Cluster Toroidal Multipoles Formed by Electric-Quadrupole and Magnetic-Octupole Trimers: A Possible Scenario for Hidden Orders in Ca$_5$Ir$_3$O$_{12}$

TL;DR

This paper proposes a theoretical framework for understanding hidden multipole orderings in Ca$_5$Ir$_3$O$_{12}$, linking cluster electric and magnetic toroidal multipoles to observed phases.

Contribution

It introduces a symmetry-based analysis connecting high-rank multipole moments to hidden phases in a 5d-electron compound, revealing new microscopic origins.

Findings

Cluster electric quadrupole ordering corresponds to electric toroidal dipole with ferroaxial moment.

Low-temperature phase involves coexisting electric and magnetic toroidal multipoles.

The study explains two hidden phases in Ca$_5$Ir$_3$O$_{12}$ through multipole orderings.

Abstract

Cluster multipole orderings composed of atomic high-rank multipole moments are theoretically investigated with a 5$d$-electron compound Ca$_5$Ir$_3$O$_{12}$ in mind. Ca$_5$Ir$_3$O$_{12}$ exhibits two hidden orders: One is an intermediate-temperature phase with time-reversal symmetry and the other is a low-temperature phase without time-reversal symmetry. By performing the symmetry and augmented multipole analyses for a $d$-orbital model under the hexagonal point group $D_{\rm 3h}$, we find that the 120$^{\circ}$-type ordering of the electric quadrupole corresponds to cluster electric toroidal dipole ordering with the electric ferroaxial moment, which can become the microscopic origin of the intermediate-temperature phase in Ca$_5$Ir$_3$O$_{12}$. Furthermore, based on ${}^{193}$Ir synchrotron-radiation-based M\"{o}ssbauer spectroscopy, we propose that the low-temperature phase in Ca$_5$Ir$_3$O$_{12}$ is regarded as a coexisting state with cluster electric toroidal dipole and cluster magnetic toroidal quadrupole, the latter of which is formed by the 120$^{\circ}$-type ordering of the magnetic octupole and accompanies a small uniform magnetization as a secondary effect. Our results provide a clue to two hidden phases in Ca$_5$Ir$_3$O$_{12}$.