Flat-band engineering in tight-binding models: Beyond the nearest-neighbor hopping

In typical flat-band models, defined as nearest-neighbor tight-binding models, flat bands are usually pinned to the special energies, such as top or bottom of dispersive bands, or band-crossing points. In this paper, we propose a simple method to tune the energy of flat bands without losing the exact flatness of the bands. The main idea is to add farther-neighbor hoppings to the original nearest-neighbor models, in such a way that the transfer integral depends only on the Manhattan distance. We apply this method to several lattice models including the two-dimensional kagome lattice and the three-dimensional pyrochlore lattice, as well as their breathing lattices and non-line graphs. The proposed method will be useful for engineering flat bands to generate desirable properties, such as enhancement of $T_c$ of superconductors and nontrivial topological orders.