Electron interactions in graphene through an effective Coulomb potential

A recent numerical work [H.-K. Tang {\it et al}, PRL 115, 186602 (2015)] considering graphene's $\pi$-electrons interacting through an effective Coulomb potential that is finite at short-distances, stressed the importance of the $sp^{2}$-electrons in determining the semimetal to Mott insulator phase transition in graphene. Some years ago, I. F. Herbut [PRL 97, 146401 (2006)] studied such a transition by mapping graphene's $\pi$-electrons into a Gross-Neveu model. From a different perspective, D. T. Son [PRB 75, 235423 (2007)] put the emphasis on the long-range interactions by modelling graphene as Dirac fermions interacting through a bare Coulomb potential. Here we build on these works and explore the phase diagram of Dirac fermions interacting through an effective Coulomb-like potential screened at short-distances. The interaction potential used allows for analytic results that controllably switch between the two perspectives above.