Emergence of flat bands in mean field limit of Tight Binding Models

We investigate the emergence of flat bands in simple Tight Binding Models (TBM) with variable connectivity between the sites and periodic boundary conditions. As a first step, we consider a one-dimensional TBM with sites and nearest neighbor (nn) interaction , to which we then add random bonds with the same strength linking different pairs of sites of the chain. When the number of additional bonds is (that is, for large ), every site is pairwise linked with every other site. The TBM model has two limits which yield an analytical solution, namely, the simple one-band solution in the limit , and the mean field (MF) limit for . In the former case, the density of states (DOS) takes the familiar square root singularity form of the one dimensional TMB, while in the MF limit it has a singular state, in addition to a highly degenerate flat band which contains the remaining eigenstates. For close to the MF limit, the flat band obtains a Wigner-like distribution. The transition from the one-band regime to the MF regime occurs at only a few percent of additional bonds , compared to the single-band limit. This demonstrates that only a small degree of long range interactions can in fact induce a flat band-like behavior in this simple TBM. In addition to the DOS, we study how the eigen-functions as well as the inverse participation ratio changes as a function of . Finally, we comment of the extension of the model to the two-dimensional case, which may share some features with more realistic systems where flat bands emerge, such as twisted bilayers of 2D materials like graphene.