Transport, multifractality, and the breakdown of single-parameter scaling at the localization transition in quasiperiodic systems

There has been a revival of interest in localization phenomena in quasiperiodic systems with a view to examining how they differ fundamentally from such phenomena in random systems. Motivated by this, we study transport in the quasiperiodic, one-dimensional Aubry-Andre model and its generalizations to two and three dimensions. We study the conductance of open systems, connected to leads, as well as the Thouless conductance, which measures the response of a closed system to boundary perturbations. We find that these conductances show signatures of a metal-insulator transition from an insulator, with localized states, to a metal, with extended states having (a) ballistic transport (one dimension), (b) superdiffusive transport (two dimensions), or (c) diffusive transport (three dimensions); precisely at the transition, the system displays subdiffusive critical states. We calculate the beta function beta(g) = d ln(g)/d ln(L) and show that, in one and two dimensions, single-parameter scaling is unable to describe the transition. Furthermore, the conductances show strong nonmonotonic variations with L and an intricate structure of resonant peaks and subpeaks. In one dimension the positions of these peaks can be related precisely to the properties of the number that characterizes the quasiperiodicity of the potential; and the L dependence of the Thouless conductance is multifractal. We find that, as dimension increases, this nonmonotonic dependence of g on L decreases and, in three dimensions, our results for beta(g) are reasonably well approximated by single-parameter scaling