pairing of time-reversed states. Hence, the density-wave analog for the Nambu spinors is given by $$\Psi_{{\boldsymbol{k}}}=\begin{pmatrix} \psi^\dagger_{{\boldsymbol{k}},\uparrow} & \psi^\dagger_{{\boldsymbol{k}},\downarrow} & \psi^\dagger_{{\boldsymbol{k}}+{\boldsymbol{Q}},\uparrow} &\psi^\dagger_{{\boldsymbol{k}}+{\boldsymbol{Q}},\downarrow}\end{pmatrix}^T {~ ,}\label{eq:spinor}$$ allowing us to represent the mean-field ground-state energy as $E_\mathrm{MF}(\Phi_0)=2\Phi_0^2/V-\int_k\operatorname{tr}\ln[-(\mathcal{G}_k^\mathrm{MF})^{-1}]$. The corresponding matrix Green’s function in Matsubara representation is given by $$\begin{aligned}
 &\big(\mathcal{G}_k^{\mathrm{MF}}\big)^{-1}=\frac{1}{2}\left(\mathrm{i}\nu_n\tau_0\sigma_0-\varepsilon_{{\boldsymbol{k}}}\tau_3\sigma_0\right) \nonumber \\ 
  &\qquad +\frac{1}{2}\tau_1\left\{\begin{matrix}\Phi_0\sigma_0 & \text{for CDW order} {~ ,}\\ {\boldsymbol{\Phi}}_0\cdot {\boldsymbol{\sigma}}& \text{for Heisenberg SDW order}{~ ,}\end{matrix}\right.\end{aligned}$$ where we abbreviated $k\equiv({\boldsymbol{k}},\nu_n)$ with the fermionic Matsubara frequency $\nu_n=(2n+1)\pi T$. The Pauli matrices $\sigma_i$ refer to the spin sector whereas the $\tau_i$ denote Pauli matrices in the band space that emerges due to a doubling of the unit cell. We assume $2{\boldsymbol{Q}}$ to be a reciprocal lattice vector.

Fluctuation corrections to the gap equation
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The validity of the mean-field approximation as discussed in the previous section relies on the assumption that deviations of the order parameter from its mean-field value lead to negligible corrections to the mean-field theory in the sense that fluctuation corrections to physical observables are small compared to their mean-field value.

Of course, there are regimes in which the role of long-wavelength ($|{\boldsymbol{q}}|\ll2\xi^{-1}$) fluctuations is well-understood. Firstly, thermal fluctuations drive the phase transition. The Ginzburg regime in which these thermal fluctuations 