Let us now analyze the structure of fluctuation corrections in the case of density-wave instabilities due to nesting following the logic of Ref. , also building on the calculations presented in Ref. . More details specific to our derivation for both charge-density wave order and spin-density wave order are presented in Appendix \[app:calculation\]. The Gaussian fluctuation corrections can be expressed in terms of the fluctuation propagator $\boldsymbol{\mathcal{D}}_q$ as $E_\mathrm{fluct}(\Phi_0)=\frac{L^d}{2}\int_q\ln\det({\boldsymbol{\mathcal{D}}}^{-1}_q)$, where the dimensionality of the order parameter translates to the dimensionality of the fluctuation propagator. Accordingly, the Gaussian fluctuation corrections to the zero-temperature gap equation take the form $$\frac{1}{{\rho_{\mathrm{F}}}L^d}\frac{{\mathrm{d}}E_\mathrm{fluct}(\Phi_0)}{{\mathrm{d}}\Phi_0^2}=\frac{1}{2{\rho_{\mathrm{F}}}}\int_q\,\frac{\frac{{\mathrm{d}}}{{\mathrm{d}}\Phi_0^2}\det(\boldsymbol{\mathcal{D}}^{-1}_q)}{\det(\boldsymbol{\mathcal{D}}^{-1}_q)}{~ ,}\label{eq:fluctuation-corrections}$$ where we introduced $q\equiv({\boldsymbol{q}},\omega)$ and the integration $\int_q\ldots=\int\frac{{\mathrm{d}}\omega}{2\pi}\int\frac{{\mathrm{d}}^d{\boldsymbol{q}}}{(2\pi)^d}\ldots$ runs over all external frequencies and momenta up to the cutoff. Consequently, an expansion of the fluctuation spectrum in small momenta and frequencies as often discussed in the context of collective modes is not sufficient here. Instead, the fluctuation propagator has to be determined for all frequencies and momenta, and in fact, the short-wavelength fluctuations with momenta $2\xi^{-1}=2\Delta_0/v_\mathrm{F}\ll|{\boldsymbol{q}}|\ll k_\mathrm{F}$ turn out to be crucial, as they give rise to an additional logarithmic contribution to the zero-temperature gap equation as discussed in Sec. \[sec:results\].

The inverse matrix of the fluctuation propagator for density-wave order 