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

TL;DR

This paper extends electric-magnetic duality to quantum double models with gapped boundaries, mapping them to Levin-Wen models and revealing boundary dualities and anyon condensation phenomena.

Contribution

It introduces a generalized EM duality for quantum double models with boundaries, characterizes boundary conditions via Frobenius algebras, and maps these models to Levin-Wen models.

Findings

Boundary conditions characterized by Frobenius algebras after transformation

Mapping of quantum double models with boundaries to Levin-Wen models

Elucidation of anyon splitting in anyon condensation phenomena

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.