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The topologically nontrivial magnetic phase with a $Z_2$-vortex (Z$_2$V)lattice is investigated by simulation in the Kitaev-Heisenberg honeycomb modelexpanded by considering the second- and third-nearest-neighbor Heisenberginteractions ($J_{H2}$ and $J_{H3}$). On the parameter region of the Z$_2$Vphase, a gradual modulation of vortex density is observed, together with atransition from single-Z$_2$V to triple-Z$_2$V state driven by the variation offrustration. Additionally, $Z_2$ vortices are arranged in different manners onthe whole honeycomb structure for these two types of Z$_2$V states. Moreover,topologically equivalent states are revealed to exist in single-Z$_2$V dominantand triple-Z$_2$V dominant styles on different parameter points, which can becontrolled to switch between each other without energy consumption. It is worthnoting that $J_{H3}$ plays a key role in expanding the Z$_2$V phase, and alsoin stabilizing the single-Z$_2$V state.

physical review. b./physical review. b

Stabilization and modulation of the topological magnetic phase with a  $Z_2$-vortex lattice in the Kitaev-Heisenberg honeycomb model: The key role  of the third-nearest-neighbor interaction