Impurity induced magnetic ordering in Sr$_2$RuO$_4$

Ti substituting Ru in Sr$_2$RuO$_4$ in small concentrations induces incommensurate spin density wave order with a wave vector $\boldsymbol{Q} \simeq (2 \pi /3, 2 \pi /3)$ corresponding to the nesting vector of two out of three Fermi surface sheets. We consider a microscopic model for these two bands and analyze the correlation effects leading to magnetic order through non-magnetic Ti-doping. For this purpose we use a position dependent mean field approximation for the microscopic model and a phenomenological Ginzburg-Landau approach, which both deliver consistent results and allow us to examine the inhomogeneous magnetic order. Spin-orbit coupling additionally leads to spin currents around each impurity, which in combination with the magnetic polarization produce a charge current pattern. This is also discussed within a gauge field theory in both charge and spin channel. This spin-orbit coupling effect causes an interesting modification of the magnetic structure, if currents run through the system. Our findings allow a more detailed analysis of the experimental data for Sr$_{2}$Ru$_{1-x}$Ti$_{x}$O$_{4}$. In particular, we find that the available measurements are consistent with our theoretical predictions.