Abstract
We compare three available models of graphene conductivity: a non-local Kubo model, a local model derived by Fialkovsky, and finally a non-local Quantum Field Theory based (QFT-b) model. The first two models are extensively used in the nanophotonic community. All these models are not ab-initio since they contain phenomenological parameters (like chemical potential and/or mass gap parameters), and are supposed to provide coherent results since they are derived from the same starting Hamiltonian. While we confirm that the local model is a proper limit of the non-local Kubo model, we find some inconsistencies in the QFT-b model as derived and used in literature. In particular, differently from the Kubo model, the QFT-b model does not satisfy the required Gauge invariance, and as a consequence it shows a plasma-like behavior for the interband transversal conductivity at low frequencies instead of the expected behavior (an almost constant conductivity as a function of frequency $\omega$ with a gap for frequencies $\hbar\omega < \sqrt{(\hbar v_{F}q)^{2} + 4m^{2}}$). The inconsistencies of QFT-b model predictions are due to a non-correct regularization-scheme which allows for the gauge invariance violation. We show how to correctly regularize the QFT-b model in order to satisfy the gauge invariance and, once also losses are correctly included, we show that the Kubo and QFT-b model exactly coincide. Our finding can be of relevant interest for both theory predictions and experimental tests in both the nanophotonic and Casimir effect communities.