Distribution of the local density of states as a criterion for Anderson localization: Numerically exact results for various lattices in two and three dimensions

TL;DR

This paper demonstrates that finite-size scaling of the local density of states distribution effectively identifies Anderson localization across various lattice types and dimensions, including graphene, using numerically exact data.

Contribution

It introduces refined numerical methods for analyzing the LDOS distribution and confirms its effectiveness as a criterion for Anderson localization across different lattice structures.

Findings

LDOS distribution follows a log-normal form over ten orders of magnitude

System-size dependence of LDOS distribution indicates Anderson localization

Results agree with non-linear sigma-model symmetry relations

Abstract

Numerical approaches to Anderson localization face the problem of having to treat large localization lengths while being restricted to finite system sizes. We show that by finite-size scaling of the probability distribution of the local density of states (LDOS) this long-standing problem can be overcome. To this end we reexamine the approach, propose numerical refinements, and apply it to study the dependence of the distribution of the LDOS on the dimensionality and coordination number of the lattice. Particular attention is given to the graphene lattice. We show that the system-size dependence of the LDOS distribution is indeed an unambiguous sign of Anderson localization, irrespective of the dimension and lattice structure. The numerically exact LDOS data obtained by us agree with a log-normal distribution over up to ten orders of magnitude and thereby fulfill a nontrivial symmetry relation previously derived for the non-linear $\sigma$-model.