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.
The multifractal analysis of disorder induced localization-delocalization transitions is reviewed. Scaling properties of this transition are generic for multi parameter coherent systems which show broadly distributed observables at criticality. The multifractal analysis of local measures is extended to more general observables including scaling variables such as the conductance in the localization problem. The relation of multifractal dimensions to critical exponents such as the order parameter exponent β and the correlation length exponent ν is investigated. We discuss a number of scaling relations between spectra of critical exponents, showing that all of the critical exponents necessary to characterize the critical phenomenon can be obtained within the generalized multifractal analysis. Furthermore we show how bounds for the correlation length exponent ν are obtained by the typical order parameter exponent α0 and make contact between the multifractal analysis and the finite size scaling approach in 2-d by relying on conformal mapping arguments. 1.
The multifractal analysis of disorder-induced localization-delocalization transitions is reviewed. Scaling properties of this transition are generic for multi parameter coherent systems which show broadly-distributed observables at criticality. The multifractal analysis of local measures is extended to more general observables including scaling variables such as the conductance in the localization problem. The relation of multifractal dimensions to critical exponents such as the order parameter exponent β and the correlation length exponent ν is investigated, We discuss a number of scaling relations between spectra of critical exponents, showing that all of the critical exponents necessary to characterize the critical phenomenon can be obtained within the generalized multifractal analysis. Furthermore we show how bounds for the correlation length exponent ν are obtained by the typical order parameter exponent α0 and make contact between the multifractal analysis and the finite size scaling approach in 2-d by relying on conformal mapping arguments.
We consider the model of the directed polymer in a random medium of dimension
1+3, and investigate its multifractal properties at the localization-delocalization transition. In close analogy with models of the quantum Anderson localization transition, where the multifractality of critical wavefunctions is well established, we analyze the statistics of the position weights ${w}_{L}(\stackrel{P\vec}{r})$ of the endpoint of the polymer of length
L via the moments ${Y}_{q}(L)={\ensuremath{\sum}}_{\stackrel{P\vec}{r}}\phantom{\rule{0.2em}{0ex}}{[{w}_{L}(\stackrel{P\vec}{r})]}^{q}$. We measure the generalized exponents $\ensuremath{\tau}(q)$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}(q)$ governing the decay of the typical values ${Y}_{q}^{\mathrm{typ}}(L)={e}^{\overline{\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{Y}_{q}(L)}}\ensuremath{\sim}{L}^{\ensuremath{-}\ensuremath{\tau}(q)}$ and disorder-averaged values $\overline{{Y}_{q}(L)}\ensuremath{\sim}{L}^{\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}(q)}$, respectively. To understand the difference between these exponents, $\ensuremath{\tau}(q)\ensuremath{\ne}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}(q)$ above some threshold $q>{q}_{c}\ensuremath{\sim}2$, we compute the probability distributions of
y=Yq(L)∕Yqtyp(L) over the samples: We find that these distributions becomes scale invariant with a power-law tail
1∕y1+xq. These results thus correspond to the Evers-Mirlin scenario [Phys. Rev. Lett. 84, 3690 (2000)] for the statistics of inverse participation ratios at the Anderson localization transitions. Finally, the finite-size scaling analysis in the critical region yields the correlation length exponent $\ensuremath{\nu}\ensuremath{\sim}2$.
For Anderson localization models, there exists an exact real-space renormalization procedure at fixed energy which preserves the Green's functions of the remaining sites [H. Aoki, J. Phys. C 13, 3369 (1980)]. Using this procedure for the Anderson tight-binding model in dimensions
d=2,3, we study numerically the statistical properties of the renormalized on-site energies
ϵ and of the renormalized hoppings
V as a function of the linear size
L. We find that the renormalized on-site energies
ϵ remain finite in the localized phase in
d=2,3 and at criticality
(d=3), with a finite density at
ϵ=0 and a power-law decay
1/ϵ2 at large
∣ϵ∣. For the renormalized hoppings in the localized phase, we find: $\text{ln}\text{ }{V}_{L}\ensuremath{\simeq}\ensuremath{-}\frac{L}{{\ensuremath{\xi}}_{loc}}+{L}^{\ensuremath{\omega}}u$, where ${\ensuremath{\xi}}_{loc}$ is the localization length and
u a random variable of order one. The exponent $\ensuremath{\omega}$ is the droplet exponent characterizing the strong disorder phase of the directed polymer in a random medium of dimension $1+(d\ensuremath{-}1)$, with $\ensuremath{\omega}(d=2)=1/3$ and $\ensuremath{\omega}(d=3)\ensuremath{\simeq}0.24$. At criticality
(d=3), the statistics of renormalized hoppings
V is multifractal, in direct correspondence with the multifractality of individual eigenstates and of two-point transmissions. In particular, we measure ${\ensuremath{\rho}}_{typ}\ensuremath{\simeq}1$ for the exponent governing the typical decay $\overline{\text{ln}\text{ }{V}_{L}}\ensuremath{\simeq}\ensuremath{-}{\ensuremath{\rho}}_{typ}\text{ }\text{ln}\text{ }L$, in agreement with previous numerical measures of ${\ensuremath{\alpha}}_{typ}=d+{\ensuremath{\rho}}_{typ}\ensuremath{\simeq}4$ for the singularity spectrum $f(\ensuremath{\alpha})$ of individual eigenfunctions. We also present numerical results concerning critical surface properties.
We present results of Monte Carlo simulations of random bond Potts models in two dimensions, for different numbers of Potts states q. We introduce a simple scheme which yields continuous self-dual distributions of the interactions. As expected, we find multifractal behavior of the correlation functions at the critical point and obtain estimates of the exponent ${\ensuremath{\eta}}_{n}$ for several moments n of the correlation functions, including typical $(\stackrel{\ensuremath{\rightarrow}}{n}0)$, average
(n=1), and others. In addition, for
q=8, we find that there is only a single correlation length exponent $\ensuremath{\nu}$ describing the correlation length away from criticality. This is numerically very close to the pure Ising value $\ensuremath{\nu}=1$.