Non-collinear magnetic structure and multipolar order in Eu$_2$Ir$_2$O$_7$

The magnetic properties of the pyrochlore iridate material Eu$_2$Ir$_2$O$_7$ (5$d^5$) have been studied based on the first principle calculations, where the crystal field splitting $\Delta$, spin-orbit coupling (SOC) $\lambda$ and Coulomb interaction $U$ within Ir 5$d$ orbitals are all playing significant roles. The ground state phase diagram has been obtained with respect to the strength of SOC and Coulomb interaction $U$, where a stable anti-ferromagnetic ground state with all-in/all-out (AIAO) spin structure has been found. Besides, another anti-ferromagnetic states with close energy to AIAO have also been found to be stable. The calculated nonlinear magnetization of the two stable states both have the d-wave pattern but with a $\pi/4$ phase difference, which can perfectly explain the experimentally observed nonlinear magnetization pattern. Compared with the results of the non-distorted structure, it turns out that the trigonal lattice distortion is crucial for stabilizing the AIAO state in Eu$_2$Ir$_2$O$_7$. Furthermore, besides large dipolar moments, we also find considerable octupolar moments in the magnetic states.