Plasmonic and excitionic effects in restricted geometries

Many electron systems are well described by Landau's Fermi liquid theory. However, such a decent theory breaks down in one dimension due to the correspondingly \unusual" shape of a Fermi surface. A one-dimensional electron system is thus called a Luttinger liquid, which was first proposed by Tomonaga in 1950 and later developed by Luttinger, Mattis, Haldane and others. The most notable difference in 1D is that low energy excitations are massless fermions with linear dispersion. Correspondingly, electron-electron interaction significantly alters transport properties in 1D. Graphene is a two-dimensional layer material with a honeycomb structure, and its low energy excitations are massless fermions. Accordingly, carbon nanotube, as the roll-up sheet of graphene, has been regarded as an ideal platform for testing Luttinger liquid effects. In Chapter 2, we studied the excitonic eect in metallic nanotubes. Exciton is formed when a particle and a hole is bounded by their attractive Coulomb interaction. In bulk metals, exciton hardly exists due to the strong screening effects. However, the situation changes in 1D where Coulomb interaction remains largely unscreened. The relatively ineffective screening yields a sufficiently large radius of excitons that is about 10 times larger than the radius of nanotube Rex  10R. Therefore, excitons in metallic nanotube become a well-defined 1D problem. The problem has been studied in an analytical manner. In Chapter 3, we discussed many-body effects in the depolarization effect. For a cylinder exposed in an external electric field perpendicular to its axis, the electric field inside the cylinder is found to be \suppressed." The suppression is due to the depolarization effect. Same phenomena occur in nanotube, which originates from a dipolar Coulomb interaction. The prediction is within the theory electrostatics or Random Phase Approximation(RPA) in quantum theory. However, what makes the problem more complex is Luttinger liquid effects, which enter the RPA series as vertex correction. Post-RPA effects, or many-body effects, are often hard to treat exactly. One often resorts to numerical approach or renormalization group in the qualitative fashion. However, by virtue of a hybrid approach, which combines the advantage of bosonization and perturbation theory, we examined many-body modification of the line shape in a depolarization effect. In addition to single particle excitations in the electron system, plasmon excitations as a novel excitation also attract researchers. Under the external field, electrons in metals are oscillating back and force to gain a balance against the external field. Such oscillations exist even without the presence of external fields, which are called plasmon excitations. Therefore, plasmon waves are collective oscillation, namely all electrons are moving as a whole. For the plasmon waves that are localized near the interface of two materials while propagating along the surface, they are called surface plasmon or surface plasmon polariton if retardation eects are considered. The geometry of the boundary mostly determines the property of the surface plasmon. In Chapter 5, a novel surface plasmon in hyperbolic boundary was discussed in many perspectives.