spatial inversion symmetry is a prerequisite for any of them to be nonzero. Collective, we denote the sum of these interactions in the spin-density or triplet channel by $S_{\text{int}}^{\text{t}}$.

### Structure of a magnetic order parameter description {#subsubsec:III.C.1}

For applications such as fermionic cold gases one will want to work directly with the fermionic action. For applications to magnets it is convenient to introduce a composite field ${\bm M}(x)$ whose expectation value is proportional to the magnetization. To this end we write \[eqs:3.19\] S\_\^ = dxdyn\_\^i(x)\_[ij]{}(x-y) n\_\^j(y), \[eq:3.19a\] with \_[ij]{}(x-y) &=& (x-y)\_ + \^2(\_x - \_y)d\_[ij]{}([x]{}-[y]{})\
&&-60pt + (x-y).\
\[eq:3.19b\] Here $(a\partial)_k = a_k\,\partial_k$ (no summation convention). We now decouple Eq. (\[eq:3.19a\]) by means of a Hubbard-Stratonovich transformation with a bosonic field ${\bm M}(x)$. Neglecting a constant contribution to the action, this allows us to write S\_\[[|]{},,[M]{}\] &=& S\_0’\[[|]{},\]\
&& - dxdy M\_i(x)\_[ij]{}(x-y)M\_j(y)\
&& + dxdy.\
\[eq:3.20\]

If one neglects the interacting part of $S_0'$ this action depends only bilinearly on the fermion fields, and one can formally integrate out the fermions in order to obtain a theory entire in terms of the order-parameter field ${\bm M}$. This is a generalization of, and replaces, the Hertz-Millis theory.[@Hertz_1976; @Millis_1993] However, in 