We study the localization properties and the Anderson transition in the three-dimensional Lieb lattice ${\mathcal{L}}_3\left(1\right)$ and its extensions ${\mathcal{L}}_3\left(n\right)$ 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 $\nu$. We find that the critical disorder $W_c$ decreases from $\sim 16.5$ for the cubic lattice, to $\sim 8.6$ for ${\mathcal{L}}_3\left(1\right), \sim 5.9$ for ${\mathcal{L}}_3\left(2\right)$, and $\sim 4.8$ for ${\mathcal{L}}_3\left(3\right)$. Nevertheless, the value of the critical exponent $\nu$ for all Lieb lattices studied here and across various disorder and energy transitions agrees within error bars with the generally accepted universal value $\nu =1.590(1.579,1.602)$.

• Received 20 April 2020
• Revised 20 September 2020
• Accepted 25 October 2020

DOI:https://doi.org/10.1103/PhysRevB.102.174207