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

## TL;DR

This study uses first-principles calculations to explore the magnetic phases of Eu$_2$Ir$_2$O$_7$, revealing stable all-in/all-out antiferromagnetic order, the importance of lattice distortion, and the presence of multipolar moments.

## Contribution

It provides a detailed phase diagram and highlights the role of lattice distortion and multipolar moments in the magnetic properties of Eu$_2$Ir$_2$O$_7$, which is a novel insight.

## Key findings

- Stable all-in/all-out antiferromagnetic ground state identified
- Trigonal lattice distortion stabilizes the AIAO state
- Presence of significant octupolar moments in magnetic states

## Abstract

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.