quantities if we Fourier transform the time and space. For simplicity of notation, we consider spinless electrons with a single band, $\epsilon_{\bf k}$. Since the system is not in thermal equilibrium, we cannot evoke the usual fluctuation-dissipation theorem, but something very close to it. We use the Kadanoff-Baym ansatz [@kadanoffbaym], i.e., $$G^< = -f (G^r - G^a),\quad  G^> = (1 -f) (G^r - G^a),$$ where $f$ is given by the solution of Boltzmann equation. The spectrum function, $A = i (G^r - G^a)$, is assumed to be evaluated in thermal equilibrium. In wavevector and angular frequency domain, we can write [@mahan00; @duppen16] $$\label{eqpir}
\Pi^r_{\bf k}(\omega) = - e^2 \int \frac{d^2\bf p}{(2\pi)^2} \frac{ f_{\bf p} - f_{{\bf p} - {\bf k}}}{\hbar \omega + i \eta - 
\epsilon_{\bf p} + \epsilon_{{\bf p} - {\bf k}}}.$$ Here the integration is over the first Brillouin zone, and $\eta$ is a small damping parameter inversely proportional to the relaxation time.

As the relaxation mechanism for the electrons can be very complicated due to different scattering possibilities – impurity scatterings, electron-electron and electron-phonon scatterings – we do not attempt to solve the Boltzmann equation, and just use a single-mode relaxation time approximation [@ziman60]. In such a 