Mapping Kitaev’s quantum double lattice models to Levin and Wen’s string-net models

2009/07/31 by Oliver Buerschaper, M. Aguado, Miguel Aguado · 3 citations
Mathematics · Physics and Astronomy · #Advanced Condensed Matter Physics #Combinatorics #Discrete mathematics #Equivalence (formal languages) #Geometry #Hilbert space #Lattice (music) #Mathematics #Morita equivalence #Net (polyhedron) #Physics #Physics of Superconductivity and Magnetism #Pure mathematics #Quantum #Quantum many-body systems #Quantum mechanics #String (physics) #Theoretical physics #cond-mat.str-el #quant-ph

paper · pdf · doi:10.1103/physrevb.80.155136

published as Phys. Rev. B 80, 155136 (2009) · 8 pages, 6 eps figures; v2: matches published version

openalex publication_date 2009/10/28 · arxiv created 2009/10/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We exhibit a mapping identifying Kitaev's quantum double lattice models explicitly as a subclass of Levin and Wen's string-net models via a completion of the local Hilbert spaces with auxiliary degrees of freedom. This identification allows one to carry over to these string-net models the representation-theoretic classification of the excitations in quantum double models as well as define them in arbitrary lattices and provides an illustration of the abstract notion of Morita equivalence. The possibility of generalizing the map to broader classes of string nets is considered.

Citations

• Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters
• Criticality, the area law, and the computational power of PEPS
• String-net condensation: A physical mechanism for topological phases
• Fault-tolerant quantum computation by anyons
• Anyons in an exactly solved model and beyond
• Extended Hubbard Model with Ring Exchange: A Route to a Non-Abelian Topological Phase
• Topological quantum field theory
• Rings and Categories of Modules

Cited by

• Gapped Boundaries of Kitaev's Quantum Double Models: A Lattice Realization of Anyon Condensation from Lagrangian Algebras
• Electric–magnetic duality of lattice systems with topological order
• Fusion basis for lattice gauge theory and loop quantum gravity

Related

• From String Nets to Nonabelions
• Twisted Kitaev quantum double model as local topological order
• Electric–magnetic duality of lattice systems with topological order
• Toric code-like models from the parameter space of 3D lattice gauge theories
• Boundary algebras of the Kitaev Quantum Double model