Condensation Completion and Defects in 2+1D Topological Orders
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| Format: | Preprint |
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2024
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| author | Yue, Gen Wang, Longye Lan, Tian |
| author_facet | Yue, Gen Wang, Longye Lan, Tian |
| contents | We review the condensation completion of a modular tensor category $\mathcal{C}$, which yields a fusion 2-category $Σ\mathcal{C}$ of separable algebras, bimodules over algebras and bimodule maps in $\mathcal{C}$. Physically, $Σ\mathcal{C}$ 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 $\mathcal{C}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_19253 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Condensation Completion and Defects in 2+1D Topological Orders Yue, Gen Wang, Longye Lan, Tian Strongly Correlated Electrons High Energy Physics - Theory We review the condensation completion of a modular tensor category $\mathcal{C}$, which yields a fusion 2-category $Σ\mathcal{C}$ of separable algebras, bimodules over algebras and bimodule maps in $\mathcal{C}$. Physically, $Σ\mathcal{C}$ 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 $\mathcal{C}$. |
| title | Condensation Completion and Defects in 2+1D Topological Orders |
| topic | Strongly Correlated Electrons High Energy Physics - Theory |
| url | https://arxiv.org/abs/2402.19253 |