Conductance of Mesoscopic Rings

In the studies of conductance of closed mesoscopic systems, semi-linear response theory (SLRT) offers a novel unified framework that goes beyond the traditional Kubo formulation. In this Thesis we apply SLRT to two types of mesoscopic systems. The first is disordered quasi-1d rings, where we build an analytical model to understand the departure of SLRT results from Kubo results. Guided by the numerical analysis on the statistical properties of the current operator matrix elements, we introduce a random matrix theory (RMT) model which leads to a generalized variable range hopping (VRH) picture of the conductance. Both of these models capture the essential aspects of the mesoscopic conductance for this system. The second system is the Harper model, which is a one-dimensional model that exhibits a Metal-Insulator transition—at the critical point, it possesses fractal structures in both the eigenfunctions and the eigenvalues. We are interested in studying how the fractal structures might have an effect on the mesoscopic conductance calculations.