A significant portion of this dissertation is devoted to the study of the effects of impurities in substances whose low energy modes can be described by fermions obeying the gapless Dirac equation in 2+1 dimensions.

First, we examine the case of a spin vacancy in the staggered flux spin liquid whose excitations are Dirac fermions coupled to a U(1) gauge field. This vacancy leads to an anomalous Curie susceptibility and does not induce any local orders. Next, a Coulomb charge impurity placed on clean graphene is considered. We find that the Dirac quasiparticles in graphene do not screen the impurity charge, to all orders in perturbation theory. However, electronic correlations are found to induce a cloud of charge having the same sign as the impurity charge. We also analyze the case of a local impurity in graphene in the presence of a magnetic field and derive the spatial fourier transform of tunneling spectroscopy data obtained on an almost-clean graphene sheet in a magnetic field.

We then move on to Strong Topological Insulators (STI) whose surfaces are inhabited by an odd number of Dirac fermion species. Local impurities on the STI surface are found to induce resonance(s) in the local density of states (LDOS) in their vicinity.

In the case of magnetic impurities that may be approximated by classical spins, we show the existence of RKKY interactions that favor ferromagnetic aligning of randomly placed impurity spins perpendicular to the STI surface when the doping is insignificant. Finally we also consider the effects of a step edge on the STI surface and calculate the spatial decay of LDOS perturbations away from it.

The last chapter of this dissertation contains the theory for a new kind of spin liquid in a system of spin halves arranged in a triangular lattice, whose excitations are spin one Majorana fermions. This gapless spin liquid is found to possess certain properties that are exhibited by the recently discovered spin liquid state in the organic charge transfer salt EtMe ₃Sb(Pd(dmit)₂)₂.