and $\delta$-distributed in momentum space, which we denote ${\boldsymbol{\Phi}}_0$.

The mean-field theories for density-wave order resulting from nesting bear the same structure as the BCS theory of superconductivity. The ground-state energy $$E_\mathrm{MF}(\Phi_0)=E_0-2{\rho_{\mathrm{F}}}L^d\Phi_0^2\bigg[\ln\bigg(\frac{2E_\mathrm{F}}{\Phi_0}\bigg)+\frac{1}{2}-\frac{1}{{\rho_{\mathrm{F}}}V}\bigg]$$ deep inside the ordered phase (where $\Phi_0\equiv|{\boldsymbol{\Phi}}_0|$ is the magnitude of the order parameter) is reduced as compared to the high-temperature ground-state energy $E_0$. For the instantaneous electronic interaction  considered here, the energy window of the attractive pairing goes up to the Fermi energy. That means, in contrast to the BCS theory where the energetic cut-off $\omega_0$ of the theory is given by the Debye energy, we consider $\omega_0\simeq E_\mathrm{F}$ in the remainder. Furthermore, ${\rho_{\mathrm{F}}}$ refers to the density of states at the Fermi level and $L^d$ denotes the system’s volume. This structure results in the characteristic logarithm appearing in the zero-temperature mean-field gap equation $$0=\frac{1}{{\rho_{\mathrm{F}}}L^d}\frac{{\mathrm{d}}E_\mathrm{MF}(\Phi_0)}{{\mathrm{d}}\Phi_0^2} 
\approx \frac{1}{{\rho_{\mathrm{F}}}V}-\ln\bigg(\frac{2E_\mathrm{F}}{\Phi_0}\bigg){~ ,}\label{eq:gap-equation2}$$ which self-consistently determines the magnitude of the mean-field value of the order parameter. Details on the derivation can be found in Appendix \[app:calculation\]. Note that Eq.  is equivalent to Eq.  at $T=0$. The logarithmic contribution arises from a relative sign provided by the nesting condition $\varepsilon_{{\boldsymbol{k}}+{\boldsymbol{Q}}}=-\varepsilon_{{\boldsymbol{k}}}$ in the case of density-wave order, whereas for superconductivity it results from the 