Disordered, Interacting Electrons: Two Tractable Limits

The focus of this theoretical research is to investigate the interplay of interactions and disorder in two tractable limits, with clear implications for the more generic problem. The first tractable case is the mesocopic one. Here there are two energy scales, the single-particle level spacing and the Thouless energy, which is connected by the uncertainty principle to the time for an electron to sample the system. The ratio of these two is the dimensionless conductance, g. In the limit of large g, the problem of disorder and (Fermi-liquid) interactions is completely solvable in the same sense as a conventional large-N theory. This approach is nonperturbative in both disorder and interactions, thanks to the small parameter 1/g. Interesting phase transitions and slow collective modes also emerge in this new framework, which allows one to understand experimental and numerical results on large Coulomb blockade fluctuations in quantum dots, and potentially the sign and magnitude of persistent currents in mesoscopic rings. The second tractable limit is in the integer/fractional quantum Hall elects, where strong interaction results in gaps which enable a controlled incorporation of the effects of disorder. A new Hamiltonian approach to the fractional quantum Hall regime makes it possible to calculate physical quantities using very simple approximations (because the nonperturbative properties of the quasiparticles, such as their fractional charge, have been incorporated into the theory). This formalism will be used to treat the gapped fractional quantum Hall states and the very interesting Fermi liquid state in the half-filled Landau level in the presence of disorder. The project will have a broad impact in training of a postdoctoral research associate; in stimulating experimental activity to verify the predictions of the theory; and it may have implications for the use of quantum dots in quantum computation. %%% The focus of this theoretical research is to investigate the interplay of interactions and disorder in two tractable limits, with clear implications for the more generic problem. The first tractable case is the mesocopic one, which includes quantum dots. The second case is in the quantum Hall regime. The project will have a broad impact in training of a postdoctoral research associate; in stimulating experimental activity to verify the predictions of the theory; and it may have implications for the use of quantum dots in quantum computation.