order to clarify our notation, in this section we sketch the derivation of the quantum Boltzmann formalism for the Dirac liquid, which was developed in Ref. [@Fritz2008]. We begin with the Hamiltonian of graphene electrons at the charge neutrality point: $$H=H_{0}+H_{\mathrm{int}},\label{eq:nl_Hamiltonian}$$ where the free part is given by $$H_{0}=v\hbar\int_{\mathbf{k}}\sum_{a,b,i}\psi_{a,i}^{\dagger}\left(\mathbf{k}\right)\left(\mathbf{k\cdot\sigma}\right)_{ab}\psi_{b,i}\left(\mathbf{k}\right),$$ and the interaction part reads $$H_{\mathrm{int}}=\frac{1}{2}\int_{\mathbf{k},\mathbf{k}',\mathbf{q}}\sum_{a,b,i,j}V\left(\mathbf{q}\right)\psi_{\mathbf{k+q},a,i}^{\dagger}\psi_{\mathbf{k}^{\prime}-\mathbf{q},b,j}^{\dagger}\psi_{\mathbf{k}^{\prime},b,j}\psi_{\mathbf{k,}a,i}.$$ $V\left(\mathbf{q}\right)=\frac{2\pi e^{2}}{\varepsilon\left\vert \mathbf{q}\right\vert }$ is the 2D Coulomb potential. The indices $i,j=1,2\,...,N=4$ refer to the spin and valley quantum numbers of an electron, whereas the two sub-lattices are labelled by the indices $a,b$. The free particle Hamiltonian $H_{0}$ is diagonalized by the unitary transformation $$U_{\mathbf{k}}=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}
1 & o_{\mathbf{k}}^{*}\\
1 & -o_{\mathbf{k}}^{*}
\end{array}\right],\label{eq:free_transform}$$ where $o_{\mathbf{k}}=\left(k_{x}+ik_{y}\right)/\sqrt{k_{x}^{2}+k_{y}^{2}}$.

For the derivation of the quantum Boltzmann equation, it is convenient to use the band representation of Dirac spinors $\psi_{\lambda,\mathbf{k}}=U_{\mathbf{k},\lambda a}\psi_{\mathbf{k},a}$ with $\lambda=\pm1$ labeling the upper and lower Dirac cones. In this way, one can easily distinguish between processes that involve the creation of particle-hole pairs and those which do not. The thermally excited electron-hole pairs occupy states in a window of $k_{B}T$ around the Dirac point. Thus, if the applied fields have frequencies $\omega<2k_{B}T/\hbar$, which is true in the hydrodynamic regime, processes that create electron-hole pairs are unlikely and can 