## Appendix A Perturbation Theory  
In order to demonstrate the effectiveness of the introduced approach, we will reproduce some well known results in the framework of perturbation theory for relatively small external fields, \(E<E_{0}\). We begin with the analysis of currents in the low density approximation (30) and (32) which corresponds to the one-photon excitation mechanism [33]. In the minimal leading approximation \(\varepsilon(\vec{p},t)\rightarrow\varepsilon_{0}(\vec{p})=v_{F}|\vec{p}|\) we have  
\[f_{LD}^{(2)}(\vec{p},t) = \frac{1}{2}\int^{t}dt^{\prime}\lambda^{(1)}(\vec{p},t^{\prime}) \,u_{LD}^{(1)}(\vec{p},t^{\prime}) \tag{101}\] \[u_{LD}^{(1)}(\vec{p},t) = \int^{t}dt^{\prime\prime}\lambda^{(1)}(\vec{p},t^{\prime\prime}) \cos[\eta p(t-t^{\prime\prime})], \tag{102}\]  
with  
\[\lambda^{(1)}(\vec{p},t)=\lambda_{0}(\vec{p})\left[E_{1}(t)P_{2}-E_{2}(t)P_{1} \right]\, \tag{103}\]where \(\lambda_{0}(\vec{p})=ev_{F}^{2}/2\epsilon_{0}^{2}(p)=e/2p_{2}\) and \(\eta=2v_{F}/\hbar\). The upper indices at the functions \(f^{(2)}\) and \(u^{(1)}\) indicate the order of perturbation theory. The relations (A1) and (A2) indicate the dominant role of polarization effects in the considered approximation, \(|j^{(1)\text{pol}}(t)|\gg|j^{(2)\text{cond}}(t)|\). Let us consider the case of linear polarization \(\vec{E}(t)=(E(t),0)\) with arbitrary time dependence of the electric field. The polarization current in lowest order perturbation theory according to Equation (47) is  
\[j_{1}^{(1)\text{pol}}(t) = 8 ev_{F}\int[dp]u\sin\varkappa,\] (A4) \[j_{2}^{(1)\text{pol}}(t) = -8 ev_{F}\int[dp]u\cos\varkappa,\]  
where \(\varkappa\) is defined in explanation to Equation (7),  
\[\varkappa=\arctan(P_{2}/P_{1})\approx\arctan(p_{2}/p_{1}),\] (A5)  
where the last step corresponds to the leading approximation. From Equation (A5) follows  
\[\sin\varkappa\approx p_{2}/p=\sin\Phi,\quad\cos\varkappa\approx p_{1}/p=\cos\Phi,\] (A6)  
where \(\Phi\) is the polar angle in the polar representation of the momentum space. Integration over the momentum \(p\) in the neighborhood of the Dirac points is limited by the cutoff parameter \(\Lambda\). It is implied that it can be defined by the limits of the validity range of the linear dispersion law \(\varepsilon_{0}(p)=v_{F}p\). However, the results obtained below are universal and do not depend on the choice of \(\Lambda\). Taking these remarks into account, one can thus write the polarization current after integration over the angle (here \(\mathfrak{t}_{0}\rightarrow-\infty\)),  
\[j_{1}^{(1)\text{pol}}(t) = \frac{e^{2}v_{F}}{\pi\hbar^{2}}\int\limits_{-\infty}^{t}dt^{ \prime}E(t^{\prime})\int_{0}^{\Lambda}dp\cos[\eta p(t-t^{\prime})],\] \[j_{2}^{(1)\text{pol}}(t) = 0\;.\] (A7)  
Let us now perform a Fourier transformation of the function \(E(t)\) and after that integrate over the momentum \(p\),  
\[j_{1}^{(1)\text{pol}}(t) = \frac{e^{2}}{\pi\hbar}\int d\omega E(\omega)\int\limits_{-\infty} ^{t}dt^{\prime}\frac{\sin[\Lambda\eta(t-t^{\prime})]}{t-t^{\prime}}e^{i\omega t ^{\prime}}\] \[u_{LD}^{(1)}(\vec{p},t) = \int^{t}dt^{\prime\prime}\lambda^{(1)}(\vec{p},t^{\prime\prime}) \cos[\eta p(t-t^{\prime\prime})], \tag{102}\]0  
The last integral does not depend on the parameter \(\gamma=2v_{F}\Lambda/\hbar\omega\),  
\[u_{LD}^{(1)}(\vec{p},t) = \int^{t}dt^{\prime\prime}\lambda^{(1)}(\vec{p},t^{\prime\prime}) \cos[\eta p(t-t^{\prime\prime})], \tag{102}\]1 (A9)  
so that  
\[u_{LD}^{(1)}(\vec{p},t) = \int^{t}dt^{\prime\prime}\lambda^{(1)}(\vec{p},t^{\prime\prime}) \cos[\eta p(t-t^{\prime\prime})], \tag{102}\]2 (A10)  
This result does not depend on the choice of the field model. In order to calculate the conduction current, it is necessary to find the distribution function. To this end we use perturbation theory as a first step and consider the case of a constant electric field (57) switched on at the time \(t_{0}=0\).