The ability to isolate Graphene, a one-atom-thick two-dimensional crystal made by carbon atoms arranged in a honeycomb pattern, and control its electrical properties  has led to a revolution in material science and condensed matter physics . Electrons in graphene behave as massless Dirac fermions, the role of the speed of light c being played by the Fermi velocity vF ≈ c/300 . This essential feature of electrons in graphene derives from its unusual linear band dispersion , ultimately stemming from the underlying honeycomb topology. This basic solid-state effect has led to the unique possibility to test quantum relativistic phenomena in table-top experiments .
In the first part of the talk I will review the salient properties of graphene, summarize the main directions of fundamental and applied research and present recent results obtained by our group [6,7]. In the second part of the talk I’ll focus on the issue of spontaneous symmetry breaking effects linked to lattice-scale order. These many-body phenomena yield unusual ground states where Dirac quasiparticles are non-uniformly distributed between the two triangular sub-lattices forming the honeycomb pattern . I’ll show how these phenomena can be simulated in nanofabricated artificial lattices with honeycomb topology [9,10,11].