Symmetric tensor network studies of Kitaev spin liquids

We study the spin-1/2 star and spin-1 honeycomb Kitaev models using $\mathbb{Z}$_2 invariant Projected Entangled Pair States (PEPS). We find that the transition from Abelian to non-Abelian phase of the star lattice Kitaev model can be understood as charge and flux transmuting into $\sigma$ anyon in the $\mathbb{Z}$_2 invariant PEPS framework. Besides, we show that the ground state always has infinite correlation length in the non-Abelian regime, consistent with the no-go theorem that a chiral PEPS has a gapless parent Hamiltonian. On the other hand, we confirm the $\mathbb{Z}$_2 spin liquid nature of isotopic spin-1 Kitaev honeycomb model by evaluating the virtual order parameters defined on the virtual Hilbert space of the tensor network. Using the correspondence between the transfer matrix spectrum and low-lying excitations, we find that contrary to the dispersive Majorana excitation in the spin-1/2 case, the system has a dispersive charge anyon excitation. Bottom of the gapped single-particle charge anon excitations are also identified with the corresponding correlation length $\xi \approx 6.7$ unit cells.