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We propose a unified scheme to identify phase transitions out of the$\mathbb{Z}_2$ Abelian topological order, including the transition to anon-Abelian chiral spin liquid. Using loop gas and and string gas states [H.-Y.Lee, R. Kaneko, T. Okubo, N. Kawashima, Phys. Rev. Lett. 123, 087203 (2019)] onthe star lattice Kitaev model as an example, we compute the overlap ofminimally entangled states through transfer matrices. We demonstrate that,similar to the anyon condensation, continuous deformation of a$\mathbb{Z}_2$-injective projected entangled-pair state (PEPS) also allows usto study the transition between Abelian and non-Abelian topological orders. Weshow that the charge and flux anyons defined in the Abelian phase transmuteinto the $\sigma$ anyon in the non-Abelian topological order. Furthermore, weshow that contrary to the claim in [Phys. Rev. B 101, 035140 (2020)], both theLG and SG states have infinite correlation length in the non-Abelian regime,consistent with the no-go theorem that a chiral PEPS has a gapless parentHamiltonian.

Detecting transition between Abelian and non-Abelian topological orders through symmetric tensor networks