In this talk I will review recent advances on the understanding of the ground state properties of interacting electrons on the honeycomb lattice. In the case of weak short range interactions, renormalization group methods allowed us to give a complete construction of the ground state of the half-filled system and to prove analyticity in the coupling constant of the thermodynamic functions and of the equilibrium correlations. In the case of retarded Coulomb interactions, the ground state can be constructed order by order in renormalized perturbation theory, with the n-th order admitting n!-bounds. Ward Identities are needed to control the flow of the effective charges. Lorentz invariance is not dynamically restored, unless a fine tuning of the bare charges is made. This talk is based on joint work with V. Mastropietro and M. Porta.

Univ. Roma Tre