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

We study the localization properties and the Anderson transition in the 3D Lieb lattice $\mathcal{L}_3(1)$ and its extensions $\mathcal{L}_3(n)$ in the presence of disorder. We compute the positions of the flat bands, the disorder-broadened density of states and the energy-disorder phase diagrams for up to 4 different such Lieb lattices. Via finite-size scaling, we obtain the critical properties such as critical disorders and energies as well as the universal localization lengths exponent $\nu$. We find that the critical disorder $W_c$ decreases from $\sim 16.5$ for the cubic lattice, to $\sim 8.6$ for $\mathcal{L}_3(1)$, $\sim 5.9$ for $\mathcal{L}_3(2)$ and $\sim 4.8$ for $\mathcal{L}_3(3)$. Nevertheless, the value of the critical exponent $\nu$ for all Lieb lattices studied here and across disorder and energy transitions agrees within error bars with the generally accepted universal value $\nu=1.590 (1.579,1.602)$.