Designing flat-band materials through compact localized state clusters

Flat-band systems, featuring dispersionless electronic bands that localize carriers and amplify interaction effects, provide an ideal platform for realizing exotic quantum states such as superconductivity and fractional quantum Hall phases. However, designing such systems remains a major challenge because of the limited dimensional scope, poor generalizability, and complex constraints of existing construction methods. Here, an efficient and general method for constructing flat-band lattices is proposed, which assembles of compact localized state (CLS) clusters under coherent cancellation regularization, free from dimensional restrictions. Various CLS clusters were collected and designed as initial building blocks with different arrangements for constructing flat-band lattices. The existence of flat-bands in the resulting lattices, including the widely studied zero-energy flat-bands and the so-called ``Yin-Yang'' flat-bands, was confirmed based on tight-binding models. Finally, a series of graphdiyne allotropes with flat-bands were designed, based on these lattice models, including a superconductor (${\mathrm{C}}_{44}$) and a ferromagnetic metal (${\mathrm{C}}_{56}$) with flat-bands that remain at the Fermi level despite the inclusion of multiple-neighbor interactions under first-principles calculations. These results demonstrate the feasibility of designing flat-band materials by starting from CLS cluster analysis and lattice model construction. Potential metal-free light-element superconductors and magnetic materials can be designed under the guidance of the CLS theory.