Magnons dispersion and anisotropies in SrCu$_2$(BO$_3$)$_2$

We study the dispersion of the lowest excited states in the 2d Shastry-Sutherland system, SrCu$_2$(BO$_3$)$_2$, including all relevant Dzyaloshinskii-Moriya interactions. We can reduce the complexity of the general Hamiltonian to a new simpler model at zero-field that is obtained by transformations of the spin operators. The resulting Hamiltonian is studied by means of exact numerical diagonalization on a 32-site cluster and the couplings are extracted. The Dzyaloshinskii-Moriya interactions affect the dispersion of the magnons (triplet states) to linear-order because they partially lift the frustration of the lattice. We argue that earlier perturbative techniques have overestimated the dispersion and missed the dominant interactions responsible for the dispersion.