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 $\nu =1.590(1.579,1.602)$.

中文翻译：

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三维扩展Lieb晶格中的局部化，相变和过渡

我们研究三维Lieb晶格中的定位特性和Anderson跃迁 ${\mathcal{大号}}_3（\mathrm{1个}）$ 及其扩展 ${\mathcal{大号}}_3（ñ）$在混乱的情况下。我们计算了平坦带的位置，无序扩展的状态密度以及直至$ñ=4$。通过有限尺寸缩放，我们获得了关键特性，例如关键失常和能量以及通用的局部化长度指数$\nu$。我们发现严重疾病${\mathrm{w\; ^}}_C$ 从 $〜16.5$ 对于立方晶格， $〜8.6$ 对于 ${\mathcal{大号}}_3（\mathrm{1个}）， 〜5.9$ 对于 ${\mathcal{大号}}_3（2）$和 $〜4.8$ 对于 ${\mathcal{大号}}_3（3）$。尽管如此，关键指数的价值$\nu$ 对于本文研究的所有利勃（Lieb）晶格以及跨各种无序和能量跃迁，在误差条内与公认的普适值一致 $\nu =1.590（1.579，1.602）$。