to the Fermi energy, but the corresponding contributions stemming from fluctuations of phase and amplitude of the complex order parameter even cancel exactly. Nonetheless, the corrections due to the longitudinal mode and the $N-1$ transverse modes enter the result  with opposite signs, and hence the prefactor depends on the number of transverse modes. Only for XY spins ($N=2$), the large fluctuation corrections stemming from the regime $r\gg1$ cancel – which is consistent with previous results in the context of superconductivity [@KosMillisLarkin-PRB2004; @FischerEtAl2017]. In conclusion, the analysis of the role of fluctuation corrections in the case of density-wave order reveals that mean-field approaches are generally not justified in this situation, the only exception being spin-density wave order of XY spins.

Furthermore, the sign of the fluctuation corrections allows us to judge whether the presence of fluctuations is advantageous or detrimental to the formation of density-wave order: The effective interaction $$\begin{aligned}
 \lambda_\mathrm{eff} &= \lambda\bigg[1+\frac{N-2}{16\pi(d-1)}\bigg]{~ ,}\label{eq:effective-interaction}
$$ which governs the ordering in the presence of fluctuations, either decreases (${\mathrm{d}}E_\mathrm{fluct}/{\mathrm{d}}\Phi_0^2>0$) or increases (${\mathrm{d}}E_\mathrm{fluct}/{\mathrm{d}}\Phi_0^2<0$), and the same is true for the solution of the zero-temperature gap equation  in the presence of fluctuations. For $N=1$ or $N=3$ and $d=3$, the relative change in $\lambda$ is $1/(32\pi)\simeq0.01$, 