Cavity effects on the Fermi velocity renormalization in a graphene sheet

Recently, in the literature, it was shown that the logarithmic renormalization of the Fermi velocity in a plane graphene sheet (which, in turn, is related to the Coulombian static potential associated to electrons in the sheet) is inhibited by the presence of a single parallel conducting plate. In the present paper, we investigate the situation of a suspended graphene sheet in a cavity formed by two conducting plates parallel to the sheet. The effect of a cavity on the interaction between electrons in the graphene is not merely the addition of the effects of each plate individually. From this, one can expect that the inhibition of the renormalization of the Fermi velocity generated by a cavity is not a mere addition of the inhibition induced by each single plate. In other words, the simple addition of the result for the inhibition of the renormalization of the Fermi velocity found in the literature for a single plate could not be used to predict the exact behavior of the inhibition for the graphene between two plates. Here, we show that, in fact, this is what happens and calculate how the presence of a cavity formed by two conducting plates parallel to the suspended graphene sheet amplifies, in a non-additive manner, the inhibition of the logarithmic renormalization of the Fermi velocity. In the limits of a single plate and no plates, our formulas recover those found in the literature.