Superstructures at magnetization plateaus in ${\mathrm{SrCu}}_2({\mathrm{BO}}_3{)}_2$

Magnetization plateaus observed in ${\mathrm{SrCu}}_2({\mathrm{BO}}_3{)}_2$ are discussed theoretically. We map the two-dimensional Heisenberg model for ${\mathrm{SrCu}}_2({\mathrm{BO}}_3{)}_2$ to a hard-core boson model and study first the latter in the classical limit and then compare the results with fully quantum ones for finite systems. Plateaus are expected at 1/2, 1/3, 1/4, $1/8,\dots$ of the full moment. Translational symmetry of the lattice is spontaneously broken at the plateaus except for the 1/2 plateau. The 1/3 and 1/4 plateaus have magnetic superstructures of stripe orders, while the others have square unit cells.