The Anderson localization transition with long-ranged hoppings: analysis of the strongmultifractality regime in terms of weighted Lévy sums

Abstract

For Anderson tight-binding models in dimensiond with random on-site energies and critical long-ranged hoppings decaying typically asVtyp(r) ∼ V/rd, we show that the strong multifractality regime corresponding to smallV can bestudied via the standard perturbation theory for eigenvectors in quantum mechanics. The inverse participationratios Yq(L), which are the order parameters of Anderson transitions, can be written in terms ofweighted Lévy sums of broadly distributed variables (as a consequence of the presence ofon-site random energies in the denominators of the perturbation theory). Wecompute at leading order the typical and disorder-averaged multifractal spectraτtyp(q) andτav(q) as afunction of q. For q < 1/2, we obtain the non-vanishing limiting spectrumτtyp(q) = τav(q) = d(2q − 1) as . For q > 1/2, this method yields the same disorder-averaged spectrumτav(q) of orderO(V) asobtained previously via the Levitov renormalization method by Mirlin and Evers (2000 Phys. Rev. B 627920). In addition, it allows us to compute explicitly the typical spectrum, also of orderO(V), but with a differentdependence on q, for all q > qc = 1/2. As a consequence, we find that the corresponding singularity spectraftyp(α) andfav(α) differ even in the positiveregion f > 0, and vanishat different values α + typ > α + av, in contrast to the standard picture. We also obtain that the saddle valueαtyp(q) of the Legendre transform reaches the termination pointα + typ whereftyp(α + typ) = 0 only in the limit .