Topological phase transition to Abelian anyon phases due to off-diagonal exchange interaction in the Kitaev spin liquid state

We investigate how the Kitaev spin liquid state described by Majorana fermions coupled with $Z_2$ gauge fields is affected by non-Kitaev interactions which exist in real candidate materials of the Kitaev magnet. It is found that the off-diagonal exchange interaction referred to as the ${\mathrm{\Gamma }}^{\mathrm{\prime }}$ term dramatically changes the Majorana band structure in the case of the antiferromagnetic Kitaev interaction, and gives rise to a topological phase transition from a non-Abelian topological phase with the Chern number equal to $\pm 1$ to an Abelian phase with the Chern number equal to $\pm 2$, in which the $Z_2$ vortices behave as Abelian anyons. On the other hand, other non-Kitaev interactions such as the Heisenberg interaction and the $\mathrm{\Gamma }$ term, only affect the bandwidth of the Majorana band as long as the spin liquid state is not destabilized.