On the valence bond solid in the presence of Dzyaloshinskii-Moriya interaction

We examine the stability of the valence bond solid (VBS) phase against the Dzyaloshinskii-Moriya (DM) interaction in the bipartite lattice. Despite the VBS is vulnerable against the antiferromagnetic interaction, for example in the Q-J model proposed by Sandvik, where the quantum phase transition occurs at $J^*/Q = 0.04$, we found that on the contrary the VBS is very stable against the DM interaction. The quantum phase transition does not occur until D/Q goes to infinity, where D is the strength of the DM interaction. The VBS in the ALKT model and the Haldane gap system also exhibit the same property.