Conductance fluctuations in the localized regime: Numerical study in disordered noninteracting systems

We have studied numerically the fluctuations of the conductance $g$ in two-dimensional, three-dimensional, and four-dimensional disordered noninteracting systems. We have checked that the variance of $\ln g$ varies with the lateral sample size as $L^{2∕5}$ in three-dimensional systems, and as a logarithm in four-dimensional systems. The precise knowledge of the dependence of this variance with system size allows us to test the single-parameter scaling hypothesis in three-dimensional systems. We have also calculated the third cumulant of the distribution of $\ln g$ in two- and three-dimensional systems, and have found that in both cases it diverges with the exponent of the variance times $3∕2$, remaining relevant in the large size limit.

Figure 1

${\sigma }^2$ as a function of $L^{2∕5}$ for NSS model in 3D systems.

Figure 2

(Color online) ${\sigma }^2$ versus ${⟨-\ln g⟩}^{2∕5}$ for the Anderson model in 3D systems. Each symbol corresponds to a different disorder: 20 (squares), 25 (solid dots), 27 (up triangles), and 30 (down triangles).

Figure 3

${\sigma }^2$ as a function of $L$ in a logarithmic scale for NSS model in 4D systems. Inset: the same data as in the main part of the figure as a function of $L^{2∕9}$.

Figure 4

Third cumulant of $\ln g$ versus $L^{3∕5}$ for the model of NSS in 3D systems. The corresponding skewness is represented in the inset.

Figure 5

Fourth cumulant of the distribution of $⟨-\ln g⟩$ versus $L^{4∕3}$ for the model of NSS in 2D systems.