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In this work we study the complex entanglement between spin interactions,electron correlation and Janh-Teller structural instabilities in the 5d$^1$$J_{eff}=\frac{3}{2}$ spin-orbit coupled double perovskite $\rm Ba_2NaOsO_6$using first principles approaches. By combining non-collinear magneticcalculations with multipolar pseudospin Hamiltonian analysis and many-bodytechniques we elucidate the origin of the observed quadrupolar cantedantifferomagnetic. We show that the non-collinear magnetic order originatesfrom Jahn-Teller distortions due to the cooperation of Heisenberg exchange,quadrupolar spin-spin terms and both dipolar and multipolarDzyaloshinskii-Moriya interactions. We find a strong competition betweenferromagnetic and antiferromagnetic canted and collinear quadrupolar magneticphases: the transition from one magnetic order to another can be controlled bythe strength of the electronic correlation ($U$) and by the degree ofJahn-Teller distortions.

Interplay between multipolar spin interactions, Jahn-Teller effect and electronic correlation in a $J_{eff}=\frac{3}{2}$ insulator