Electrons and fractons on percolation structures at criticality: Sublocalization and superlocalization

Abstract

We discuss the localization of electronic eigenfunctions and vibrational excitations (``fractons'') on self-similar percolation clusters at criticality on the Cayley tree, in $d=2$, and in $d=3$, both in topological $(\mathcal{l})$ and in Euclidian $(r)$ space. We find that the localization behavior of electrons and fractons is very similar, but does depend on the type of average performed and on the number of configurations $N$ taken into account. It can be characterized by three different localization regimes. In the first regime, at small distances $\mathcal{l}$ and $r$ from the localization center, electrons and fractons are superlocalized in $\mathcal{l}$ space, while the localization is not exponential in $r$ space. In the intermediate regime, which is the most important one, we find stretched exponential localization (``sublocalization''), $\mathrm{ln}〈\ensuremath{\psi}〉\ensuremath{\sim}\ensuremath{-}{\mathcal{l}}^{{d}_{\ensuremath{\Psi},\mathcal{l}}}$ or $\ensuremath{\sim}\ensuremath{-}{r}^{{d}_{\ensuremath{\Psi},r}}$ with effective localization exponents ${d}_{\ensuremath{\Psi},\mathcal{l}}\ensuremath{\approx}{d}_{\ensuremath{\Psi},r}\ensuremath{\approx}0.6$. In the third regime, for large $\mathcal{l}$ and $r$, the averages strongly depend on the number of configurations, even in $\mathcal{l}$ space, and superlocalization ${(d}_{\ensuremath{\Psi},\mathcal{l}}{,d}_{\ensuremath{\Psi},r}>1)$ is observed, converging to simple exponential behavior asymptotically. To understand this complicated localization behavior, we present an analytical calculation of $\mathrm{ln}〈\ensuremath{\psi}〉$ for the third regime, which is based on the fact that the local amplitudes at large distances from the localization center obey a log-normal distribution.