Multipolar effects in Eu2Ir2O7

We use the density functional theory plus the rotationally invariant Hartree-Fock mean-field method to study the magnetic properties of the pyrochlore iridate material Eu$_2$Ir$_2$O$_7$ (5d$^5$), where the crystal field splitting $\Delta$, spin-orbit coupling (SOC) $\lambda$ and Coulomb interaction $U$ of Ir atoms are all playing significant roles. We have constructed a $t_{2g}$ Wannier tight-binding Hamiltonian and calculated the $U$-$\lambda$ phase diagram, from which we find a very stable all-in/all-out antiferromagnetic ground state for moderate SOC (0.2-0.5 eV). In this magnetic state, except for the dipole moments, we also find considerable multipolar moments (octupole) and large non-linear magnetic susceptibility. With strong enough SOC, the system reduces to a $j_{\text{eff}}=\frac{1}{2}$ single band Hubbard model, and the ground state changes to another antiferromagnetic configuration without multipolar moments. Our results indicate that the coexisting multipolar order is crucial to stabilize the all-in/all-out state and contributes a lot to the non-linear magnetic susceptibility.