Localization properties in Lieb lattices and their extensions

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 $W=0.01t$ 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}

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• Kaleidoscopes of Hofstadter Butterflies and Aharonov-Bohm caging from $2^n$-root topology in decorated square lattices
• Topological $n$-root Su-Schrieffer-Heeger model in a non-Hermitian photonic ring system
• Quantum percolation on Lieb Lattices
• Complete escape from localization on a hierarchical lattice: A Koch fractal with all states extended
• Quantum engineering for compactly localized states in disordered Lieb lattices
• Localization and Flat Bands in Bond-Inflated Lattices
• Exceptional horns in $n$-root graphene and Lieb photonic ring lattices