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 ( I $_{q}$) typ ∝ L − ( q − 1) D$_{q}$ , (15) 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 D$_{2}$ = { 2 b , b ≪ 1 1 − ( πb ) − 1 , b ≫ 1 . (16) Moreover, in Ref. [44] the following phenomenological an- alytical expression for D$_{2}$ as a function of b was proposed: D$_{2}$ ( b ) = 1 1 + ( γb ) − 1 , (17) 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$_{2}$ : 〈 | S$_{11}$ | $^{2}$〉 D$_{2}$ = 1 − 〈 | S$_{12}$ | $^{2}$〉 D$_{2}$ , (18) | S$_{12}$ | D$_{2}$ = 1 3 1 1 + ( γ/δ ) 2 ( D − 1 2 − 1 ) 2 . (19) 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 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 B. Conductance probability distribution 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). which might be relevant for systems at the MIT where D$_{2}$ can be tuned. 〈 | S$_{12}$ | $^{2}$〉 D$_{2}$ ∝ { D 2 , D$_{2}$ → 0 1 − ( γ/δ ) 2 ( D − 1 2 − 1 ) 2 , D$_{2}$ 1 , (20) is interesting to note that FIG. 3: (Color online) (a) Black [red (gray)] curves: Probabil- ity distribution ρ (ln T ) for the periodic [non-periodic] PBRM model at criticality for several values of b < 1 ( b = 0 . 01, 0.02, 0.04, 0.1, 0.2, and 0.4 from left to right) in the case M = 1. (b) Black [red (gray)] symbols: 〈 ln T 〉 as a function of b for the periodic [non-periodic] PBRM model. The blue (gray) dashed line is the best fit of the data to the logarith- mic function A + ln b 2 with A ≈ − 1 . 44. (c) ρ (ln T ) for b < 1 scaled to T typ = exp 〈 ln T 〉 ∼ b $^{2}$. The blue (gray) dashed line proportional to ( T/T typ ) 1 / 2 is plotted to guide the eye. -25 -20 -15 -10 -5 0 ln T 10 -4 10 -3 10 -2 10 -1 (ln T) -15 -10 -5 0 5 ln (T/T$_{typ}$) 10 -4 10 -3 10 -2 10 -1 (ln T) 0.01 0.1 1 10 b -10 -5 0 <ln T> ~ ln b 2 (a) (b) (c) ~ (T/T$_{typ}$) 1/2 4