Electric-magnetic duality in the quantum double models of topological orders with gapped boundaries

Abstract

We generalize the Electric-magnetic (EM) duality in the quantum double (QD) models to the case of topological orders with gapped boundaries. We also map the QD models with boundaries to the Levin-Wen (LW) models with boundaries. To this end, we Fourier transform and rewrite the extended QD model with a finite gauge group $G$ on a trivalent lattice with a boundary. Gapped boundary conditions of the model before the transformation are known to be characterized by the subgroups $K\subseteq G$. We find that after the transformation, the boundary conditions are then characterized by the Frobenius algebras $A_{G,K}$ in ${\mathrm{Rep}}_G$. An $A_{G,K}$ is the dual space of the quotient of the group algebra of $G$ over that of $K$, and ${\mathrm{Rep}}_G$ is the category of the representations of $G$. The EM duality on the boundary is revealed by mapping the $K$'s to $A_{G,K}$'s. We also show that our transformed extended QD model can be mapped to an extended LW model on the same lattice via enlarging the Hilbert space of the extended LW model. Moreover, our transformed extended QD model elucidates the phenomenon of anyon splitting in anyon condensation.

30+11 pages

References in corpus (7)

• A classification of symmetry enriched topological phases with exactly solvable models
• Twisted Quantum Double Model of Topological Phases in Two--Dimension
• A theory of topological edges and domain walls
• Mapping Kitaev's quantum double lattice models to Levin and Wen's string-net models
• Boundary Hamiltonian theory for gapped topological phases on an open surface
• Twisted Quantum Double Model of Topological Orders with Boundaries
• Boundary Hamiltonian theory for gapped topological orders

Cited by in corpus (12)

• Boundary and domain wall theories of 2d generalized quantum double model
• Electric-Magnetic duality in twisted quantum double model of topological orders
• Extend The Levin-Wen Model To Two-dimensional Topological Orders With Gapped Boundary Junctions
• Weak Hopf symmetry and tube algebra of the generalized multifusion string-net model
• Characteristic Properties of a Composite System of Topological Phases Separated by Gapped Domain Walls via an Exactly Solvable Hamiltonian Model
• Electric-magnetic duality and ${\mathbb{Z}}_2$ symmetry enriched Abelian lattice gauge theory
• Nonabelian Anyon Condensation in 2+1d topological orders: A String-Net Model Realization
• Classification of Gapped Domain Walls of Topological Orders in 2+1 dimensions: A Levin-Wen Model Realization
• Noninvertible Gauge Symmetry in (2+1)d Topological Orders: A String-Net Model Realization
• Finite-group gauge theories on lattices as Hamiltonian systems with constraints
• Nonlinear Symmetry-Fragmentation of Nonabelian Anyons In Symmetry-Enriched Topological Phases: A String-Net Model Realization
• Fourier-transformed gauge theory models of three-dimensional topological orders with gapped boundaries