Fig. 11. Schematic diagrams of Lieb’s theorem for a 2D bipartite lattice. (a) The rhombus unit cell of graphene, shown by a dashed box, has two A and B sub-lattice points that localize with opposite spins for energy minimization. These opposite spins result in antiferromagnetic interactions with the same number of A and B sub-lattice points. Nevertheless, the bipartite characteristics collapse in the case of (b) Stone–Wales defects, (c) vacancies, adatoms and substitution defects in graphene, (d) positive Gaussian curvature due to pentagons, and (e) negative Gaussian curvature resulting from heptagons, which lead to unusual magnetic properties.