  The effects of doping on the electronic evolution of the Mott insulating
state have been extensively studied in efforts to understand mechanisms of
emergent quantum phases of materials. The study of these effects becomes ever
more intriguing in the presence of entanglement between spin and orbital
degrees of freedom. Here, we present a comprehensive investigation of charge
doping in the double perovskite Ba$_2$NaOsO$_6$, a a complex Mott insulator
where such entanglement plays an important role. We establish that the
insulating magnetic ground state evolves from canted antiferromagnet (cAF)to
N\'eel order for dopant levels exceeding ~ 10 %. Furthermore, we determine that
a broken local point symmetry (BLPS) phase, precursor to the magnetically
ordered state, occupies an extended portion of the (H-T) phase diagram with
increased doping. This finding reveals that the breaking of the local cubic
symmetry is driven by a multipolar order, most-likely of the
antiferro-quadrupolar type.
