Magnetization Plateaus of ${\mathrm{SrCu}}_2({\mathrm{BO}}_3{)}_2$ from a Chern-Simons Theory

The antiferromagnetic Heisenberg model on the frustrated Shastry-Sutherland lattice is studied by a mapping onto spinless fermions carrying one quantum of statistical flux. Using a mean-field approximation these fermions populate the bands of a generalized Hofstadter problem. Their filling leads to the magnetization curve. For ${\mathrm{SrCu}}_2({\mathrm{BO}}_3{)}_2$ we reproduce plateaus at $1/3$ and $1/4$ of the saturation moment and predict a new one at $1/2$. Gaussian fluctuations of the gauge field are shown to be massive at these plateau values.