nLab anyonic topological order via braided fusion categories -- references

Anyonic topological order in terms of braided fusion categories

Claim and status

In condensed matter theory it is folklore that species of anyonic topological order correspond to braided unitary fusion categories/modular tensor categories.

The origin of the claim may be:

Alexei Kitaev, Section 8 and Appendix E of: Anyons in an exactly solved model and beyond, Annals of Physics 321 1 (2006) 2-111 $[$ doi:10.1016/j.aop.2005.10.005 $]$

Early accounts re-stating this claim (without attribution):

Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, Sankar Das Sarma, pp. 28 of: Non-Abelian Anyons and Topological Quantum Computation, Rev. Mod. Phys. 80 1083 (2008) $[$ arXiv:0707.1888, doi:10.1103/RevModPhys.80.1083 $]$
Zhenghan Wang, Section 6.3 of: Topological Quantum Computation, CBMS Regional Conference Series in Mathematics 112, AMS (2010) $[$ ISBN-13: 978-0-8218-4930-9, pdf $]$

Further discussion (all without attribution):

Eric C. Rowell, Zhenghan Wang, Mathematics of Topological Quantum Computing, Bull. Amer. Math. Soc. 55 (2018), 183-238 (arXiv:1705.06206, doi:10.1090/bull/1605)
Tian Lan, A Classification of (2+1)D Topological Phases with Symmetries $[$ arXiv:1801.01210 $]$
From categories to anyons: a travelogue $[$ arXiv:1811.06670 $]$
Colleen Delaney, A categorical perspective on symmetry, topological order, and quantum information (2019) $[$ pdf, uc:5z384290 $]$
Colleen Delaney, Lecture notes on modular tensor categories and braid group representations (2019) $[$ pdf, pdf $]$
Liang Wang, Zhenghan Wang, In and around Abelian anyon models, J. Phys. A: Math. Theor. 53 505203 (2020) $[$ doi:10.1088/1751-8121/abc6c0 $]$
Parsa Bonderson, Measuring Topological Order, Phys. Rev. Research 3, 033110 (2021) $[$ arXiv:2102.05677, doi:10.1103/PhysRevResearch.3.033110 $]$
Zhuan Li, Roger S.K. Mong, Detecting topological order from modular transformations of ground states on the torus $[$ arXiv:2203.04329 $]$
Eric C. Rowell, Braids, Motions and Topological Quantum Computing $[$ arXiv:2208.11762 $]$
Willie Aboumrad, Quantum computing with anyons: an F-matrix and braid calculator $[$ arXiv:2212.00831 $]$

Emphasis that the expected description of anyons by braided fusion categories had remained folklore, together with a list of minimal assumptions that would need to be shown:

Sachin J. Valera, Fusion Structure from Exchange Symmetry in (2+1)-Dimensions, Annals of Physics 429 (2021) $[$
arXiv:2004.06282, doi:10.1016/j.aop.2021.168471 $]$

Exposition and review:

Sachin Valera, A Quick Introduction to the Algebraic Theory of Anyons, talk at CQTS Initial Researcher Meeting (Sep 2022) $[$ pdf $]$

An argument that the statement at least for SU(2)-anyons does follow from an enhancement of the K-theory classification of topological phases of matter to interacting topological order:

Hisham Sati, Urs Schreiber, Anyonic topological order in TED K-theory, Rev. Math. Phys. (20223) $[$ arXiv:2206.13563, doi:10.1142/S0129055X23500010 $]$

Further discussion

Relation to ZX-calculus:

Fatimah Rita Ahmadi, Aleks Kissinger, Topological Quantum Computation Through the Lens of Categorical Quantum Mechanics $[$ arXiv:2211.03855 $]$

On detection of topological order by observing modular transformations on the ground state:

Zhuan Li, Roger S. K. Mong, Detecting topological order from modular transformations of ground states on the torus, Phys. Rev. B 106 (2022) 235115 $[$ doi:10.1103/PhysRevB.106.235115, arXiv:2203.04329 $]$