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

Abstract

We propose a unified scheme to identify phase transitions out of the Z2 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 Z2-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 σ anyon in the non-Abelian topological order. Furthermore, we show that both the LG and SG states have infinite correlation length in the non-Abelian regime, consistent with the recent claim that a chiral PEPS has a gapless parent Hamiltonian.

Excitation spectrum of spin-1 Kitaev spin liquids

“…Interestingly, we find that the LG projector for the integer spins only supports dispersive excitations belonging to the vacuum-and charge-sectors, while the flux-and fermion-sector excitations are static. This is intrinsically different from the half-integer LG projector, where only vacuum and fermion anyon excitations are dispersive [37]. Interestingly, this is consistent with the argument put forth in Ref.…”

“…[12] up to a phase factor. Note that a Z 2 -invariant PEPS does not necessarily guarantee a Z 2 topologically ordered phase, for the system can be driven into a trivial [30,35] or even a non-Abelian [36,37] phase by a physical deformation of the local tensor. This property makes it as a suitable ansätz to study whether the system harbors a QSL phase and identify possible topological transitions [36,38,39].…”