Anderson localization transition with long-ranged hoppings : analysis of the strong multifractality regime in terms of weighted Levy sums

Abstract

For Anderson tight-binding models in dimension $d$ with random on-site energies ${\epsilon }_{\vec{r}}$ and critical long-ranged hoppings decaying typically as $V^{typ}(r)\sim V/r^d$, 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 $Y_q(L)$, which are the order parameters of Anderson transitions, can be written in terms of weighted Lévy 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 ${\tau }_{typ}(q)$ and ${\tau }_{av}(q)$ as a function of $q$. For $q<1/2$, we obtain the non-vanishing limiting spectrum ${\tau }_{typ}(q)={\tau }_{av}(q)=d(2q-1)$ as $V\to 0^{+}$. For $q>1/2$, this method yields the same disorder-averaged spectrum ${\tau }_{av}(q)$ 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 ${\tau }_{typ}(q)\ne {\tau }_{av}(q)$ for all $q>q_c=1/2$. As a consequence, we find that the corresponding singularity spectra $f_{typ}(\alpha )$ and $f_{av}(\alpha )$ differ even in the positive region $f>0$, and vanish at different values ${\alpha }_{+}^{typ}>{\alpha }_{+}^{av}$, in contrast to the standard picture. We also obtain that the saddle value ${\alpha }_{typ}(q)$ of the Legendre transform reaches the termination point ${\alpha }_{+}^{typ}$ where $f_{typ}({\alpha }_{+}^{typ})=0$ only in the limit $q\to +\infty$.

13 pages, 2 figures, v2=final version

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