Condensation Completion and Defects in 2+1D Topological Orders

TL;DR

This paper reviews the mathematical process of condensation completion in 2+1D topological orders, illustrating its physical realization and applications to various models and defect classifications.

Contribution

It introduces the concept of condensation completion of modular tensor categories and applies it to analyze defects and boundaries in topological orders.

Findings

Explicit enumeration of defects in Toric Code, $3 extbf{F}$, semion, and $bZ_4$ models.

Realization of walls in the Toric Code via Hamiltonian deformation.

Application of condensation completion to classify gapped boundaries and defects.

Abstract

We review the condensation completion of a modular tensor category $\mathcal{C}$, which yields a fusion 2-category $\Sigma\mathcal{C}$ of separable algebras, bimodules over algebras and bimodule maps in $\mathcal{C}$. Physically, $\Sigma\mathcal{C}$ is the fusion 2-category of codimension-1 defects, codimension-2 defects and instantons in the $2+1$D topological order $\mathcal{C}$. We realize the rough-rough wall and $e$-$m$ exchange wall in Toric Code model on the lattice by deforming the Hamiltonian based on the corresponding algebraic data. We apply condensation completion to Toric Code, $3\mathbf{F}$, two-laryer semion and $\mathbb{Z}_4$ topological orders, and explicitly enumerate their $1$d and $0$d defects along with fusion rules. We also mention other applications of condensation completion: alternative interpretations of condensation completion of a braided fusion category; condensation completion of the category of symmetry charges and its correspondence to gapped phases with symmetry; for a topological order $\mathcal{C}$, one can find all gapped boundaries of the stacking of $\mathcal{C}$ with its time-reversal conjugate through computing the condensation completion of $\mathcal{C}$.