3 II. PBRM MODEL A. Average scattering matrix elements In the following sections we focus on 〈 | S$_{aa}$ ′ | $^{2}$〉 , 〈 T 〉 , var( T ), and P for the PBRM model and the 3D Anderson model, both at the MIT. In all cases we set the coupling strength ϵ such that 〈 S 〉 ≈ 0 in order to compare our results, in the proper limits, with the RMT predictions introduced above. We attach the 2 M leads to the first 2 M sites of the 1D sample described by the PBRM model. That is, in the non-periodic version of the PBRM model, Eq. (1), we attach the leads at the boundary of the system. See Fig. 1(a). While in the periodic version, Eq. (2), we attach them to the bulk . See Fig. 1(b). In the latter case, finite size effects are considerably reduced. However, the quantities we analyse below are L -independent once L is much larger than the number of attached leads for both versions of the PBRM model. We point out that our setup is significantly different from the one used in Refs. [28,29], where the conductance has also been studied using the PBRM model. There, for example, the leads in the M = 1 case are attached to sites which are separated a distance of L/ 2 and L in the periodic and non-periodic versions of the PBRM model, respectively. In such situation the scattering quantities are strongly L -dependent. First we consider the case M = 1, where the S -matrix is a 2 × 2 matrix. In Fig. 2(a) we plot the ensemble average of the elements | S$_{11}$ | 2 and | S$_{12}$ | 2 as a function of the bandwidth parameter b , 〈 | S$_{11}$ | $^{2}$〉 b and 〈 | S$_{12}$ | $^{2}$〉 $_{b}$, for the periodic (black symbols) and the non-periodic ered] for b ≥ 4. Moreover, we have found that 〈 | S$_{11}$ | $^{2}$〉 and 〈 | S$_{12}$ | $^{2}$〉 , as a function of the bandwidth b , are well described by (red symbols) PBRM model. We concentrate on these two matrix elements since the other two elements give no additional information: 〈 | S$_{22}$ | $^{2}$〉 b = 〈 | S$_{11}$ | $^{2}$〉 b and 〈 | S$_{21}$ | $^{2}$〉 b = 〈 | S$_{12}$ | $^{2}$〉 $_{b}$. Notice a strong b -dependence of the average S -matrix elements driving them from a localized-like regime [ 〈 | S$_{11}$ | $^{2}$〉 ≈ 1 and 〈 | S$_{12}$ | $^{2}$〉 ≈ 0; i.e. the average conductance is close to zero] for b ≪ 1, to a delocalized-like or ballistic-like regime [ 〈 | S$_{11}$ | $^{2}$〉 ≈ 2 / 3 and 〈 | S$_{12}$ | $^{2}$〉 ≈ 1 / 3; i.e., RMT results are already recov- ered] for b 4. where δ is a fitting parameter. Eq. (13) is a consequence of the unitarity of the scattering matrix, SS † = 1 , while the factor 1/3 in Eq. (14) comes from Eq. (7) with M = 1. In Fig. 2(a) we plot Eqs. (13) and (14) (dashed lines) and compare them with the corresponding results from 〈 | S$_{11}$ | $^{2}$〉 b = 1 − 〈 | S$_{12}$ | $^{2}$〉 b , (13) | S$_{12}$ | b = 1 3 [ 1 1 + ( δb ) − 2 ] , (14) FIG. 2: (Color online) Black [red (gray)] symbols: Average S -matrix elements 〈 | S 11 | 2 〉 and 〈 | S 12 | 2 〉 for the periodic [non- periodic] PBRM model at criticality as a function of (a) b and (b) D 2 for M = 1. The blue (gray) dot-dashed lines correspond to 2/3 and 1/3; the RMT prediction for 〈 | S 11 | 2 〉 and 〈 | S 12 | 2 〉 , respectively, given by Eq. (7). The black [red (gray)] dashed lines in (a) are Eqs. (13) and (14) with δ ≈ 2 . 5 [ δ ≈ 2 . 2]. The black [red (gray)] dashed lines in (b) are Eqs. (18) and (19) with δ ≈ 2 . 5 [ δ ≈ 2 . 2]. Error bars are not shown since they are much smaller than symbol size. FIG. 1: (Color online) Scattering setup. 2 M leads, shown as blue (gray) lines, are attached to 2 M sites (black dots) of a 1D sample described by the (a) non-periodic and (b) periodic versions of the PBRM model. The case M = 2 with L = 11 is shown as example. (a) (b) 0.01 0.1 1 10 b 0 0.5 1 <|S$_{11}$|$^{2}$>$_{b}$ <|S$_{12}$|$^{2}$>$_{b}$ 0 0.25 0.5 0.75 1 D$_{2}$ 0 0.5 1 <|S$_{11}$|$^{2}$>$_{D}$ 2 <|S$_{12}$|$^{2}$>$_{D}$ 2 (a) (b)