We review the problem of electron-electron interactions in graphene. Starting from the screening of long range interactions in these systems, we discuss the existence of an emerging Dirac liquid of Lorentz invariant quasi-particles in the weak coupling regime, and strongly correlated electronic states in the strong coupling regime. We also analyze the analogy and connections between the many-body problem and the Coulomb impurity problem. The problem of the magnetic instability and Kondo effect of impurities and/or adatoms in graphene is also discussed in analogy with classical models of many-body effects in ordinary metals. We show that Lorentz invariance plays a fundamental role and leads to effects that span the whole spectrum, from the ultraviolet to the infrared. The effect of an emerging Lorentz invariance is also discussed in the context of finite size and edge effects as well as mesoscopic physics. We also briefly discuss the effects of strong magnetic fields in single layers and review some of the main aspects of the many-body problem in graphene bilayers. In addition to reviewing the fully understood aspects of the many-body problem in graphene, we show that a plethora of interesting issues remain open, both theoretically and experimentally, and that the field of graphene research is still exciting and vibrant. 42 B. States at vacancies and cracks 43 C. Midgap States and Random Gauge Fields 43 VII. Interaction effects in mesoscopic systems 44 A. Magnetism in quantum dots 44

We study the superconducting phases of the two-dimensional honeycomb lattice of graphene. We find two spin singlet pairing states; s wave and an exotic p+ip that is possible because of the special structure of the honeycomb lattice. At half filling, the p+ip phase is gapless and superconductivity is a hidden order. We discuss the possibility of a superconducting state in metal coated graphene.

In this article, we derive an effective theory of graphene on a hexagonal Boron Nitride (h-BN) substrate. We show that the h-BN substrate generically opens a spectral gap in graphene despite the lattice mismatch. The origin of that gap is particularly intuitive in the regime of strong coupling between graphene and its substrate, when the low-energy physics is determined by the topology of a network of zero energy modes. For twisted graphene bilayers, where inversion symmetry is present, this network percolates through the system and the spectrum is gapless. The breaking of that symmetry by h-BN causes the zero energy modes to close into rings. The eigenstates of these rings hybridize into flat bands with gaps in between. The size of this band gap can be tuned by a gate voltage and it can reach the order of magnitude needed to confine electrons at room temperature.

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We are as interested as the the commenters [1] about the possibility of observing a Dirac loop in a real physical system. In our work [2] we showed a simple class of lattice models that possess a Dirac loop at theier Fermi level and detailed some of the physical consequences of that loop, such as the possibility of a 3D anomalous Hall effect and topological surface states in the presence of spin-orbit coupling. nodal lines. It is remarkable that these loops can occur in simple hyper-honeycomb lattices.