i.e., rather small. More important than the numerical value of this correction is the fact that there is no guarantee that even higher-order processes give rise to equally non-negligible corrections. Our analysis of fluctuation corrections shows that contributions stemming from the longitudinal mode lead to a decrease of the gap compared to its mean-field value, whereas transverse fluctuations increase the gap. Hence, if the latter dominate, fluctuations are favorable to the formation of density-wave order, as it is the case for spin-density wave order of Heisenberg spins. If, on the other hand, transverse fluctuations cannot compensate for the effect of the longitudinal mode, the ordered state is weakened. This applies to mean-field theories for charge-density wave order as well as for spin-density wave order of Ising spins, see also Table \[tab:summary-sdw-fluctuations\] for an overview of our results.

Channel interference in the renormalization group approach {#sec:rg}
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![image](channel-interference-repulsive){width="\textwidth"}

Our analysis of the role of fluctuations for density-wave instabilities has been performed at zero temperature, i.e., deep inside the ordered phase. In doing so, we concentrated on a single channel and neglected the potential presence of competing instabilities. Another, complementary perspective is provided by a renormalization group (RG) analysis [@Shankar-RevModPhys1994], in which 