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

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