Nonuniversality of Anderson Localization in Short-Range Correlated Disorder

We provide an analytic theory of Anderson localization on a lattice with a weak short-range correlated disordered potential. Contrary to the general belief, we demonstrate that already next-neighbor statistical correlations in the potential can give rise to strong anomalies in the localization length and the density of states, and to the complete violation of single-parameter scaling. Such anomalies originate in additional symmetries of the lattice model in the limit of weak disorder. The results of numerical simulations are in full agreement with our theory, with no adjustable parameters.

Figure 1

A lack of phase randomization leads to a distinctive band center anomaly in the conductance $g_n$ through a disordered wire (1) of length $n=12000$ with next-neighbor correlations (18) of the potential and $\sigma =1/150$ (RPR localization length ${\xi }_{\varphi }=2/\sigma =300$). The main panel shows the mean logarithm $c_1=-(1/2n)⟨\ln g_n⟩$ and its variance $c_2=(1/4n)\mathrm{var}\ln g_n$ from numerical simulations (data points) and the analytical expressions (22, 25a) (solid lines), scaled to ${\xi }_{\varphi }^{-1}$. The inset shows the distribution of the reflection phase $P(\alpha )$ at the band center, calculated numerically as well as analytically (19).

Figure 2

The mean logarithm of the conductance and its variance are calculated numerically for the system (1) of the large length $n$. The coefficients $c_1=-(1/2n)⟨\ln g_n⟩$ and $c_2=(1/4n)\mathrm{var}\ln g_n$ are plotted as the function of energy near the quarter band $E=1$. The disorder is generated by $U_n=2u_n-u_{n+1}-u_{n-1}$ (top panel) and $U_n=2u_{n+1}-u_n-u_{n-1}$ (bottom panel), where the random numbers $u_n=0$ unless $n$ is a multiple of 3 and $⟨u_{3m}{u}_{3m^{\prime }}⟩=\sigma {\delta }_{m{m}^{\prime }}$, $\sigma =1/150$. The insets show the distribution function of the reflection phase. The solid lines are obtained from the solution of Eq. (11) with $p=-1$, $q=3$. The RPR localization length ${\xi }_{\varphi }$ is obtained from Eq. (14b).