the gap equation evaluates to

\[eq:result\]$$\begin{aligned}
  &\frac{1}{{\rho_{\mathrm{F}}}L^d}\frac{{\mathrm{d}}E_\mathrm{fluct}(\Phi_0)}{{\mathrm{d}}\Phi_0^2}\Big|_{r\gg1} \approx \frac{1}{2{\rho_{\mathrm{F}}}}\int_q\frac{[(N-1)-1]\frac{{\mathrm{d}}\Pi^\mathrm{a}_q}{{\mathrm{d}}\Phi_0^2}}{\frac{1}{V}-\Pi^\mathrm{n}_q} \\ 
  &\qquad \qquad \approx -\frac{1}{2}\frac{(N-1)-1}{8\pi (d-1)}\ln\Big(\frac{E_\mathrm{F}}{\Phi_0}\Big){~ .}\end{aligned}$$

This reveals that Gaussian fluctuations indeed quite generically entail an additional logarithmic contribution to the gap equation for density-wave instabilities due to nesting. This logarithmic divergence can be traced back to the fermionic triangle depicted in Fig. \[fig:triangles\](a), which results from the derivative of the anomalous part of the polarization function that enters the numerator of ${\mathrm{d}}E_\mathrm{fluct}(\Phi_0)/{\mathrm{d}}\Phi_0^2$, while the two other contributions \[Figs. \[fig:triangles\](b) and \[fig:triangles\](c)\] do not give rise to additional logarithms. Thus, the precarious process that invalidates the mean-field approach in the case of density-wave order is the interaction of quasiparticles through fluctuations.

                     fluctuations   
  ------------------ -------------- ----------------
  Heisenberg SDW     ($N=3$)            increase gap
  XY SDW             ($N=2$)          are negligible
  Ising SDW or CDW   ($N=1$)            decrease gap

  : Gaussian fluctuation corrections in $\mathrm{O}(N)$ models. Depending on the dimensionality of the order parameter, fluctuation corrections can increase or decrease the gap value compared to its mean-field value.[]{data-label="tab:summary-sdw-fluctuations"}

The above result is in stark contrast to the insignificant role that fluctuations play in the context of superconductivity: Analogous contributions to the BCS gap equation are not only suppressed by the smallness of the Debye energy as compared 