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
⟨ϕfr⟩∼exp[−(r/Λ(ω))dϕ], where
Λ(ω)∼ω−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.
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
(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
l and
r from the localization center, electrons and fractons are superlocalized in
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
l and
r, the averages strongly depend on the number of configurations, even in
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.
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.
Disordered systems present multifractal properties at criticality. In particular, as discovered by Ludwig [A.W.W. Ludwig, Nucl. Phys. B 330, 639 (1990)] in the case of a diluted two-dimensional Potts model, the moments rho(q) (r) of the local order parameter rho(r) scale with a set x(q) of nontrivial exponents x(q) not = qx(1). We reexamine these ideas to incorporate more recent findings: (i) whenever a multifractal measure w(r) normalized over space sum(r) w(r) = 1 occurs in a random system, it is crucial to distinguish between the typical values and the disorder-averaged values of the generalized moments Y(q) = sum(r) w(q) (r), since they may scale with different generalized dimensions D(q) and D(q), and (ii), as discovered by Wiseman and Domany [S. Wiseman and E. Domany, Phys. Rev. E 52, 3469 (1995)], the presence of an infinite correlation length induces a lack of self-averaging at critical points for thermodynamic observables, in particular for the order parameter. After this general discussion, valid for any random critical point, we apply these ideas to random polymer models that can be studied numerically for large sizes and good statistics over the samples. We study the bidimensional wetting or the Poland-Scheraga DNA model with loop exponents c = 1.5 (marginal disorder) and c = 1.75 (relevant disorder). Finally, we argue that the presence of finite Griffiths-ordered clusters at criticality determines the asymptotic value x(q-->infinity) = d and the minimal value alpha(min) = D(q-->infinity) = d - x(1) of the typical multifractal spectrum f(alpha).
Based on the assumptions of the existence of a one-paramter scaling law for a self-averaging scaling variable, and on a finite correlation length for the randomness, the lower bound for the critical exponent of the Anderson transition derived earlier by Chayes et al. [Phys. Rev. Lett. 57, 2999 (1986)] for uncorrelated disorder is generalized to the case of spatially correlated disorder. The bound is determined by the system-size dependence of the number of the random variables that are necessary to describe the transition. It is speculated how to relate the critical exponent to the multifractal properties of the states near the critical point. Exponents of several models are estimated and compared with numerical scaling calculations.
We describe a new multifractal finite-size scaling (MFSS) procedure and its application to the Anderson localization-delocalization transition. MFSS permits the simultaneous estimation of the critical parameters and the multifractal exponents. Simulations of system sizes up to
L3=1203 and involving nearly
106 independent wave functions have yielded unprecedented precision for the critical disorder
Wc=16.530(16.524,16.536) and the critical exponent $\ensuremath{\nu}=1.590(1.579,1.602)$. We find that the multifractal exponents ${\ensuremath{\Delta}}_{q}$ exhibit a previously predicted symmetry relation and we confirm the nonparabolic nature of their spectrum. We explain in detail the MFSS procedure first introduced in our Letter [Phys. Rev. Lett. 105, 046403 (2010)] and, in addition, we show how to take account of correlations in the simulation data. The MFSS procedure is applicable to any continuous phase transition exhibiting multifractal fluctuations in the vicinity of the critical point.