Interacting electrons present one of the most fundamental problems in condensed matter physics: The competition of kinetic and Coulomb energies leads to the emergence of rich phases in solid state materials ranging from Fermi liquids in normal metals to heavy fermion, magnetically ordered or superconducting phases. Models of interacting electrons often start from particles with quadratic dispersion or a finite Fermi surface around which low energy excitations are possible. The rise of graphene brought up a material which does not fit into this "standard model".

Electrons in graphene (should) behave like massless Dirac quasiparticles and in absence of doping the Fermi surface shrinks to two Fermi points. In the simplest low energy model, graphene presents a realization of two dimensional quantum electrodynamics (QED) but with large effective fine structure constant alpha~1 instead of alpha=1/137 in usual QED. Thus, graphene is very likely not a weakly interacting electron system leading to the fundamental question of how the electronic system in graphene and related materials is controlled by many electron correlation phenomena. To date, it is heavily debated whether and how the electrons in graphene might be driven into insulating, magnetic, exotic quantum-disordered or topologically non-trivial strongly correlated phases.

The single particle band structure of graphene is known for more than 60 years [1]: Graphene consisting of two sublattices leads to quasiparticles being described by a Dirac equation rather than a normal Schrödinger equation. Density functional theory (DFT) being an effective single particle theory confirms this prediction. However, there is growing evidence that the single electron /simple Fermi liquid picture of graphene might be incomplete, sometimes even qualitatively wrong and that a careful investigation of the electrons interacting in graphene is required.

Generally, the extent to which Coulomb interactions determine the physics of graphene is controversial: Experiments reported ferromagnetic ordering in nanographene [2], in disordered graphite samples [3] and at grain boundaries in highly oriented pyrolytic graphite (HOPG) [4]. Ferromagnetism in pristine graphene, however, has been excluded experimentally for temperatures down to 2K [5]. Theoretically, the possibility of magnetism in defect free graphene has been predicted: An antiferromagnetic insulating ground state has been obtained for the local Coulomb interactions exceeding a critical value U_AF > 4.5t in Quantum Monte Carlo (QMC) calculations [6–8] and an exotic gapped spin-liquid has been predicted to emerge for the on-site repulsion between U_sl = 3.5t and U_AF [9]. Sizable non-local Coulomb interactions can make the phase diagram even richer and lead to a competition between spin- and charge-density-wave phases [10, 11], topologically non-trivial phases [12] or an excitonic insulator [13]. Recent realistic calculations within the constraint random phase approximation indicate the graphene is indeed close to different instabilities which might be triggered by external perturbations like strain [14].

Moreover, the Coulomb interaction retains its long range tails in graphene related systems which poses a similarly important and challenging problems: Whether or not this could lead to excitonic instabilities [13] depends on the efficiency of screening. Inelastic x-ray scattering experiments [15] suggest a fully screened dielectric constant of epsilon~15. However, electron transport experiments [16] as well as a simple Dirac electron model [17] and GW predictions [18] yield less efficient screening (epsilon~4). It remains to be understood how excitonic instabilities [13] due to the long range part of the Coulomb interaction compete with effects due to the local Coulomb interactions.

Graphene's response to defects, particularly defect induced magnetism [3], is determined by the delicate interplay of pseudorelativistic electrons with strong mutual interactions and strong local inhomogeneities like atomic scale defects. In narrow impurity bands or edge states of graphene, the Coulomb interaction can present the dominating energy scale and, thus, trigger many body instabilities including magnetism [19]. For instance, recent scanning tunnelling spectroscopy experiments on graphene nanoribbons indicate opening of many body gaps at the graphene edges [20]. However, as magnetism at graphene edges can be very sensitive to atomic structure of the defects [21, 22], the issue of defect induced electron correlation effects in graphene, how to possibly control them and how to exploit them presents an open problem, both, to experimental and theoretical solid state physics.

Not only low energy and ground state properties of graphene but also, general excitations and their decay are largely determined by interactions. Electron-electron interactions in undoped graphene lead to a marginal Fermi liquid with a quasiparticle decay rate which is linear in energy and the quasiparticle velocity being increased by many body renormalization effects as the Dirac point is approached [23, 24]. Hence, understanding experiments like angular resolved photoemission [25] or scanning tunnelling spectroscopy [26] requires accounting for electron-electron and electron-phonon interactions. Realistic GW calculations can be helpful in this context [27, 28]. Now, there is experimental evidence for electrons and collective plasma oscillations being coupled to form novel quasiparticles called plasmarons [29], which demonstrates how electron-electron interactions can change the low energy behaviour of graphene. Generally, the impact of quantum fluctuations (particularly close to ground state instabilities of graphene) on excitations in this material remains to be explored.

The delicate interplay of Dirac quasiparticles with, both, long and short range interactions hosts rich physics. Its realistic theoretical description presents a challenge that needs to be tackled for, both, understanding the basic graphene physics and optimizing this material for applications.