equivalently, if ${\cal D}({\bm x},{\bm
y};{\bm z})$ is odd under ${\bm z} \to -{\bm z}$. This implies that the DM interaction requires a lattice that is not invariant under spatial inversion; in any continuum model, where the electron-electron interaction is necessarily even under parity, it vanishes. See Appendix \[app:D\] and Sec. \[sec:IV\] for further discussions of this point.

Fermionic action, and magnetic order parameter {#subsec:III.C}
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We now have the following result. After integrating out the photons, the effective fermionic action reads S\_\[|,\] &=& S\_0 + S\_’ + S\_ + S\_ + S\_ + S\_’ + S\_”\
&& S\_0’ + S\_\^. \[eq:3.18\] Here $S_0$ describes non-interacting electrons (either free electrons or band electrons, depending on the model considered), and $S_{\text{int}}'$ contains all interactions between modes other than the spin density, which we have not explicitly considered with the exception of the Coulomb interaction, Eqs.(\[eqs:3.3\]). Collectively we denote these two terms by $S_0'$. $S_{\text{ex}}$, $S_{\text{d-d}}$, $S_{\text{DM}}$, $S_{\text{DM}}$, and $S_{\text{DM}}''$ are the exchange, dipole-dipole, and Dzyaloshinsky-Moriya interactions given by Eqs. (\[eq:3.4\]), (\[eq:3.5\]), and (\[eqs:3.17\]), respectively. $S_{\text{ex}}$ and $S_{\text{d-d}}$ are always present; which, if any, of the terms in $S_{\text{DM}}$ are nonzero depends on the details of the lattice structure and absence of 