D[q]{} + i(

[c]{} a\_x q\_x\
a\_y q\_y\
a\_z q\_z

) + O([q]{}\^3), \[eq:3.15\] and ([q]{}) = [D]{}\^[(+)]{}([q]{}) + [D]{}\^[(-)]{}([q]{}). \[eq:3.16\]

Combining our results, and transforming back to real space, we now have \[eqs:3.17\] S\_ = S\_ + S\_’ + S\_”, \[eq:3.17a\] where S\_ &=& D dx [n]{}\_(x)(\_(x)),\
\[eq:3.17b\]\
S\_’ &=& dx [n]{}\_(x),\
\[eq:3.17c\]\
S\_” &=& dx (\_(x))([d]{}\_(x)). \[eq:3.17d\] $S_{\text{DM}}$ is the Dzyaloshinsky-Moriya interaction that is believed to be responsible for the helimagnetism observed in MnSi and FeGe.[@Bak_Jensen_1980] $S_{\text{DM}}'$ is a closely related term but absent in systems with cubic lattices (e.g., MnSi or FeGe) due to the constraint, Eq. (\[eq:3.14b\]). Finally, $S_{\text{DM}}''$ is another term that is allowed by symmetry, and is generated by the above derivation. All three of these terms are contained in Moriya’s general result,[@Moriya_1960] which takes the form of our Eq. (\[eq:3.9a\]) above, but the effects of $S_{\text{DM}}'$ and $S_{\text{DM}}''$ have, to our knowledge, not been discussed explicitly.

Note that a necessary condition for any of these interactions to be nonzero is that the system is not invariant under parity: Both ${\bm D}^{(1)}$ and ${\bm
D}^{(2)}$ can be nonzero only if ${\cal D}_{{\bm k},{\bm p}}({\bm q})$ is odd under ${\bm q} \to -{\bm q}$, or, 