Citation: Chang, Liang, et al. "On enriching the Levin-Wen model with symmetry." Journal of Physics A: Mathematical and Theoretical 48.12 (2015): 12FT01.
Web: https://arxiv.org/abs/1412.6589
Tags: Modular-tensor-categories, Monoidal-categories, Kitaev-quantum-double, SPT/SETs
This paper introduces the idea that Levin-Wen models based on multi-fusion categories are a natural place to study topological order. The idea is as follows. Given every multi-fusion category, you can construct a Levin-Wen model just like you would ordinarily. The resulting anyon theory is described by the Drinfeld center Z(C). The key point to note is that Z(C) is equivalent to Z(C_{i,i}) where C_{i,i} is the component of C corresponding to some irreducible subobject of the tensor unit. Hence, the resulting anyon theory is not new - it is the Drinfeld center of an ordinary fusion category!
It might seem, then, like multi-fusion Levin-Wen models are not interesting. The interesting feature, however, comes from the fact that even though the algebraic theory is the same the on-site Hamiltonian is different! This on-site Hamiltonian has more degrees of freedom and will often naturally have symmetries that the C_{i,i} component does not. For instance, given any 3-cocycle based on a finite abelian group G one can define a multi-fusion category whose C_{i,i} component is equivalent to Vec_C. The anyon content of the resulting Levin-Wen model is thus trivial, and it naturally has an on-site G symmetry. This recovers the H^3 classification of SPTs, at least in the abelian case! In general, multi-fusion categories are a great way of creating SETs with on-site symmetry.
If you want to go past the Levin-Wen model and into the Turaev-Viro model, you can. If you don't want to assume unitarity you do get a small problem - you need to come up with the right definition of a "spherical multi-fusion category". This has been done, and it has been shown that everything works out just like you'd expect:
> Cui, Shawn X., and Zhenghan Wang. "State sum invariants of three manifolds from spherical multi-fusion categories." Journal of Knot Theory and Its Ramifications 26.14 (2017): 1750104.