arXiv subjects

Longye Wang

Publications and source records attributed to Longye Wang.

3 recordsLinked to original sources

arXiv2024

Condensation Completion and Defects in 2+1D Topological Orders

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 $\mathcal{C}$.

cond-mat.str-el↗

arXiv2023

On a class of fusion 2-category symmetry: condensation completion of braided fusion category

Recently, many studies are focused on generalized global symmetry, a mixture of both invertible and non-invertible symmetries in various space-time dimensions. The complete structure of generalized global symmetry is described by higher fusion category theory. In this paper, We first review the construction of fusion 2-category symmetry $\Sigma \cal B$ where $\cal B$ is a a braided fusion category. In particular, we elaborate on the monoidal structure of $\Sigma \cal B$ which determines fusion rules and controls the dynamics of topological operators/defects. We then take $\Sigma \mathrm{sVec}$ as an example to demonstrate how we calculate fusion rule, quantum dimension and 10j-symbol of the fusion 2-category. With our algorithm, all these data can be efficiently encoded and computed in computer program. The complete program will be uploaded to github soon. Our work can be thought as explicitly computing the representation theory of $\cal B$, in analogy to, for example the representation theory of $SU(2)$. The choice of basis bimodule maps are in analogy to the Clebsch-Gordon coefficients and the 10j-symbol are in analogy to the 6j-symbol.

hep-th↗

arXiv2023

Category of SET orders

We propose the representation principle to study physical systems with a given symmetry. In the context of symmetry enriched topological orders, we give the appropriate representation category, the category of SET orders, which include SPT orders and symmetry breaking orders as special cases. For fusion n-category symmetries, we show that the category of SET orders encodes almost all information about the interplay between symmetry and topological orders, in a natural and canonical way. These information include defects and boundaries of SET orders, symmetry charges, explicit and spontaneous symmetry breaking, stacking of SET orders, gauging of generalized symmetry, as well as quantum currents (SymTFT or symmetry TO). We also provide a detailed categorical algorithm to compute the generalized gauging. In particular, we proved that gauging is always reversible, as a special type of Morita-equivalence. The explicit data for ungauging, the inverse to gauging, is given.

cond-mat.str-el↗