Optical response in two model systems of interacting electrons: The 1D Hubbard model and 2D magnetoexcitons

In Part I of this thesis, we study the real part σ(ω) of the optical conductivity of a repulsive Hubbard ring of circumference L. At zero temperature, and in the absence of disorder, the dc part of σ(ω) has the form σ(ω) = (e2/h)Dcδ(ℎω). The charge stiffness Dc tends to a finite, positive value as L → ∞ in the metallic phase of the model; however, in the Mott-insulating phase of the model, which occurs at a mean electron density n = 1 (half filling), Dc ∼ O(L1/2exp(-L/ξ(U)) in the large-L limit, which serves to define the localization length ξ of the nondisordered Mott insulator. We obtain an analytic expression for ξ(U) as a function of the on-site repulsion U, and show that ξ is the correlation length of the equal-time single-particle Green's function, evaluated at half filling. We interpret ξ as the correlation length of the electron-hole pairing correlations in the ground state of the Mott insulator. We next consider the metallic phase of the model in the vicinity of the metal-insulator transition. We obtain analytic expressions, valid to leading order in the doping δ = |1 - n|, for the charge stiffness Dc = δ/|2m*| and the low-temperature thermopower S = -(k2BT/3e)m*/δ2 near half filing, where |m*| is a function of U which we calculate, and sign(m*) = sign(1 - n). We interpret these results in terms of a physical picture of the charge carriers in the lightly doped Mott insulator as spinless fermionic solitons of size ξ, effective mass m*, and number density δ, which become noninteracting in the limit ξδ → 0