Flatband generator in two dimensions

Abstract

Dispersionless bands -flatbands -provide an excellent testbed for novel physical phases due to the fine-tuned character of flatband tight-binding Hamiltonians. The accompanying macroscopic degeneracy makes any perturbation relevant, no matter how small. For short-range hoppings flatbands support compact localized states, which allowed to develop systematic flatband generators in d = 1 dimension in Phys. Rev. B 95 115135 (2017) and Phys. Rev. B 99 125129 (2019). Here we extend this generator approach to d = 2 dimensions. The shape of a compact localized state turns into an important additional flatband classifier. This allows us to obtain analytical solutions for classes of d = 2 flatband networks and to re-classify and re-obtain known ones, such as the checkerboard, kagome, Lieb and Tasaki lattices. Our generator can be straightforwardly generalized to three lattice dimensions as well.

Flat bands by latent symmetry

“…III D) as the unit cell of the generated lattice H . The induced CLSs then occupy U = 1 unit cell each, using the number of occupied unit cells U as a flat band classifier [27] (recently generalized accordingly for lattice dimensions d > 1 [61]). One could in principle, however, start with a supercell H of a target lattice H , consisting of U > 1 interconnected copies of H, and look for new cospectral pairs {u , v } which are not cospectral in H. Then, CLSs induced by {u , v }-odd eigenstates of H will generally occupy U > 1 primitive unit cells within the supercell.…”

Designing flat-band tight-binding models with tunable multifold band touching points

“…Such schemes make use of graph theory [6,7,[21][22][23], Origami rules [24], repeated miniarrays [25], a bipartite lattice structure [26,27], generic existence conditions [28,29], or the extension of known flat-band lattices [29][30][31]. Also, over the last years, the close relation between flat bands and compact localized states (CLSs) [32] was increasingly exploited [33][34][35][36][37][38][39]. A CLS is a wave function strictly localized to some finite region of the lattice, with zero probability amplitude outside this region.…”

Flat-band solutions in $D$-dimensional decorated diamond and pyrochlore lattices: Reduction to molecular problem

“…So far, various tight-binding models with flat bands have been explored [14,[49][50][51][52][53][54][55][56][57][58], and many insights on the model construction have been accumulated. It was also found that some flat-band models have large sublattice degrees of freedom, resulting in multiple flat bands with different energies [59][60][61][62].…”