Chaos and interactions in quantum dots.

摘要

Random-matrix theory has proved to be a successful tool in understanding the statistics of transport measurements in mesoscopic systems. Fluctuations of conductance peak-heights due to the transfer of single electrons through diffusive and chaotic ballistic quantum dots in the Coulomb-blockade regime can be modeled by fluctuations of random-matrix wavefunctions at low temperatures in the simple constant interaction model. On the other hand, statistics of the spacing between successive peaks do not show the behaviour predicted by the simple model, indicating that interactions beyond constant charging energy play an important role. The central theme of this thesis is the study of interaction effects on the mesoscopic fluctuations in a weakly disordered or chaotic ballistic quantum dot. Mean-field calculations suggest that upon addition of electrons into the dot, single-particle wavefunctions are not significantly altered while their eigenvalues "scramble" with the change in the self-consistent potential, thus affecting the peak-spacing statistics. We model this variation in the mean field by an extension of random-matrix theory: Gaussian processes, where the ensemble of random matrices depends on a discrete parameter, the number of electrons. Using a Gaussian process we can explain the saturation of the peak-to-peak correlator versus temperature. Another experimental signature of interactions is the enhanced peak-height correlation length versus an experimental magnetic field. We calculate the parametric peak-height correlator in the random interaction matrix model whose single-particle Hamiltonian is modeled by a Gaussian process. We find an increase in peak-height correlation length with the interaction fluctuation strength, that explains qualitatively the experimental observations. A microscopic understanding of the peak-spacing distribution requires the inclusion of the spin degrees of freedom. We introduce the Hartree-Fock-Koopmans approach for electrons with spin and calculate the Coulomb-blockade spacing statistics in quantum dots with a large number of electrons and at low temperatures. We account for the exchange interaction, as well as the fluctuations of interaction matrix elements to leading order in inverse Thouless conductance. We explain various features of the experimental peak-spacing distributions, including the absence of bimodality.