Transport, multifractality and breakdown of single-parameter scaling in quasiperiodic systems.

We study the transport in 1d Aubry-Andre model, and its generalizations to 2d and 3d, which show significant deviations from single-parameter scaling theory providing a broad demarcation between QPS and RS. We study the conductance of open systems connected to leads as well as the Thouless conductance. Depending on dimension, the conductances show metal-insulator transition from localized to either ballistic, superdiffusive or diffusive transport typically through subdiffusive critical states. We show that, even though a beta function, β(g) =dln(g)/dln(L), can be constructed separately for individual phases based on a overall length (L) dependence of typical dimensionless conductance g, in 1d and 2d, the single-parameter scaling is unable to describe the transition. Moreover, the conductances show strong non-monotonic variations (multifractal scaling in 1d) with system size and intricate resonant peak and subpeak structures of number theoretic origin, invalidating a strict definition of β(g). We show that the non-monotonicity is very weak in 3d with a critical point having localization length exponent close to that of 3d Anderson transition and the single-parameter scaling is almost restored providing a good description of the metal-insulator transition.