Interplay between Lattice Distortion and Spin-Orbit Coupling in Double Perovskites

We develop anisotropic pseudo-spin nearest-neighbour antiferromagnetic Heisenberg models for monoclinically distorted double perovskites. We focus on these A$_2$BB$'$O$_6$ materials that have magnetic moments on the $4d$ or $5d$ transition metal B$'$ ions, which form a face-centred cubic lattice. In these models, we consider tetragonal distortion of B$'$-O octahedra, affecting relative occupancy of $t_{2g}$ orbitals, along with geometric effects of the distortion and spin-orbit coupling. The resulting pseudo-spin-$1/2$ models are solved in the saddle-point limit of the Sp(N) generalization of the Heisenberg model. The spin $S$ in the SU(2) case generalizes as a parameter $\kappa$ controlling quantum fluctuation in the Sp(N) case. We consider two different models that may be appropriate for these systems. In particular, using Heisenberg exchange parameters for La$_2$LiMoO$_6$ from a spin-dimer calculation [T. Aharen \textit{et al.}, Phys. Rev. B {\textbf 81}, 224409 (2010)], we conclude that this $S=1/2$ system may order, but must be very close to a disordered spin liquid state.