the Heisenberg spins introduced in Eq.  by restricting ourselves to ${\boldsymbol{\sigma}}=(\sigma_1)$ or ${\boldsymbol{\sigma}}=(\sigma_1,\sigma_2)$, respectively.

When formulating the mean-field theories for charge-density wave (CDW) order and spin-density wave (SDW) order, we consider these two channels separately. The interaction projected onto the respective channels takes the form

\[eq:interaction\] $$\begin{aligned}
 \mathcal{H}_\mathrm{ch}&=-\frac{V_\mathrm{ch}}{8}\sum_{\sigma\sigma^\prime}\sum_{{\boldsymbol{k}}{\boldsymbol{k}}^\prime{\boldsymbol{q}}} (\psi^\dagger_{{\boldsymbol{k}},\sigma}\psi^{}_{{\boldsymbol{k}}+{\boldsymbol{q}},\sigma})(\psi^\dagger_{{\boldsymbol{k}}^\prime,\sigma^\prime}\psi^{}_{{\boldsymbol{k}}^\prime-{\boldsymbol{q}},\sigma^\prime}) {~ ,}\\ 
 \mathcal{H}_\mathrm{sp}&=-\frac{V_\mathrm{sp}}{8}\sum_{\sigma_1^{}\sigma_2^{}}\sum_{\sigma_1^\prime\sigma_2^\prime}\sum_{{\boldsymbol{k}}{\boldsymbol{k}}^\prime{\boldsymbol{q}}} (\psi^\dagger_{{\boldsymbol{k}},\sigma_1}{\boldsymbol{\sigma}}_{\sigma_1\sigma_2}\psi^{}_{{\boldsymbol{k}}+{\boldsymbol{q}},\sigma_2}) \nonumber \\ 
 &\qquad \qquad  \cdot (\psi^\dagger_{{\boldsymbol{k}}^\prime,\sigma^\prime_1}{\boldsymbol{\sigma}}_{\sigma_1^\prime\sigma_2^\prime}\psi^{}_{{\boldsymbol{k}}^\prime-{\boldsymbol{q}},\sigma^\prime_2}){~ ,}\end{aligned}$$

where we assumed the couplings $U_{\mathrm{ch},\mathrm{sp}}$ to be independent of momenta and thus introduced $U_{\mathrm{ch},\mathrm{sp}}({\boldsymbol{k}}_1,{\boldsymbol{k}}_2;{\boldsymbol{k}}_3,{\boldsymbol{k}}_4)\equiv -V_{\mathrm{ch},\mathrm{sp}}/4$.

The corresponding mean-field theories can be derived from the microscopic Hamiltonian by the usual procedure of introducing an effective bosonic field ${\boldsymbol{\Phi}}$ via a Hubbard-Stratonovich decoupling $$\begin{aligned}
 &\mathrm{e}^{\frac{V}{2}\sum_{{\boldsymbol{q}}}{\boldsymbol{b}}_{{\boldsymbol{q}}}\cdot{\boldsymbol{b}}_{-{\boldsymbol{q}}}}= \nonumber  \\ 
 &\quad \int\mathcal{D}{\boldsymbol{\Phi}}\,\mathrm{e}^{-\frac{1}{2V}\sum_{{\boldsymbol{q}}}{\boldsymbol{\Phi}}_{{\boldsymbol{q}}}\cdot{\boldsymbol{\Phi}}_{-{\boldsymbol{q}}}+\frac{1}{2}\sum_{{\boldsymbol{q}}}{\boldsymbol{\Phi}}_{{\boldsymbol{q}}}\cdot{\boldsymbol{b}}_{-{\boldsymbol{q}}}+\frac{1}{2}\sum_{{\boldsymbol{q}}}{\boldsymbol{b}}_{{\boldsymbol{q}}}\cdot{\boldsymbol{\Phi}}_{-{\boldsymbol{q}}}}\label{eq:decoupling}\end{aligned}$$ of the interaction  in the channel of interest and subsequently integrating out the fermions. This requires $V>0$ for the interaction in the respective channel, where we skip the subindex ch or sp in what follows. The effective field ${\boldsymbol{\Phi}}\in\mathds{R}^N$ plays the role of an order parameter and its dimensionality $N$ depends on the channel in which the decoupling of the interaction is performed. Formulated in the language of field integrals, the corresponding mean-field theory follows immediately at the level of the saddle-point approximation, i.e., assuming that the order parameter be temporally homogeneous 