– in accordance with the cancellation of amplitude and phase fluctuations in superconductors [@KosMillisLarkin-PRB2004; @FischerEtAl2017].

Mean-field theory for density-wave instabilities
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Density-wave order arises naturally for systems in which the Fermi surface is perfectly nested, i.e., different parts of the Fermi surface are connected by the vector ${\boldsymbol{Q}}$ such that $\varepsilon_{{\boldsymbol{k}}+{\boldsymbol{Q}}}=-\varepsilon_{{\boldsymbol{k}}}$ holds for the energy dispersion. This nesting condition suggests we formulate mean-field theories for density-wave order in complete structural analogy to the BCS theory of superconductivity. The noninteracting part of the Hamiltonian $\mathcal{H}=\mathcal{H}_0+\mathcal{H}_\mathrm{int}$ is of the usual form $$\label{eq:H0}
 \mathcal{H}_0= \sum_{{\boldsymbol{k}},\sigma}\varepsilon_{{\boldsymbol{k}}}\psi^\dagger_{{\boldsymbol{k}},\sigma}\psi^{}_{{\boldsymbol{k}},\sigma}{~ ,}$$ where $\psi_{{\boldsymbol{k}},\sigma}^{(\dagger)}$ annihilates (creates) a fermionic state with momentum ${\boldsymbol{k}}$ and spin $\sigma$, where the nesting condition for the dispersion will be implied in the remainder. The most general $\mathrm{SU}(2)$-invariant interaction can be written as $$\begin{aligned}
 \mathcal{H}_\mathrm{int}&= \frac{1}{2}\sum\psi^\dagger_{{\boldsymbol{k}}_1,\sigma_1}\psi^\dagger_{{\boldsymbol{k}}_2,\sigma_2} U_{\sigma_1\sigma_2;\sigma_3\sigma_4}({\boldsymbol{k}}_1,{\boldsymbol{k}}_2;{\boldsymbol{k}}_3,{\boldsymbol{k}}_4) \nonumber \\ &\qquad \times  \psi^{}_{{\boldsymbol{k}}_3,\sigma_3}\psi^{}_{{\boldsymbol{k}}_4,\sigma_4}\delta({\boldsymbol{k}}_1+{\boldsymbol{k}}_2-{\boldsymbol{k}}_3-{\boldsymbol{k}}_4){~ ,}\label{eq:H_interaction}\end{aligned}$$ where the summation is over momenta ${\boldsymbol{k}}_i$ and spins $\sigma_i$. The spin sector of the interaction can be decomposed into charge (ch) and spin (sp) channel according to $$U_{\sigma_1\sigma_2;\sigma_3\sigma_4} = U_\mathrm{ch}\,\delta_{\sigma_1\sigma_4}\delta_{\sigma_2\sigma_3}  +U_\mathrm{sp}\,{\boldsymbol{\sigma}}_{\sigma_1\sigma_4}\cdot{\boldsymbol{\sigma}}_{\sigma_2\sigma_3}{~ ,}\label{eq:decomposition}$$ where we introduced ${\boldsymbol{\sigma}}=(\sigma_1,\sigma_2,\sigma_3)$ as the vector of Pauli matrices in spin space. Furthermore, we could allow for spin-orbit interaction and consider Ising spins or XY spins instead of 