Aspects of interacting electrons on graphene honeycomb lattice

In this thesis, we study the electron-electron interaction on the graphene honeycomb lattice theoretically. Even though pristine graphene behaves like a semimetal, sufficiently strong interactions can place the system in ordered phases. First, we derive a general Lagrangian for repulsive, short-ranged quartic interactions. The number of parameters in the Lagrangian is restricted by the symmetries present in the lattice, and the emergent ones. Then, we study the interacting theory in the framework of the renormalization group. All the critical points describing the transitions from the semimetallic phase to insulating phases reside in a Lorentz symmetric subspace. All the transitions are continuous and weak Lorentz symmetry breaking is irrelevant near the critical points. We also study the behaviour of various physical observables near the criticality. In the presence of an attractive interaction, we study the superconducting ground state, when fermions living on the nearest-neighbour sites of the honeycomb lattice attract each other strongly. A spatially inhomogeneous, spin-triplet, odd under sublattice exchange, Kekule superconductor turns out to be the variational ground state. Within the mean field approximation, the Kekule superconductor is energetically the best solution at and close to filling one-half. Even though all the transitions in neutral graphene can only take place at strong couplings, penetration of either real or pseudo magnetic field lowers the critical strength for insulation to zero. We study the problem of interacting fermions in the presence of the two magnetic fields, as well as when both of them are present. Moreover, our analysis includes the formation of insulators in the presence of inhomogeneous fields. We take analytical and numerical approaches to convey the central message: irrespective of the form of the fields, as long as there exists a finite density of states near zero energy, graphene finds itself in an ordered phase even at infinitesimal interactions. However, in the presence of real (pseudo) magnetic field the order parameter breaks the chiral (time reversal) symmetry. We present a thorough study of the scaling of the interaction induced gap, universal amplitudes and finite size effects.