Anisotropic exchange and spin-wave damping in pure and electron-doped Sr$_2$IrO$_4$

The collective magnetic excitations in the spin-orbit Mott insulator (Sr$_{1-x}$La$_x$)$_2$IrO$_4$ ($x=0,\,0.01,\,0.04,\, 0.1$) were investigated by means of resonant inelastic x-ray scattering. We report significant magnon energy gaps at both the crystallographic and antiferromagnetic zone centers at all doping levels, along with a remarkably pronounced momentum-dependent lifetime broadening. The spin-wave gap is accounted for by a significant anisotropy in the interactions between $J_\text{eff}=1/2$ isospins, thus marking the departure of Sr$_2$IrO$_4$ from the essentially isotropic Heisenberg model appropriate for the superconducting cuprates.