of quasiparticles inside the coherence volume which enhances the kinetic energy of quasiparticle excitations. In contrast, longitudinal fluctuations can only lead to an increase of the energy since the potential energy is already minimized by the mean-field configuration.

Because of the different nature of longitudinal and transverse fluctuations, the case $N=2$ is an interesting exception: The logarithmic contributions to the gap equation stemming from the longitudinal mode and the single transverse mode cancel exactly which renders the overall fluctuation corrections negligible. This cancellation legitimates the mean-field approach to density-wave order of XY spins. This is in accordance with the analysis of fluctuations in the context of superconductivity [@KosMillisLarkin-PRB2004; @FischerEtAl2017], where fluctuation corrections from phase and amplitude mode cancel analogously, providing the justification of the BCS mean-field theory.

Acknowledgments
===============

We thank M. Bard, A. V. Chubukov, S. Fischer, M. Hecker, N. Kainaris, and M. S. Scheurer for helpful discussions.

Calculation of fluctuation corrections {#app:calculation}
======================================

This Appendix provides technical details on our calculation of fluctuation corrections, closely following the logic and notation introduced in the context of superconductivity in Ref.  and extended by Ref. .

In what follows, we consider the partition function $\mathcal{Z}=\int\mathcal{D}[\bar{\psi},\psi]\,\exp(-\mathcal{S}[\bar{\psi},\psi])$ with the appropriate action $\mathcal{S}[\bar{\psi},\psi]=\int_0^\beta{\mathrm{d}}\tau\,(\sum_\sigma\int{\mathrm{d}}^d{\boldsymbol{x}}\,\bar{\psi}_\sigma({\boldsymbol{x}})\partial_\tau\psi_\sigma({\boldsymbol{x}})+\mathcal{H})$ corresponding 