Polaron-driven switching of octupolar order in doped 5d${}^2$ double perovskite

We investigate how doping-induced small polarons impact the low-temperature multipolar orders of the $5d^2$ double perovskite Ba${}_2$CaOsO${}_6$. By computing intersite exchange interactions between 5d${}^1$ localized hole polarons and 5d${}^2$ magnetic ions from first principles, we demonstrate the reversal of the dominant octupolar exchange from ferromagnetic to antiferromagnetic. Solving the corresponding effective Hamiltonian we find this reversal to account for the progressive suppression of the ferro-octupolar order and the reduction of the ordering temperature upon Na doping. These findings clarify previously ambiguous experimental observations and demonstrate that charge doping in the form of small polarons offers a viable route to tuning intersite exchange interactions in spin-orbit-entangled materials, enabling the emergence of novel quantum orders.