Magnetic reorientation transition in a three orbital model for $\mathrm{C}{\mathrm{a}}_2\mathrm{R}\mathrm{u}{\mathrm{O}}_4$ -- Interplay of spin-orbit coupling, tetragonal distortion, and Coulomb interactions

Including the orbital off-diagonal spin and charge condensates in the self consistent determination of magnetic order within a realistic three-orbital model for the $4d^4$ compound $\mathrm{C}{\mathrm{a}}_2\mathrm{R}\mathrm{u}{\mathrm{O}}_4$, reveals a host of novel features including strong and anisotropic spin-orbit coupling (SOC) renormalization, coupling of strong orbital magnetic moments to orbital fields, and a magnetic reorientation transition. Highlighting the rich interplay between orbital geometry and overlap, spin-orbit coupling, Coulomb interactions, tetragonal distortion, and staggered octahedral tilting and rotation, our investigation yields a planar antiferromagnetic (AFM) order for moderate tetragonal distortion, with easy $a-b$ plane and easy $b$ axis anisotropies, along with small canting of the dominantly $yz,xz$ orbital moments. With decreasing tetragonal distortion, we find a magnetic reorientation transition from the dominantly planar AFM order to a dominantly $c$ axis ferromagnetic (FM) order with significant $xy$ orbital moment.