Localization transition in random Levy matrices : multifractality of eigenvectors in the localized phase and at criticality

For random L\'evy matrices of size $N \times N$, where matrix elements are drawn with some heavy-tailed distribution $P(H_{ij}) \propto N^{-1} |H_{ij} |^{-1-\mu}$ with $0<\mu<2$ (infinite variance), there exists an extensive number of finite eigenvalues $E=O(1)$, while the maximal eigenvalue grows as $E_{max} \sim N^{\frac{1}{\mu}}$. Here we study the localization properties of the corresponding eigenvectors via some strong disorder perturbative expansion that remains consistent within the localized phase and that yields their Inverse Participation Ratios (I.P.R.) $Y_q$ as a function of the continuous parameter $0<q<+\infty$. In the region $0<\mu<1$, we find that all eigenvectors are localized but display some multifractality : the IPR are finite above some threshold $q>q_c$ but diverge in the region $0<q<q_c$ near the origin. In the region $1<\mu<2$, only the sub-extensive fraction $N^{\frac{3}{2+\mu}}$ of the biggest eigenvalues corresponding to the region $| E| \geq N^{\frac{(\mu-1)}{\mu(2+\mu)}}$ remains localized, while the extensive number of other states of smaller energy are delocalized. For the extensive number of finite eigenvalues $E=O(1)$, the localization/delocalization transition thus takes place at the critical value $\mu_c=1$ corresponding to Cauchy matrices : the Inverse Participation Ratios $Y_q$ of the corresponding critical eigenstates follow the Strong-Multifractality Spectrum characterized by the generalized fractal dimensions $D^{criti}(q) = \frac{1-2q}{1-q} \theta (0 \leq q \leq \frac{1}{2})$, which has been found previously in various other Localization problems in spaces of effective infinite dimensionality.