Anomalous minimum and scaling behavior of localization length near an isolated flat band

TL;DR

This paper demonstrates that the anomalous minimum of localization length near an isolated flat band is a universal phenomenon in disordered one-dimensional lattices, revealing unique scaling behaviors and potential multiple minima in band gaps formed by flat bands.

Contribution

It provides an analytical and numerical study showing the universality of localization length minima near flat bands in disordered systems and introduces a summation rule relating localization and density of states.

Findings

Localization length exhibits an anomalous minimum near flat bands.

The scaling of localization length with disorder strength is characterized.

Multiple minima can occur inside a band gap formed by flat bands.

Abstract

Using one-dimensional tight-binding lattices and an analytical expression based on the Green's matrix, we show that anomalous minimum of the localization length near an isolated flat band, previously found for evanescent waves in a defect-free photonic crystal waveguide, is a generic feature and exists in the Anderson regime as well, i.e., in the presence of disorder. Our finding reveals a scaling behavior of the localization length in terms of the disorder strength, as well as a summation rule of the inverse localization length in terms of the density of states in different bands. Most interesting, the latter indicates the possibility of having two localization minima inside a band gap, if this band gap is formed by two flat bands such as in a double-sided Lieb lattice.