Probing Band-center Anomaly with the Kernel Polynomial Method

TL;DR

This paper uses the Kernel Polynomial Method to accurately compute the localization length at the band center in a 1D Anderson model, confirming the Thouless formula and revealing a cusp-like anomaly.

Contribution

It demonstrates the effectiveness of KPM in correctly capturing the localization length and band-center anomaly, resolving previous numerical discrepancies.

Findings

KPM accurately reproduces the Thouless localization length.

A cusp-like singularity is observed at the band center.

The results confirm the validity of the Thouless formula in this context.

Abstract

We investigate the anomalous behavior of localization length of a non-interacting one-dimensional Anderson model at zero temperature. We report numerical calculations of the Thouless expression of localization length, based on the Kernel polynomial method (KPM), which has an O(N ) computational complexity, where N is the system size. The KPM results show an excellent agreement with perturbative result in large system size limit, confirming the validity of Thouless formula. Thus, contrary to the previous numerical results, the KPM approximation of the Thouless expression produce the correct localization length at the band center. The Thouless expression relates localization length in terms of density of states in a one-dimensional disordered system. By calculating the KPM estimates of density of states, we find a cusp-like behavior around the band center in the perturbative regime. This cusp-like singularity can not be obtained by an approximate analytical calculations with in the second-order approximations, reflects the band-center anomaly.