wave (CDW) or spin-density wave (SDW) ordering, it corresponds to the amplitude of the modulation of the commensurate charge density or spin density, respectively. At zero temperature, the gap equation  can be integrated straightforwardly, resulting for ${\rho_{\mathrm{F}}}V\ll1$ in the well-known expression for the zero-temperature energy gap $$\Delta_0=2\omega_0\mathrm{e}^{-\frac{1}{{\rho_{\mathrm{F}}}V}}{~ .}\label{eq:gap}$$

Experimentally, deviations from mean-field behavior have been observed for density-wave systems in the quantum regime [@JaramilloEtAl-Nature2009; @SokolovEtAl-PRB2014; @FreitasEtAl-PRB2015; @FengEtAl-NPhys2015]. The aim of this paper is to theoretically clarify to what extent CDW and SDW systems fulfill the second characteristic of the BCS theory, namely a lack of fluctuation corrections as discovered in the case of superconductivity [@GeorgesYedidia-PRB1991; @vanDongen-PRL1991; @Martin-RoderoFlores-PRB1992; @KosMillisLarkin-PRB2004; @EberleinMetzner-PRB2013; @Eberlein-PRB2014; @FischerEtAl2017]. We find that the result depends on the order-parameter manifold (in particular, the number of components), and on whether longitudinal or transverse fluctuations, i.e., fluctuations of the amplitude or of the phase of the order parameter, dominate. In Fig. \[fig:visu-fluctuations\], the different types of fluctuating modes are visualized for three examples. In all cases, we show that the characteristic length scales of these fluctuations are within the coherence volume $\xi^d$, with coherence length $\xi\sim v_\mathrm{F}/\Delta_0$.

Our analysis focuses on the stability of the ordered state at $T=0$. A complementary 