Electronic transport in mesoscopic structures: The quantum Brownian motion picture

We study the quantum transport of electrons in mesoscopic structures using the picture of quantum Brownian motion in a periodic potential.; First, resonant tunneling between fractional quantum Hall edge states is considered in the Luttinger liquid picture. For the {dollar}nu=1/3{dollar} quantum Hall liquid, the resonance line shape is described by a universal function whose width scales to zero at T = 0. We calculate the scaling function using a Monte Carlo simulation method. Our results confirm the scaling theory and predict the explicit form of the scaling function over the entire width of the resonance.; Then, we investigate a quantum dot in the integer quantum Hall effect regime that is strongly coupled to a lead via a point contact. We find that even when the point contact is perfectly transmitting, important features of Coulomb blockade persist. In particular, the tunneling into the dot from a second weakly coupled lead is suppressed, showing features that can be ascribed to cotunneling. Weak backscattering at the point contact gives rise to oscillations in both the tunneling conductance G and the differential capacitance C as a function of gate voltage. We point out that the dimensionless ratio {dollar}xi{dollar} between the fractional oscillations in G and C is an intrinsic property of the dot. We compute {dollar}xi{dollar} within two models of electron-electron interactions. The role of additional channels is also discussed.; Finally, we study the general problem of quantum Brownian motion in a periodic potential. In D = 1 dimension, there are two T = 0 phases: a localized phase with zero-temperature mobility {dollar}mu{dollar} = 0 and an extended phase with {dollar}mu{dollar} unaffected by the periodic potential. For D {dollar}>{dollar} 1, however, non-symmorphic lattices such as honeycomb lattice and its D-dimensional generalization have an intermediate phase with a universal mobility {dollar}musp*{dollar} between 0 and the maximum perfect value. We study this intermediate fixed point in perturbatively accessible regimes. In addition, by mapping this problem to the Toulouse limit of the (D + 1)-channel Kondo model we can exactly compute {dollar}musp*{dollar} using results known from conformal field theory. Experimental implications are discussed in connection to resonant tunneling in Coulomb blockade structures.