since the coherence length is large compared to the inverse Fermi momentum in conventional superconductors, $\xi\gg  k_\mathrm{F}^{-1}$, resulting in a narrow Ginzburg regime.

![Visualization of the order parameter and fluctuations around its mean-field value (red dot) for three examples: (a) commensurate charge-density wave order, (b) superconductivity, and (c) commensurate spin-density wave order of Heisenberg spins. The longitudinal mode is indicated in green while the $N-1$ transverse modes are shown in blue. In case of (a) and (b), the manifold of degenerate ground states (red) is shown within the corresponding energy landscape (gray) below the phase transition, whereas for (c) only the former.[]{data-label="fig:visu-fluctuations"}](visu-fluct-OP){width="\columnwidth"}

There are instabilities in the particle-hole channel with perfect nesting [@Peierls1955; @Overhauser-PR1962] which share the first characteristic with the BCS theory [@FeddersMartin-PR1966; @ChanHeine-JPhysF1973; @Gorkov-JETPLett1973; @Gorkov-JETP1974; @RiceScott-PRL1975; @HirschScalapino-PRL1986], and which are governed by the same self-consistent mean-field equation, $$\Delta_0={\rho_{\mathrm{F}}}V\int_{-\omega_0}^{\omega_0}\mathrm{d}\varepsilon_{{\boldsymbol{k}}}\,\frac{[1-2n_\mathrm{F}(E_{{\boldsymbol{k}}})]\Delta_0}{2E_{{\boldsymbol{k}}}}{~ ,}\label{eq:gap-equation}$$ which determines the order parameter $\Delta_0$. Here, we abbreviated $E_{{\boldsymbol{k}}}=\sqrt{\varepsilon_{{\boldsymbol{k}}}^2+\Delta_0^2}$, with the energy dispersion $\varepsilon_{{\boldsymbol{k}}}$. Furthermore, ${\rho_{\mathrm{F}}}V$ is the dimensionless interaction leading to long-range order, $n_\mathrm{F}(E_{{\boldsymbol{k}}})$ is the Fermi distribution, and $\omega_0$ refers to the energetic cut-off of the theory. The physical interpretation of $\Delta_0$ is the pairing gap in the case of superconductivity. In the case of charge-density 