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There has been a revival of interest in localization phenomena inquasiperiodic systems with a view to examining how they differ fundamentallyfrom such phenomena in random systems. Mo- tivated by this, we study transportin the quasiperiodic, one-dimentional (1d) Aubry-Andre model and itsgeneralizations to 2d and 3d. We study the conductance of open systems,connected to leads, as well as the Thouless conductance, which measures theresponse of a closed system to boundary perturbations. We find that theseconductances show signatures of a metal-insulator transition from an insulator,with localized states, to a metal, with extended states having (a) ballistictransport (1d), (b) superdiffusive transport (2d), or (c) diffusive transport(3d); precisely at the transition, the system displays sub-diffusive criticalstates. We calculate the beta function $\beta(g) = dln(g)/dln(L)$ and showthat, in 1d and 2d, single-parameter scaling is unable to describe thetransition. Further- more, the conductances show strong non-monotonicvariations with L and an intricate structure of resonant peaks and subpeaks. In1d the positions of these peaks can be related precisely to the prop- erties ofthe number that characterizes the quasiperiodicity of the potential; and theL-dependence of the Thouless conductance is multifractal. We find that, as dincreases, this non-monotonic de- pendence of g on L decreases and, in 3d, ourresults for $\beta(g)$ are reasonably well approximated by single-parameterscaling.

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