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.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.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsDec 1, 199723 citations

Localization properties of fractons in percolating structures

Colin John Lambert, G D Hughes

Scaling curves for the density of states and localization length of fractons near a percolation threshold are obtained for dilute and superconducting lattices in both two and three dimensions. These are used to verify the predictions of single-parameter scaling and to show that the energy dependence of the inverse localization length is consistent with known values of the spectral dimensionalities of these structures. The results demonstrate that eigenstates of an infinite, percolating, dilute structure exhibit superlocalization of the form \ensuremath{\psi}(r)\ensuremath{\sim}${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}\mathit{r}\mathrm{\ensuremath{\delta}}}$ with values of \ensuremath{\delta}=1.13\ifmmode\pm\else\textpm\fi{}0.06 and 1.39\ifmmode\pm\else\textpm\fi{}0.07 in two and three dimensions, respectively.Scaling curves for the density of states and localization length of fractons near a percolation threshold are obtained for dilute and superconducting lattices in both two and three dimensions. These are used to verify the predictions of single-parameter scaling and to show that the energy dependence of the inverse localization length is consistent with known values of the spectral dimensionalities of these structures. The results demonstrate that eigenstates of an infinite, percolating, dilute structure exhibit superlocalization of the form \ensuremath{\psi}(r)\ensuremath{\sim}${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}\mathit{r}\mathrm{\ensuremath{\delta}}}$ with values of \ensuremath{\delta}=1.13\ifmmode\pm\else\textpm\fi{}0.06 and 1.39\ifmmode\pm\else\textpm\fi{}0.07 in two and three dimensions, respectively.

Physical Review LettersFeb 25, 199131 citations

Fractional excitations in foliated fracton phases

Wilbur Shirley, Kevin Slagle, Xie Chen

arXiv:1806.08625Annals of PhysicsAug 16, 2019120 citations

Entanglement and Localization of Wavefunctions

O. Giraud, Bertrand Georgeot, John Martin

arXiv:0907.3389NATO science for peace and security series. B, Physics and biophysicsJan 1, 20090 citations

Characteristics of Fractons: from Specific Realizations to Ensemble Averages

Tsuneyoshi Nakayama, Kousuke Yakubo, R. Orbach

Specific realizations due to computer experiments show that fracton excitations possess a core in which the vibrational amplitudes fall off sharply at their edges. When constructing the ensemble average of the fracton wave functions, we find $\langle\phi_{\text{fr}}\rangle{\sim}\exp[-(r/\varLambda(\omega))^{d_{\phi}}]$ , where $\varLambda(\omega){\sim}\omega^{-0.71}$ . The exponent, -0.71, is in close agreement with the prediction of the fracton dispersion law for a two-dimensional percolating network, -0.705, and the geometrical exponent d φ is much larger than current theoretical limits for impurity electronic states. This suggests that the two problems are different.Specific realizations due to computer experiments show that fracton excitations possess a core in which the vibrational amplitudes fall off sharply at their edges. When constructing the ensemble average of the fracton wave functions, we find $\langle\phi_{\text{fr}}\rangle{\sim}\exp[-(r/\varLambda(\omega))^{d_{\phi}}]$, where $\varLambda(\omega){\sim}\omega^{-0.71}$. The exponent, -0.71, is in close agreement with the prediction of the fracton dispersion law for a two-dimensional percolating network, -0.705, and the geometrical exponent d φ is much larger than current theoretical limits for impurity electronic states. This suggests that the two problems are different.

Journal of the Physical Society of JapanJun 15, 198924 citations

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.

\langle\phi_{\text{fr}}\rangle{\sim}\exp[-(r/\varLambda(\omega))^{d_{\phi}}]

, where

\varLambda(\omega){\sim}\omega^{-0.71}

. The exponent, -0.71, is in close agreement with the prediction of the fracton dispersion law for a two-dimensional percolating network, -0.705, and the geometrical exponent d φ is much larger than current theoretical limits for impurity electronic states. This suggests that the two problems are different.