<3-5 sentence high-level summary> The paper develops a comprehensive framework for the generalized multifusion Levin-Wen string-net model with weak Hopf symmetry. It builds a macroscopic theory connecting unitary multifusion categories to weak Hopf algebras, and introduces a weak Hopf tube algebra whose representation category matches the bulk topological excitations; gapped boundaries and domain walls are described via boundary and domain-wall tube algebras, anyon condensation, and Morita theory. The authors provide explicit lattice constructions of the generalized string-net ground states, a local commutative projector Hamiltonian, and a detailed tube-algebra treatment of bulk, boundary, and domain-wall excitations, illustrating how defective string-nets arise as restricted multifusion string-nets. The work lays a foundation for analyzing SET phases, boundary phenomena, and defects within a unified weak Hopf symmetry framework, with potential impact on quantum codes and topological quantum computation.

Abstract

We investigate the multifusion generalization of string-net ground states and lattice Hamiltonians, delving into its associated weak Hopf symmetry. For the multifusion string-net, the gauge symmetry manifests as a general weak Hopf algebra, leading to a reducible vacuum string label; the charge symmetry, serving as a quantum double of gauge symmetry, constitutes a connected weak Hopf algebra. This implies that the associated topological phase retains its characterization by a unitary modular tensor category (UMTC). The bulk charge symmetry can also be captured by a weak Hopf tube algebra. We offer an explicit construction of the weak Hopf tube algebra structure and thoroughly discuss its properties. The gapped boundary and domain wall models are extensively discussed, with these phases characterized by unitary multifusion categories (UMFCs). We delve into the gauge and charge symmetries of these phases, as well as the construction of the boundary and domain wall tube algebras. Additionally, we illustrate that the domain wall tube algebra can be regarded as a cross product of two boundary tube algebras. As an application of our model, we elucidate how to interpret the defective string-net as a restricted multifusion string-net.
Paper Structure (26 sections, 24 theorems, 101 equations, 5 figures, 3 tables)

This paper contains 26 sections, 24 theorems, 101 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Macroscopic theory of multifusion string-net and weak Hopf symmetry
  3. Unitary multifusion category
  4. Weak Hopf symmetry for $2d$ topological phase
  5. Macroscopic theory of multifusion string-net
  6. Generalized multifusion string-net model
  7. Input data of multifusion string-net model
  8. Generalized multifusion string-net ground states
  9. Generalized multifusion string-net lattice Hamiltonian
  10. Topological ground state degeneracy
  11. Weak Hopf tube algebra and topological excitations
  12. Bulk tube algebra as weak Hopf algebra
  13. Tube algebra as a crossed product
  14. Topological excitation and representation of tube algebra
  15. Morita theory for the tube algebra
  16. ...and 11 more sections

Key Result

Lemma 1

For an indecomposable multifusion category $\EuScript{D}=\oplus_{i,j\in I}\EuScript{D}_{i,j}$, we have

Figures (5)

  • Figure 1: Pentagon relation for the F-symbols.
  • Figure 2: Depiction of the generalized multifusion string-net.
  • Figure 3: Depiction of the tube algebra.
  • Figure 4: Illustration of bulk-to-boundary anyon condensation based on tube algebra.
  • Figure 5: (a) A trivalent lattice in the vicinity of the boundary (bold arrows). (b) A disk with the bulk labeled by $\EuScript{D}_{i,i}=\EuScript{C}$ and the boundary labeled by $\EuScript{D}_{i,j}=\EuScript{M}$.

Theorems & Definitions (73)

  • Definition 1
  • Example 1: UMFC $\mathsf{Mat}_n$
  • Lemma 1
  • proof
  • Definition 2: Quantum double, see Ref. Jia2023weak for details
  • Example 2
  • Definition 3: Weak Hopf symmetry
  • Remark 2.1
  • Lemma 2
  • proof
  • ...and 63 more