Citation: Bauer, Andreas, Jens Eisert, and Carolin Wille. "A unified diagrammatic approach to topological fixed point models." SciPost Physics Core 5.3 (2022): 038.
Web: https://arxiv.org/abs/2011.12064
Tags: TQFT, Hopf-algebras, Tensor-networks
In this paper, the authors give a tensor-network based approach to topological order which they call "tensor-network path integrals". In this formalism, the tensor networks are not used to represent the ground state of a topological theory, in the way they are with PEPS or MPS. Instead, the tensor networks are tiled on closed manifolds to get partition functions. To compute correlation functions one can insert POVM tensors into the network.
The authors argue that this formalism can be used to appreciate the latent algebraic structure within topological order. The key condition wants out of a topological partition function is that it should not depend on the choice of geometry of the tensor network - it should only depend on the topology of the network. This means that the tensors in the network have to be invariant under certain Pachner-type moves. A tensor can be thought of as an algebra-like map on a vector space. Invariance under the Pachner moves is equivalent to certain restrictions on the algebraic structure, such as associativity. The authors claim that they are able to rederive the axioms of a weak Hopf algebra from such topological considerations, though I can't quite follow their argument.