Multifractality of eigenstates in the delocalized non-ergodic phase of some random matrix models : application to Levy matrices

Cecile Monthus

Published 2016-09-05Version 1

The delocalized non-ergodic phase existing in some random $N \times N$ matrix models is analyzed via the Wigner-Weisskopf approximation for the dynamics from an initial site $j_0$. The main output of this approach is the inverse $\Gamma_{j_0}(N)$ of the characteristic time to leave the state $j_0$ that provides some broadening $\Gamma_{j_0}(N) $ for the weights of the eigenvectors. In this framework, the localized phase corresponds to the region where the broadening $\Gamma_{j_0}(N) $ is smaller in scaling than the level spacing $\Delta_{j_0}(N) $, while the delocalized non-ergodic phase corresponds to the region where the broadening $\Gamma_{j_0}(N) $ decays with $N$ but is bigger in scaling than the level spacing $\Delta_{j_0}(N) $. Then the number $\frac{\Gamma_{j_0}(N)}{\Delta_{j_0}(N)} $ of resonances grows only sub-extensively in $N$. We describe how the multifractal spectrum of eigenstates can be then explicitly computed in two models. For the Generalized-Rosenzweig-Potter (GRP) Matrix model, the present approach allows to recover the multifractal spectrum of Ref [V.E. Kravtsov, I.M. Khaymovich, E. Cuevas and M. Amini, New. J. Phys. 17, 122002 (2015)]. For the L\'evy matrix model, where matrix elements are drawn with some heavy-tailed distribution $P(H_{ij}) \propto N^{-1} \vert H_{ij} \vert^{-1-\mu}$, we derive the multifractal spectrum in the delocalized non-ergodic phase existing in some energy region for $1<\mu<2$ (while the region $0<\mu<1$ containing only localized eigenvectors is characterized by the multifractality studied in our previous work arxiv:1606.03241).