We incorporate spin-orbit coupling (SOC) into effective Kugel-Khomskii models for the $n=1$ and $n=2$ members of the Ruddlesden-Popper series ${\mathrm{Sr}}_{n+1}{\mathrm{Cr}}_n{\mathrm{O}}_{3n+1}$. These models contain interacting spins 1 and pseudospins 1/2 at each site describing spin and orbital degrees of freedom, respectively. We solve the models at zero temperature using pseudospin bond operators and spin waves. We find that for realistic parameters, SOC dominates the physics of the compound ${\mathrm{Sr}}_2{\mathrm{CrO}}_4$ with almost decoupled single ${\mathrm{CrO}}_2$ planes. The spin ordering is antiferromagnetic, with nearest-neighbor Cr spins aligned antiparallel. The corresponding orbital configuration is $d_{xy\uparrow }^1(d_{xz\uparrow }^1-i{d}_{yz\uparrow }^1)$ or $d_{xy\downarrow }^1(d_{xz\downarrow }^1+i{d}_{yz\downarrow }^1)$ depending on the spin of the site. In contrast, for the bilayer compound ${\mathrm{Sr}}_3{\mathrm{Cr}}_2{\mathrm{O}}_7$ we find that the effect of the SOC is weak and the system prefers to form pseudospin singlets in the $z$ direction perpendicular to the planes. The spin order is antiferromagnetic within each plane and ferromagnetic between planes, in agreement with previous studies.

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• Electronic structure
• Spin-orbit coupling
• Approximation methods for many-body systems
• Holstein-Primakoff method
• Spin wave theory