There has been much recent interest in Dirac fermions due to their physical realizationas low energy excitations in graphene. In this thesis we introduce birefringent relativisticfermions, for which the chiral symmetry usually present for Dirac fermions is broken, andthere can be more than one Fermi velocity. We first introduce a lattice model of spinlessfermions that can arise from a scheme to introduce an artificial magnetic field for coldatoms. This model has an unusual Hofstadter-like spectrum as a function of the flux perplaquette. When there is an average of half a flux quantum per plaquette, the model hasDirac points in its spectrum and exhibits low energy excitations with two different “speedsof light”, i.e. birefringent fermions. We investigate the effects of several perturbations onthe spectrum such as staggered potentials and topological defects and we study the orderedphases that can arise from interactions. We find that sufficiently strong nearest neighbourinteractions lead to a charge density wave phase but that next-nearest neighbour interactionsallow the possibility of other phases. We also study the response of birefringent fermionsto a magnetic field and discuss how both Landau levels and the Integer Quantum Hall effectfor regular Dirac fermions are modified for birefringent fermions.
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Thesis advisor: Kennett, Malcolm
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