Multifractality of eigenstates in the delocalized non-ergodic phase of some random matrix models : Wigner-Weisskopf approach

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 $\Delta_{j_0}(N) \propto \frac{1}{N}$, 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 $N$. This approach allows to recover the multifractal spectrum of the Generalized-Rosenzweig-Potter (GRP) Matrix model [V.E. Kravtsov, I.M. Khaymovich, E. Cuevas and M. Amini, New. J. Phys. 17, 122002 (2015)]. We then consider the L\'evy generalization of the GRP Matrix model, where the off-diagonal matrix elements are drawn with an heavy-tailed distribution of L\'evy index $1<\mu<2$ : the dynamics is then governed by a stretched exponential of exponent $\beta=\frac{2 (\mu-1)}{\mu}$ and the multifractal properties of eigenstates are explicitly computed.