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 ${\Gamma }_{j_0}\left(N\right)$ of the characteristic time to leave the state j ₀ that provides some broadening ${\Gamma }_{j_0}\left(N\right)$ for the weights of the eigenvectors. In this framework, the localized phase corresponds to the region where the broadening ${\Gamma }_{j_0}\left(N\right)$ is smaller in scaling than the level spacing ${\Delta }_{j_0}\left(N\right)\propto \frac{1}{N}$, while the delocalized non-ergodic phase corresponds to the region where the broadening ${\Gamma }_{j_0}\left(N\right)$ decays with N but is bigger in scaling than the level spacing ${\Delta }_{j_0}\left(N\right)$. Then the number $\frac{{\Gamma }_{j_0}\left(N\right)}{{\Delta }_{j_0}\left(N\right)}$ of resonances grows only sub-extensively in N. This approach allows to recover the multifractal spectrum of the Generalized–Rosenzweig–Potter (GRP) Matrix model (Kravtsov et al 2015 New. J. Phys. 17 122 002). We then consider the Lévy generalization of the GRP Matrix model, where the off-diagonal matrix elements are drawn with an heavy-tailed distribution of Lévy 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. (paper)