Singular Behavior of Eigenstates in Anderson's Model of Localization

TL;DR

This paper uncovers a previously unreported singularity in the electronic properties of Anderson's Localization Model with bounded disorder, distinct from the known mobility edge, through numerical analysis in one to three dimensions.

Contribution

It reveals a new singular behavior in Anderson's model and compares different localization measures to understand its nature and evolution with disorder.

Findings

Identification of a singularity in electronic properties

Comparison of localization length measures

Rich behavior of Anderson's model beyond known transitions

Abstract

We observe a singularity in the electronic properties of the Anderson Model of Localization with bounded diagonal disorder, which is clearly distinct from the well-established mobility edge (localization-delocalization transition) that occurs in dimensions $d>2$. We present results of numerical calculations for Anderson's original (box) distribution of onsite disorder in dimensions $d$ = 1, 2 and 3. To establish this hitherto unreported behavior, and to understand its evolution with disorder, we contrast the behavior of two different measures of the localization length of the electronic wavefunctions - the averaged inverse participation ratio and the Lyapunov exponent. Our data suggest that Anderson's model exhibits richer behavior than has been established so far.