Localization, phases, and transitions in three-dimensional extended Lieb lattices

Abstract

We study the localization properties and the Anderson transition in the three-dimensional Lieb lattice ${\mathcal{L}}_{3}(1)$ and its extensions ${\mathcal{L}}_{3}(n)$ in the presence of disorder. We compute the positions of the flatbands, the disorder-broadened density of states, and the energy-disorder phase diagrams for up to $n=4$. Via finite-size scaling, we obtain the critical properties such as critical disorders and energies as well as the universal localization lengths exponent $\ensuremath{\nu}$. We find that the critical disorder ${W}_{c}$ decreases from $\ensuremath{\sim}16.5$ for the cubic lattice, to $\ensuremath{\sim}8.6$ for ${\mathcal{L}}_{3}(1), \ensuremath{\sim}5.9$ for ${\mathcal{L}}_{3}(2)$, and $\ensuremath{\sim}4.8$ for ${\mathcal{L}}_{3}(3)$. Nevertheless, the value of the critical exponent $\ensuremath{\nu}$ for all Lieb lattices studied here and across various disorder and energy transitions agrees within error bars with the generally accepted universal value $\ensuremath{\nu}=1.590\phantom{\rule{4pt}{0ex}}(1.579,1.602)$.