For Anderson tight-binding models in dimension d with random on-site
energies ϵr and critical long-ranged hoppings decaying
typically as Vtyp(r)∼V/rd, we show that the strong multifractality
regime corresponding to small V can be studied via the standard perturbation
theory for eigenvectors in quantum mechanics. The Inverse Participation Ratios
Yq(L), which are the order parameters of Anderson transitions, can be
written in terms of weighted L\'evy sums of broadly distributed variables (as a
consequence of the presence of on-site random energies in the denominators of
the perturbation theory). We compute at leading order the typical and
disorder-averaged multifractal spectra τtyp(q) and τav(q) as a
function of q. For q<1/2, we obtain the non-vanishing limiting spectrum
τtyp(q)=τav(q)=d(2q−1) as V→0+. For q>1/2, this method
yields the same disorder-averaged spectrum τav(q) of order O(V) as
obtained previously via the Levitov renormalization method by Mirlin and Evers
[Phys. Rev. B 62, 7920 (2000)]. In addition, it allows to compute explicitly
the typical spectrum, also of order O(V), but with a different q-dependence
τtyp(q)=τav(q) for all q>qc=1/2. As a consequence, we find
that the corresponding singularity spectra ftyp(α) and
fav(α) differ even in the positive region f>0, and vanish at
different values α+typ>α+av, in contrast to the standard
picture. We also obtain that the saddle value αtyp(q) of the Legendre
transform reaches the termination point α+typ where
ftyp(α+typ)=0 only in the limit q→+∞.Comment: 13 pages, 2 figures, v2=final versio