Low-energy excitations in a one-dimensional orthogonal dimer model with   the Dzyaloshinski-Moriya interaction
  Effects of the Dzyaloshinski-Moriya (DM) interaction on low-energy excitations in a one-dimensional orthogonal-dimer model are studied by using the perturbation expansions and the numerical diagonalization method. In the absence of the DM interaction, the triplet excitations show two flat spectra with three-fold degeneracy, which are labeled by magnetization $M=0,\pm{1}$. These spectra split into two branches with M=0 and with $M=\pm{1}$ by switching-on of the DM interaction and besides the curvature appears in the triplet excitations with $M=\pm 1$ more strongly than those of M=0. 
Kenji Sasaoka and Chikara Ishii