Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2012.Cataloged from PDF version of thesis.Includes bibliographical references.Two dimensional electron systems (2DES) provide a uniquely promising avenue for investigation of many body physics. Graphene constitutes a new and unusual 2DES, which may give rise to unexpected collective phenomena. However, the vanishing density of states in charge neutral single layer graphene suppresses many body effects, and one has to alter the system to observe strongly ordered states. We consider three ways of accessing quantum many body physics using graphene. First, we consider doping single layer graphene to a Van Hove singularity in the density of states. We show that there are strong instabilities to several strongly ordered states, with the leading instability being to a d-wave superconducting state. The superconducting state realizes chiral superconductivity, an exotic form of superconductivity wherein the phase of the order parameter winds by 4[pi] as we go around the Fermi surface. We also discuss the nature of the spin density wave state which is the principal competitor to superconductivity in doped graphene. Next, we study bilayer graphene (BLG), which has a non-vanishing density of states even at charge neutrality. We show that Coulomb interactions give rise to a zero bias anomaly in the tunneling density of states for BLG, which manifests itself at high energy scales. We also show that the quadratic band crossing in BLG is unstable to arbitrarily weak interactions, and estimate the energy scale for formation of strongly ordered states. We show that gapped states in BLG have topological properties, and we classify the various possible gapped and gapless states in terms of symmetries. We study the competition between various ordered states, and discuss how the nature of the ground state may be deduced experimentally. We also discuss recent experimental observations of strongly ordered states in bilayer graphene. Finally, we study bilayer graphene in a transverse magnetic field, focusing on the properties of the quantum Hall ferromagnet (QHF) state. We resolve an apparent discrepancy between the experimentally observed energetics and theory. We close with a discussion of the exotic topological defects that form above the QHF state.by Rahul Nandkishore.Ph.D
