Citation: Kim, Isaac H., and Daniel Ranard. "Classifying 2D topological phases: mapping ground states to string-nets." arXiv preprint arXiv:2405.17379 (2024).
Web: https://arxiv.org/abs/2405.17379
Tags: MTC-reconstruction, Defects/boundaries, Kitaev-quantum-double
In this paper, the authors prove that every state which admits a gapped boundary is equivalent to the ground state of a Levin-Wen model based on some fusion category by some local finite-depth unitary circuit.
The fusion category whose string-net they map to is described the the boundary defects. The superselection sectors and fusion rules were already defined in earlier work. One new feature is that they are also able to derive the F-symbols of the boundary theory. They do this by adapting the approach of Kawagoe-Levin the boundary superselection sectors:
> Kim, Isaac H., and Daniel Ranard. "Classifying 2D topological phases: mapping ground states to string-nets." arXiv preprint arXiv:2405.17379 (2024).
The idea of the overall proof is the introduce gapped boundaries and constrain the original degrees of freedom, until they live entirely in contractible regions so they can be ignored. This process of restricting where the degrees of freedom can live involves creating boundary defects, so the final state is described by a big superposition of boundary defects. That is, a big superposition of assignments of labels in the underlying fusion category of the boundary defects! This big superposition is exactly the string-net the authors are trying to construct.
This paper is fantastic progress. One big issue with it is that it assumes zero correlation length - this sort of assumption needs to be made because entanglement bootstrap only works on zero correlation length. As soon as entanglement bootstrap is extended this paper should give a proof of the classification of phases with gappable boundaries, however.
The previous state-of-the-art on this question was the work of Lin-Levin:
> Lin, Chien-Hung, and Michael Levin. "Generalizations and limitations of string-net models." Physical Review B 89.19 (2014): 195130.
This paper gave a strong derivation of Lagrangian subgroups and string-nets from gapped boundaries in the case of abelian anyons.