We present first principles calculations of the magnetic and orbital properties of Ba$_2$NaOsO$_6$ (BNOO), a 5$d^1$ Mott insulator with strong spin orbit coupling (SOC) in its low temperature emergent quantum phases. Our computational method takes into direct consideration recent NMR results that established that BNOO develops a local octahedral distortion preceding the formation of long range magnetic order. We found that the two-sublattice canted ferromagnetic ground state identified in Lu etal, Nature Comm. {bf 8}, 14407 (2017) is accompanied by a two-sublattice staggered orbital ordering pattern in which the $t_{2g}$ orbitals are selectively occupied as a result of strong spin orbit coupling. The staggered orbital order found here using first principles calculations asserts the previous proposal of Chen etal, Phys. Rev. B {bf 82}, 174440 (2010) and Lu etal, Nature Comm. {bf 8}, 14407 (2017), that two-sublattice magnetic structure is the very manifestation of staggered quadrupolar order. Therefore, our results affirm the essential role of multipolar spin interactions in the microscopic description of magnetism in systems with locally entangled spin and orbital degrees of freedom.
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