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Ribbon operators in the Semidual lattice code model

Fred Soglohu, Prince K. Osei, Abdulmajid Osumanu

Abstract

In this work, we provide a rigorous definition of ribbon operators in the Semidual Kitaev lattice model and study their properties. These operators are essential for understanding quasi-particle excitations within topologically ordered systems. We show that the ribbon operators generate quasi-particle excitations at the ends of the ribbon and reveal themselves as irreducible representations of the Bicrossproduct quantum group $M(H)=H^{\text{cop}}\lrbicross H$ or $M(H)^{\text{op}}$ depending on their chirality or local orientation.

Ribbon operators in the Semidual lattice code model

Abstract

In this work, we provide a rigorous definition of ribbon operators in the Semidual Kitaev lattice model and study their properties. These operators are essential for understanding quasi-particle excitations within topologically ordered systems. We show that the ribbon operators generate quasi-particle excitations at the ends of the ribbon and reveal themselves as irreducible representations of the Bicrossproduct quantum group or depending on their chirality or local orientation.
Paper Structure (13 sections, 4 theorems, 77 equations, 3 figures)

This paper contains 13 sections, 4 theorems, 77 equations, 3 figures.

Table of Contents

  1. Introduction & Motivation
  2. Background
  3. Directed graphs
  4. Hopf algebra
  5. Quantum double and Semidualisation
  6. Semidual lattice code model
  7. Ribbon Operators in the Semidual Kitaev Model
  8. Directed ribbons and ribbon types
  9. Ribbon operators for the semidual model
  10. Conclusion
  11. Multiplication of the Ribbon Operators
  12. Relationship between the Ribbon and Geometric Operators
  13. Proof that geometric operators commute with ribbon operators at sites within a ribbon

Key Result

Theorem 3.1

FOM Let $H$ be a finite-dimensional Hopf algebra satisfying $S^2 = {\rm id}$ with dual $H^*$ and the graph $\Gamma$ a square lattice as above. Then the operators $A^h(v,p)$ and $B^a(v,p)$ define an $M(H)$ representation on $H^{*\mathop{\otimes} |E|}$ associated to each site $(v,p)$. Here $(a\mathop{

Figures (3)

  • Figure 2.1: An oriented 2d lattice with examples of direct edges ($e_{i}$), dual edges ($e_{i}^{*}$), vertices ($v_{i}$), faces ($p_{i}$) and sites ($s_{i}$).
  • Figure 3.1: An illustration of the action of the edge/triangle operators on an edge.
  • Figure 4.1: An example of an arbitrary shaped oriented 2d-graph with examples of a direct (dual) triangle $\tau_{i}$($\tilde{\tau}_{i}$) and type-$A$ (type-$B$) ribbon $\rho^{A}_{i}$($\rho^{B}_{i}$). The dual ribbon paths are denoted by the thick dashed lines and the direct ribbon paths are denoted by the thick continuous lines.

Theorems & Definitions (8)

  • Theorem 3.1
  • Definition 4.1: Triangles
  • Proposition 4.2
  • proof
  • Lemma 4.3
  • proof
  • Proposition 4.4
  • proof