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Study of Nontrivial Magnetism in 3$d$-5$d$ Transition Metal based Double Perovskites

The coexistence of the strongly correlated 3$d$ transition metal (TM) atom and the strong spin-orbit coupling (SOC) of 5$d$ TM atom is potential to host exotic physical phenomena. In the present work, we have studied the magnetism resulting from such a coexistence in La$_2$CoIrO$_6$ (LCIO), a representative of bulk 3$d$-5$d$ double perovskites. In order to gain further insight into the effect of Co-$d$ states on the Ir-$d$ states, comparisons are carried out with the isostructural La$_2$ZnIrO$_6$, where nonmagnetic Zn atom replaces the Co atom. An in-depth analysis of the magnetic states in the framework of first principles calculation, using Landau theory and magnetic multipole analysis shows that the magnetism at the two constituent TM atoms in LCIO are driven by two different magnetic order parameters, viz., the spin moment as the primary order parameter responsible for the broken time-reversal state in Co and the higher order multipole: triakontadipole for the Ir magnetic state. A tight-binding analysis with the Ir-$t_{2g}$ orbitals, further, indicate that the Ir-$d$ states are hardly affected by the Co-$d$ states, in agreement with the multipole analysis. The computed heirarchy of the relevant multipoles in the present work can be probed in neutron diffraction measurements, motivating further experiments in this direction.