content_image/438607.jpg content_image/438608.jpg content_image/438612.jpg
<img src='content_image/438607.jpg'>

taking as a reference the results shown in the previous section for the PBRM model. 

We attach the 2 M leads to 2 M sites at one of the edges of the cubic lattice described by the 3D Anderson model. In this way we make a line contact as we did in the case of the PBRM model. So, we can compare the scattering properties of both models. We use Gaussian distributed on-site potentials and system sizes from L = 6 to 10 (we have verified that our results do not change by increasing L further) with 10 6 to 10 4 disorder realizations. 

We start our analysis by looking at the average scat- tering matrix elements for M = 1. For the 3D Ander- son model at criticality we found 〈 | S$_{11}$ | $^{2}$〉 ≈ 0 . 926 and 〈 | S$_{12}$ | $^{2}$〉 ≈ 0 . 074, see Table II, which are close to those of the non-periodic PBRM model with b = 0 . 24. 

In Fig. 10 we show conductance probability distribu- tions ρ (ln T ) for the 3D Anderson model at criticality in the cases M = 1 and M = 2. We found 〈 ln T 〉 ≈ − 3 . 47 and − 3 . 06 for M = 1 and M = 2, respectively. See Ta- ble III. These values of 〈 ln T 〉 for the 3D Anderson model are close to those of the PBRM model with b = 0 . 36 and b = 0 . 22, respectively. Notice also that, as well as for the PBRM model, for the 3D Anderson model ρ (ln T ) in the region of T ≪ T typ is proportional to ( T/T typ ) 1 / 2 and ( T/T typ ) 2 for M = 1 and M = 2, respectively (dashed lines in Fig. 10). 

Additionally, in Fig. 11 we plot w ( T ) for the 3D Ander- son model in the cases M = 1 and M = 2. In the same figure we have also plotted (dashed lines) the conduc- tance distributions from the PBRM model with b = 0 . 2. We can see that the conductance distributions of both models are similar at this bandwidth value. 

We also compute 〈 T 〉 , var( T ), and P for the 3D Ander- son model and plot them in Figs. 7(a), 8(a), and 9(a), re- spectively (red curves labeled as 3DAM). We observe that for the 3D Anderson model 〈 T 〉 , var( T ), and P behave as the corresponding quantities for the PBRM model with b close to 0.2. 

We want to stress that our results for the 3D Ander- son model at criticality do not seem to depend on the on-site potential distribution. Here we used Gaussian 

<img src='content_image/438608.jpg'>

<img src='content_image/438612.jpg'>

distributed potentials. However, we observe practically the same results by the use of box distributed on-site potentials (not shown here). 

We study the scattering and transport properties of the Power-law Banded Random Matrix (PBRM) model and of the three-dimensional (3D) Anderson model, both at criticality. 

We observed a smooth crossover from localized- to delocalized-like (ballistic-like) behavior in the scattering properties of the PBRM model by varying b from small ( b ≪ 1) to large ( b > 1) values. For this crossover region we proposed heuristic analytical expressions for 〈| S$_{aa}$ ′ | $^{2}$〉 b and 〈| S$_{aa}$ ′ | $^{2}$〉 $_{D$_{2}$}$. For small b , we have shown that ρ (ln T ) is scale invariant with the typical value of T , T typ , as scaling factor. We realized that RMT results, expected in the limit b → ∞ , are already recovered for relatively small values of the bandwidth: b ≥ 10. However, this fact is closely related to the small number of attached leads we used in this work: the larger the number of leads the larger the value of b needed to approach RMT behavior. 

Our conclusions are valid for leads attached to the bulk of the system as well as for leads attached at the bound- ary. In this work we assumed that time-reversal symme- try ( β = 1) is present in our disordered systems. More- over, the case of broken time-reversal symmetry ( β = 2) has been preliminarily studied in Ref. [45] where similar conclusions to this work were made. 

We have also shown that the scattering properties of the 3D Anderson model are similar to those for the PBRM model with b ∈ [0 . 2 , 0 . 4]. This result is in agree- ment with previous studies 32 were it was shown that sev- eral critical quantities related to the spectrum and eigen- states of the PBRM model with b = 0 . 3 are practically the same as for the 3D Anderson model. This makes the 

## IV. CONCLUSIONS 

