Electric conductivity in graphene: Kubo model versus a nonlocal quantum field theory model

We compare three models of graphene electric conductivity: a non-local Kubo model, a local model derived by Falkovsky, and finally, a non-local quantum field theory (QFT) polarization-based model. These models are supposed to provide consistent results since they are derived from the same Hamiltonian. While we confirm that the local model is a proper $\textbf{q}\to\textbf{0}$ limit of both the non-local Kubo and the non-local QFT model (once losses are added to this last model), we find hard inconsistencies in the non-local QFT model as derived and currently used in literature. In particular, in the genuine non-local region ($\textbf{q}\neq\textbf{0}$), the available QFT model shows an intrinsic non-physical plasma-like behavior for the interband transversal electric conductivity at low frequencies (even after introducing the unavoidable losses). The Kubo model, instead, shows the expected behavior, i.e., an almost constant electric conductivity as a function of frequency $\omega$ with a gap for frequencies $\hbar\omega<\sqrt{(\hbar v_F q)^{2}+4m^{2}}$. We show that the Kubo and QFT models can be expressed using an identical Polarization operator $\Pi_{\mu\nu}(\omega,\textbf{q})$, but they employ different expressions for the electric conductivity $\sigma_{\mu\nu}(\omega,\textbf{q})$. In particular, the Kubo model uses a standard regularized expression, a direct consequence of Ohm's Law and causality, as we rigorously re-derive. We show that, once the standard regularized expression for $\sigma_{\mu\nu}(\omega,\textbf{q})$ is used in the QFT model, and losses are included, the Kubo and QFT model coincide, and all its anomalies naturally disappear. Our findings show the necessity to appropriately define and regularize the electric conductivity to connect it with the available QFT model.