Localization properties in Lieb lattices and their extensions

TL;DR

This study investigates localization in generalized Lieb lattices across dimensions, revealing distinct scaling behaviors at flat and dispersive bands, and shows that disorder influences flat band degeneracy and localization properties.

Contribution

It provides a comprehensive analysis of localization and scaling in generalized Lieb lattices, highlighting differences between flat and dispersive bands and the impact of disorder.

Findings

Flat band states are extended at very low disorder in 3D Lieb lattices.

Critical disorder W_c decreases with increasing n in 3D Lieb lattices.

Phase diagrams are similar for periodic and hard boundary conditions.

Abstract

We study the localization properties of generalized, two- and three-dimensional Lieb lattices, $\mathcal{L}_2(n)$ and $\mathcal{L}_3(n)$, $n= 1, 2, 3$ and $4$, at energies corresponding to flat and dispersive bands using the transfer matrix method (TMM) and finite size scaling (FSS). We find that the scaling properties of the flat bands are different from scaling in dispersive bands for all $\mathcal{L}_d(n)$. For the $d=3$ dimensional case, states are extended for disorders $W$ down to $W=0.01 t$ at the flat bands, indicating that the disorder can lift the degeneracy of the flat bands quickly. The phase diagram with periodic boundary condition for $\mathcal{L}_3(1)$ looks similar to the one for hard boundaries. We present the critical disorder $W_c$ at energy $E=0$ and find a decreasing $W_c$ for increasing $n$ for $\mathcal{L}_3(n)$, up to $n=3$. Last, we show a table of FSS parameters including so-called irrelevant variables; but the results indicate that the accuracy is too low to determine these reliably. \end{abstract}