Exact Demonstration of Magnetization Pleateaus and First Order Dimer-N\'eel Phase Transitions in a Modified Shastry-Sutherland Model for SrCu${}_2$(BO${}_3$)${}_2$

We study a generalized Shastry-Sutherland model for the material SrCu${}_2$(BO${}_3$)${}_2$. Along a line in the parameter space, we can show rigorously that the model has a first order phase transition between Dimerized and N'eel-ordered ground states. Furthermore, when a magnetic field is applied in the Dimerized phase, magnetization plateaus develop at commensurate values of the magnetization. We also discuss various aspects of the phase diagram and properties of this model away from this exactly soluble line, which include gap-closing continuous transitions between Dimerized and magnetically ordered phases.