content_image/385499.jpg content_image/385500.jpg content_image/385491.jpg content_image/385494.jpg content_image/385496.jpg content_image/385509.jpg content_image/385506.jpg
the PBRM model (symbols) in the periodic and non- periodic setups. In the same figure Eq. (7) is also plotted (dot-dashed lines). 

On the other hand, it is well known that in systems at the disorder driven MIT both the energy spectra and the eigenstates exhibit multifractal characteristics. 4 The PBRM model is characterized by the effective bandwidth b that drives the system from strong ( b → 0) to weak ( b → ∞ ) multifractality. Multifractality can be quan- tified by the generalized dimensions D$_{q}$ which describe the fluctuations of the eigenfunctions. The multifractal dimensions D$_{q}$ of the σ -th eigenfunction Ψ σ (given as a linear combination of the basis states in a system with lin- ear size L , Ψ σ = ∑ L l $_{=1}$C σ l φ$_{l}$ ) are defined through the so- called inverse participation numbers, I q = ∑ L l =1 | C σ l | 2 $^{q}$, by the scaling 4,33,42 

where ( I $_{q}$) typ = exp 〈 ln I q 〉 is the typical value of I $_{q}$. However, among all dimensions, the correlation dimen- sion D$_{2}$ plays a prominent role. 43 As b transits from zero to infinity, D$_{2}$ takes values from zero to one. 

Using numerical diagonalization of matrices with sizes L = 128, 256, 512, and 1024, we extracted D$_{2}$ by the use of Eq. (15). For each system size L , we used a num- ber of disorder realizations in order to have at least 10 5 data for its analysis. For each disorder realization we used 25% of the states at the centre of the spectral band (where the density of states is almost constant) in or- der to avoid boundary effects. We found good agreement between the numerically obtained D$_{2}$ and the analytical estimation 4,42 

Moreover, in Ref. [44] the following phenomenological an- alytical expression for D$_{2}$ as a function of b was proposed: 

where γ is a fitting parameter. See also Ref. [14]. Eq. (17) describes well our numerical results for D$_{2}$ with γ = 2 . 94 and γ = 2 . 88 for the periodic and non-periodic versions of the PBRM model, respectively. 

Then, by substituting Eq. (17) into Eqs. (13) and (14) we obtain the following expressions for the averages of | S$_{11}$ | 2 and | S$_{12}$ | 2 as a function of D$_{2}$ : 

In Fig. 2(b) we compare Eqs. (18) and (19) (dashed lines) with the ensemble-average numerical results at different values of D$_{2}$ . The agreement is excellent. In addition, it 

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is interesting to note that 

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which might be relevant for systems at the MIT where D$_{2}$ can be tuned. 

Finally we want to remark that concerning 〈 | S$_{aa}$ ′ | $^{2}$〉 for the PBRM model, the RMT limit, expected for b → ∞ , is already recovered for b ≥ 4. See Fig. 2(a). 

## B. Conductance probability distribution 

Now we turn to the conductance statistics. For b ≪ 1 the conductance distribution w ( T ) is highly concentrated close to T = 0. So it is more convenient to analyse the 

