corresponding transition temperature decreases.

Summary
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Mean-field theories for density-wave order resulting from a nesting of the Fermi surface can be derived from a microscopic model of interacting fermions in full analogy to the formulation of BCS theory in the language of field integrals. The BCS theory of superconductivity is the poster child of mean-field theories since neither thermal nor quantum fluctuations lead to sizable effects in conventional superconductors and hence can be neglected. In this paper, we showed that, in contrast to superconductivity (where the order parameter is a complex scalar), the impact of fluctuations is crucial in the case of commensurate density-wave order (characterized by a real $N$-component order parameter) as long as $N\neq2$.

To be specific, we have investigated the role of fluctuations for charge-density wave order and spin-density wave order due to nesting of the Fermi surface. Our main finding is that, generally, fluctuation corrections to the zero-temperature gap equations for such density-wave instabilities are of the same order $\mathcal{O}[\ln(E_\mathrm{F}/\Phi_0)]$ as the terms already appearing at mean-field level. In conclusion, the mean-field theories for density-wave instabilities are not justified since the large fluctuation corrections imply that the respective mean-field theory cannot capture all relevant contributions. Of 