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For nearly a decade, quasi-one-dimensional conductors have been the subject of intensive study. Theoretically, much attention has been devoted to the development of one-dimensional Fermi gas models, some which may be solved exactly, and to the calculation of their response functions. After a review of this theory, a different approach is adopted in the investigation of two models. The dielectric response theory of the three-dimensional Coulomb gas has been applied to an anisotropic system in which the particles interact with an effective one-dimensional long-range potential. Within the framework of the approximation of Singwi, Tosi, Land, and Sjolander, the dielectric properties of the model are examined in order to determine the conditions under which it is unstable with respect to formation of a charge density wave state. It is found that the positive neutralizing background must be polarizable in order for such an instability to occur. The same approximation method, when applied to a one-dimensional fermion gas with a ʃ-function interaction may be compared with the exact solution of Yang. This solution, which exists in the form of coupled integral equations, has been calculated numerically, and, as predicted by the Lieb-Mattis theorem, the ground state is found to be non-magnetic. The approximation of Singwi et al. proves to give better correlation energies than other inexact methods, particularly at higher densities. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

In this thesis we study the effect of electron-electron interactions on Chiral two-dimensional electron gas (C2DEGs). C2DEGs are a very good description of the low-energy electronic properties of single layer and multilayer graphene systems. The low-energy properties of single layer and multilayer graphene are described by Chiral Hamiltoninans whose band eigenstates have definite chirality. In this thesis we focus on the effect of electron-electron interactions on two of these systems: monolayer and bilayer graphene. In the first half of this thesis we use the massless Dirac Fermion model and random-phase-approximation to study the effect of interactions in graphene sheets. The interplay of graphene's single particle chiral eigenstates along with electron-electron interactions lead to a peculiar supression of spin susceptibility and compressibility, and also to an unusual velocity renormalization. We also report on a theoretical study of the influence of electron-electron interactions on ARPES spectra in graphene. We find that level repulsion between quasiparticle and plasmaron resonances gives rise to a gap-like feature near the Dirac point. In the second half we anticipate interaction driven integer quantum Hall effects in bilayer graphene because of the near-degeneracy of the eight Landau levels which appear near the neutral system Fermi level. We predict that an intra-Landau-level cyclotron resonance signal will appear at some odd-integer filling factors, accompanied by collective modes which are nearly gapless and have approximate q[superscrit 3/2] dispersion. We speculate on the possibility of unusual localization physics associated with these modes. / text