Citation: Bridgeman, Jacob C., Benjamin J. Brown, and Samuel J. Elman. "Boundary topological entanglement entropy in two and three dimensions." Communications in Mathematical Physics 389.2 (2022): 1241-1276.
Web: https://arxiv.org/abs/2012.05244
Tags: Modular-tensor-categories, Higher-dimensional, Information-theory
This paper computes the topological entanglement entropy associated with the bulk and boundary of Levin-Wen and Walker-Wang models. In the case of the Levin-Wen model, the bulk TEE is well-known and it has the classical formula 2log(D). Boundary TEE in the Levin-Wen model is the same up to a factor of two, and is independent of the boundary-type chosen.
By far the most interesting result of this paper is the computation of bulk TEE in the Walker-Wang model. Just like before, the answer is 2log(D) where this time D is the quantum dimension of the Drinfeld center of the input premodular category. The amazing feature of this paper is that the answer is conjectural and is only proved in certain special cases! The point is that the TEE can be reduced to computing a sum of the standard von Neumann entropy form \sum_{\lambda}\lambda \cdot \log(\lambda). The \lambda index in this sum runs over the singular values of a certain matrix the authors call the "connected S-matrix". The authors then conjecturally relate the value of this sum to the quantum dimension of the Drinfeld center! Thus, this fundamental formula of TEE in 3D reduces to an algebraic conjecture about the structure of premodular categories.
This is a lovely little open problem, which I'm sure I could solve if I put my mind to it.