  Density-wave instabilities have been observed and studied in a multitude of
materials. Most recently, in the context of unconventional superconductors like
the iron-based superconductors, they have excited considerable interest. We
analyze the fluctuation corrections to the equation of state of the
density-wave order parameter for commensurate charge-density waves and
spin-density waves due to perfect nesting. For XY magnets, we find that
contributions due to longitudinal and transverse fluctuations cancel each
other, making the mean-field analysis of the problem controlled. This is
consistent with the analysis of fluctuation corrections to the BCS theory of
superconductivity [S. Kos, A. J. Millis, and A. I. Larkin, Phys. Rev. B 70,
214531 (2004)]. However, this cancellation does not occur in density-wave
systems when the order parameter is a real $N$-component object with $N\neq2$.
Then, the number of transverse fluctuating modes differs from the number of
longitudinal fluctuating modes. Indeed, in the case of charge-density waves as
well as spin-density waves with Heisenberg symmetry, we find that fluctuation
corrections are not negligible, and hence mean-field theories are not
justified. These singular fluctuations originate from the intermediate
length-scale regime, with wavelengths between the lattice constant and the
$T=0$ correlation length. The resulting logarithmic fluctuation contributions
to the gap equation originate from the derivative of the anomalous polarization
function, and the crucial process is an interaction of quasiparticles through
the exchange of fluctuations.
