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

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 $D_{3\mathrm{h}}$, 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 Mössbauer 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${}_{12}$.