Localization properties in Lieb lattices and their extensions

We study the localization properties of generalized two- and three-dimensional Lieb lattices, L2(n) and L3(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 Ld(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 L3(1) looks similar to the one for hard boundaries (Liu et al., 2020). We present the critical disorder Wc at energy E=0 and find a decreasing Wc for increasing n for L3(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.