Anisotropic spin fluctuations in Sr$_2$RuO$_4$: role of spin-orbit coupling and induced strain

We analyze the spin anisotropy of the magnetic susceptibility of Sr$_2$RuO$4$ in presence of spin-orbit coupling and anisotropic strain using quasi-two-dimensional tight-binding parametrization fitted to the ARPES results. Similar to the previous observations we find the in-plane polarization of the low ${\bf q}$ magnetic fluctuations and the out-of-plane polarization of the incommensurate magnetic fluctuation at the nesting wave vector ${\bf Q}_1 = (2/3 \pi ,2/3 \pi)$ but also nearly isotropic fluctuations near ${\bf Q}_2=(\pi/6,\pi/6)$. Furthermore, one finds that apart from the high-symmetry direction of the tetragonal Brillouin zone the magnetic anisotropy is maximal, i.e. $\chi^{xx} \neq \chi^{yy} \neq \chi^{zz}$. This is the consequence of the orbital anisotropy of the $xz$ and $yz$ orbitals in the momentum space. We also study how the magnetic anisotropy evolves in the presence of the strain and find strong Ising-like ferromagnetic fluctuations near the Lifshitz transition for the $xy$-band.