Scattering and transport statistics at the metal-insulator transition: A numerical study of the power-law banded random-matrix model

We study numerically scattering and transport statistical properties of the one-dimensional Anderson model at the metal-insulator transition described by the power-law banded random matrix (PBRM) model at criticality. Within a scattering approach to electronic transport, we concentrate on the case of a small number of single-channel attached leads. We observe a smooth crossover from localized to delocalized behavior in the average-scattering matrix elements, the conductance probability distribution, the variance of the conductance, and the shot noise power by varying $b$ (the effective bandwidth of the PBRM model) from small $\left(b⪡1\right)$ to large $\left(b>1\right)$ values. We contrast our results with analytic random matrix theory predictions which are expected to be recovered in the limit $b\to \infty$. We also compare our results for the PBRM model with those for the three-dimensional (3D) Anderson model at criticality, finding that the PBRM model with $b∊\left[0.2,0.4\right]$ reproduces well the scattering and transport properties of the 3D Anderson model.

Figure 1

(Color online) Scattering setup. $2M$ leads, shown as blue (gray) lines, are attached to $2M$ sites (black dots) of a 1D sample described by the (a) nonperiodic and (b) periodic versions of the PBRM model. The case $M=2$ with $L=11$ is shown as example.

Figure 2

(Color online) Black [red (gray)] symbols: average $S$-matrix elements $⟨{\left|S_{11}\right|}^2⟩$ and $⟨{\left|S_{12}\right|}^2⟩$ for the periodic [nonperiodic] PBRM model at criticality as a function of (a) $b$ and (b) $D_2$ for $M=1$. The blue (gray) dotted-dashed lines correspond to 2/3 and 1/3; the RMT prediction for $⟨{\left|S_{11}\right|}^2⟩$ and $⟨{\left|S_{12}\right|}^2⟩$, respectively, given by Eq. (7). The black [red (gray)] dashed lines in (a) are Eqs. (13, 14) with $\delta \approx 2.5$ $\left[\delta \approx 2.2\right]$. The black [red (gray)] dashed lines in (b) are Eqs. (18, 19) with $\delta \approx 2.5$ $\left[\delta \approx 2.2\right]$. Error bars are not shown since they are much smaller than symbol size.

Figure 3

(Color online) (a) Black [red (gray)] curves: probability distribution $\rho \left(\text{ln}T\right)$ for the periodic [nonperiodic] 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: $⟨\text{ln}T⟩$ as a function of $b$ for the periodic [nonperiodic] PBRM model. The blue (gray) dashed line is the best fit of the data to the logarithmic function $A+\text{ln}{b}^2$ with $A\approx -1.44$. (c) $\rho \left(\text{ln}T\right)$ for $b<1$ scaled to $T_{\text{typ}}=\text{exp}⟨\text{ln}T⟩\sim b^2$. The blue (gray) dashed line proportional to ${\left(T/T_{\text{typ}}\right)}^{1/2}$ is plotted to guide the eyes.

Figure 4

(Color online) Black [red (gray)] curves: conductance probability distribution $w\left(T\right)$ for the periodic [nonperiodic] PBRM model at criticality for some large values of $b$ in the case $M=1$. Blue (gray) dashed lines are $w{\left(T\right)}_{\text{COE}}$; the RMT prediction for $w\left(T\right)$ given by Eq. (8).

Figure 5

(Color online) (a) Black [red (gray)] curves: probability distribution $\rho \left(\text{ln}T\right)$ for the periodic [nonperiodic] PBRM model at criticality for several values of $b<1$ ($b=0.01$, 0.02, 0.04, 0.1, and 0.2 from left to right) in the case $M=2$. (b) Black [red (gray)] symbols: $⟨\text{ln}T⟩$ as a function of $b$ for the periodic [nonperiodic] PBRM model. The blue (gray) dashed line is the best fit of the data to the logarithmic function $A+\text{ln}{b}^2$ with $A\approx -0.057$. (c) $\rho \left(\text{ln}T\right)$ for $b<1$ scaled to $T_{\text{typ}}=\text{exp}⟨\text{ln}T⟩\sim b^2$. The blue (gray) dashed line proportional to ${\left(T/T_{\text{typ}}\right)}^2$ is plotted to guide the eyes.

Figure 6

(Color online) Black [red (gray)] curves: conductance probability distribution $w\left(T\right)$ for the periodic [nonperiodic] PBRM model at criticality for some large values of $b$ in the case $M=2$. Blue (gray) dashed lines are $w{\left(T\right)}_{\text{COE}}$; the RMT prediction for $w\left(T\right)$ given by Eq. (9).

Figure 7

(Color online) (a) Average conductance $⟨T⟩$ as a function of $M$ for the nonperiodic PBRM model at criticality for several values of $b$. $⟨T⟩$ for the 3D Anderson model at criticality (3DAM) is also shown. (b) $⟨T⟩/{⟨T⟩}_{\text{COE}}$ as a function of $b\delta$ for the periodic PBRM model at criticality for $M∊\left[1,5\right]$. Insert: $\delta$ versus $M$. $\delta$ is obtained from the fitting of Eq. (24) to the $⟨T⟩$ vs $b$ data. Thick full lines correspond to $⟨T⟩=0$. Blue (gray) dashed lines are (a) the RMT prediction for $⟨T⟩$ given by Eq. (10); and (b) one.

Figure 8

(Color online) (a) Variance of $T$ as a function of $M$ for the nonperiodic PBRM model at criticality for several values of $b$. $\text{var}\left(T\right)$ for the 3DAM at criticality is also shown. (b) $\text{var}\left(T\right)/\text{var}{\left(T\right)}_{\text{COE}}$ as a function of $b\delta$ for the periodic PBRM model at criticality for $M∊\left[1,5\right]$. Insert: $\delta$ versus $M$. $\delta$ is obtained from the fitting of Eq. (24) to the $\text{var}\left(T\right)$ vs $b$ data. Thick full lines correspond to $\text{var}\left(T\right)=0$. Blue (gray) dashed lines are (a) the RMT prediction for $\text{var}\left(T\right)$ given by Eq. (11); and (b) one.

Figure 9

(Color online) (a) Shot noise power $P$ as a function of $M$ for the nonperiodic PBRM model at criticality for several values of $b$. $P$ for the 3DAM at criticality is also shown. (b) $P/P_{\text{COE}}$ as a function of $b\delta$ for the periodic PBRM model at criticality for $M∊\left[1,5\right]$. Insert: $\delta$ versus $M$. $\delta$ is obtained from the fitting of Eq. (24) to the $P$ vs $b$ data. Thick full lines correspond to $P=0$. Blue (gray) dashed lines are (a) the RMT prediction for $P$ given by Eq. (12); and (b) one.

Figure 10

Probability distribution $\rho \left(\text{ln}T\right)$ for the 3D Anderson model at criticality in the cases $M=1$ and $M=2$. Dashed lines with slopes $T^{1/2}$ and $T^2$ are plotted to guide the eyes.

Figure 11

(Color online) Conductance probability distribution $w\left(T\right)$ for the 3D Anderson model at criticality in the cases (a) $M=1$ and (b) $M=2$. The red (gray) dashed line corresponds to $w\left(T\right)$ for the nonperiodic PBRM model with $b=0.2$.