Anomalous Wave Function Statistics on a One-Dimensional Lattice with Power-Law Disorder

Within a general framework, we discuss the wave function statistics in the Lloyd model of Anderson localization on a one-dimensional lattice with a Cauchy distribution for random on-site potential. We demonstrate that already in leading order in the disorder strength, there exists a hierarchy of anomalies in the probability distributions of the wave function, the conductance, and the local density of states, for every energy which corresponds to a rational ratio of wavelength to lattice constant. Power-law rather than log-normal tails dominate the short-distance wave-function statistics.

Figure 1

The ratio $c_3/c_1$ is plotted according to the analytical result (18) for the Lloyd model at the disorder strength $\delta =0.01t$. The ratio is never small inside the band and reveals anomalies at energies $E=-2t\cos (\pi p/q)$ with $p$ and $q$ integers. The corresponding rational number $p/q$ is indicated in the figure. The size of the anomaly depends only on the value of the denominator $q$.

Figure 2

Distribution function $P_{\ln g}(\ln g)$ obtained from the Anderson model (1) with Cauchy disorder (solid circles), box disorder (open circles), and Gaussian disorder (open squares). Parameters are in the localized regime $L/\xi =6$, with localization length $\xi =200$ lattice constants, and $E=0.2t$. In this semilogarithmic plot, the log-normal distribution function for conventional weak (box or Gaussian) disorder maps to an inverted parabola, while the straight-line asymptotics found for Cauchy disorder correspond to a power-law tail in $P_g(g)$.