Flat-band generator in two dimensions

© 2021 American Physical Society.Dispersionless bands, flat bands, provide an excellent test bed for novel physical phases due to the fine-tuned character of flat band tight-binding Hamiltonians. The accompanying macroscopic degeneracy makes any perturbation relevant, no matter how small. For short-range hoppings flat-bands support compact localized states, which allowed us to develop systematic flat-band generators in d=1 dimension in Phys. Rev. B 95, 115135 (2017)2469-995010.1103/PhysRevB.95.115135 and Phys. Rev. B 99, 125129 (2019)2469-995010.1103/PhysRevB.99.125129. Here we extend this generator approach to d=2 dimensions. The shape of a compact localized state turns into an important additional flat-band classifier. This allows us to obtain analytical solutions for classes of d=2 flat-band networks and to reclassify and reobtain known ones, such as the checkerboard, kagome, Lieb, and Tasaki lattices. Our generator can be straightforwardly generalized to three lattice dimensions as well.11Nsciescopu