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

TL;DR

This paper introduces a unified tensor network approach to detect phase transitions from Abelian to non-Abelian topological orders, demonstrating charge-flux transmutation and analyzing correlation lengths in the non-Abelian regime.

Contribution

It presents a novel method using symmetric tensor networks to identify topological phase transitions and characterizes anyon transmutation during the transition.

Findings

Charge and flux anyons transmute into sigma anyons in the non-Abelian phase.

Both loop gas and string gas states have infinite correlation length in the non-Abelian regime.

The method confirms the gapless nature of chiral PEPS in non-Abelian topological order.

Abstract

We propose a unified scheme to identify phase transitions out of the $\mathbb{Z}_2$ Abelian topological order, including the transition to a non-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)] on the star lattice Kitaev model as an example, we compute the overlap of minimally 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 us to study the transition between Abelian and non-Abelian topological orders. We show that the charge and flux anyons defined in the Abelian phase transmute into the $\sigma$ anyon in the non-Abelian topological order. Furthermore, we show that contrary to the claim in [Phys. Rev. B 101, 035140 (2020)], both the LG 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 parent Hamiltonian.