as the exchange interaction, Eq. (\[eq:3.4\]), but is much smaller, as $\muB^2
\approx \Gamma_{\text{T}}\,(\vF/c)^2$. The second one is the dipole-dipole interaction between the electron spins; if one replaces the electronic spin density by its quantum mechanical and thermal average one recovers the classical dipole-dipole term in Eq. (\[eq:2.4a\]) or (\[eq:2.7a\]).

### $O(1/c^2)$: Dzyaloshinsky-Moriya and related interactions {#subsubsec:III.B.3}

We now return to the effects of integrating out the scalar potential $\varphi$. To $O(1/c^2)$ the relevant contribution comes from the cross-term that multiplies the Coulomb ($\varphi\,n$) term and the spin-orbit (last) term in Eq. (\[eq:3.1\]). Contracting the two scalar potentials, integrating by parts, using Eq. (\[eq:C.2b\]), and keeping only terms that are bilinear in phase-space spin-density fluctuations, we obtain a contribution to the effective action S\_ &=& \_[ilm]{}\_[ijk]{} dxdy [D]{}(x-y) [|]{}\_[\_1]{}(x)\^l\_[\_1\_4]{}\
&&0 pt (\_[\_4]{}(y)) ([|]{}\_[\_3]{}(y)) \^m\_[\_3 \_2]{} \_[\_2]{}(x).\
\[eq:3.6\] After a Fourier transform, Eq. (\[eq:3.6\]) can be written

S\_ &=& ()\^2 \_[q,k,p]{} [D]{}([q]{}) (\_[\_1\_4]{}\_[\_3\_2]{}) . \[eq:3.7\] Here $k = ({\bm k},i\omega_n)$ comprises a wave vector ${\bm k}$ and a fermionic Matsubara frequency $\omega_n = 2\pi T(n+1/2)$, and $p$ and $q$ are used analogously. At this point we generalize to an effective interaction amplitude ${\cal D}_{{\bm k},{\bm p}}({\bm q})$ that depends on ${\bm 