the same bonds.

The space group symmetry of the lattice determines the pattern of relative signs of the DM vectors on each of these bonds (see Appendix \[sec:dmv\]). We find the following form $$\begin{aligned}
  \label{eq:11}
  H_D & = & \sum_{xyz} \Big\{ {\bm D}_{y,z}\cdot {\bm S}_{x,y,z}\times
  {\bm S}_{x+1,y,z} \nonumber
\\ && + {\bm D}^+_{y,z}\cdot {\bm S}_{x,y,z}\times
  {\bm S}_{x+1/2,y+1,z} \nonumber \\ && + {\bm D}^-_{y,z}\cdot {\bm S}_{x,y,z}\times
  {\bm S}_{x-1/2,y+1,z}\Big\},\end{aligned}$$ where $$\begin{aligned}
  \label{eq:18}
  {\bm D}_{y,z} & = & D_a (-1)^z \hat{a} + D_c (-1)^y \hat{c}, \label{DM_chain}\\
  {\bm D}^\pm_{y,z} & = & \pm D'_a (-1)^z \hat{a} + D'_b (-1)^{y+z}
  \hat{b} \pm D'_c (-1)^{y+z}\hat{c},\nonumber \label{DM_diagonal}\\ &&\end{aligned}$$ and $D'_a\equiv D$ is the DM term in the standard model. The relative signs of the DM vectors are graphically shown in Fig. \[fig: DM\_distribution\].

One sees that symmetry allows five distinct DM couplings: $D=D'_a$, $D'_b$, $D'_c$, $D_a$ and $D_c$. Of these, only $D'_a$ and $D_c$ will be invoked in the body of this paper. The remaining three can be safely neglected, as explained in Appendix \[sec:negl-dm-coupl\].

![Distribution of the DM vectors. Sites 1,2,3, and 4 correspond to $(x,y,z)$, $(x+\frac{1}{2},y+1,z)$, $(x,y,z+1)$, and $(x+\frac{1}{2},y+1,z+1)$, respectively. (a) The signs $\otimes$, $\odot$ refer to the direction of the DM vectors along $a$ 