Compass Impurity Model of Tb Substitution in Sr$_{2}$IrO$_{4}$

We show that upon Tb substitution the interaction between the magnetic moments on the impurity Tb$^{4+}$ ion and its surrounding Ir$^{4+}$ ions is described by a ``compass'' model, i.e., Ising-like interaction favoring the magnetic moments across each bond to align along the bond direction. Such interaction nucleates quenched magnetic vortices near the impurities and drives a reentrant transition out of the antiferromagnetic ordered phase at low temperatures hence quickly suppresses the N\'eel temperature consistent with the experiment [Phys. Rev. B \textbf{92}, 214411 (2015)]. As a by-product, we propose that the compass model can be realized in ordered double perovskites composed of the spin-orbital-coupled $d^{5}$ ions and the half-closed-shell $f^{7}$ ions.