L-point quadrupole order under magnetic field in cubic PrIr$_2$Zn$_{20}$

We study quadrupole orders in heavy-fermion compound PrIr$_2$Zn$_{20}$ under magnetic fields on the basis of the Landau theory. Assuming $E_g$ electric quadrupolar orders in the cubic symmetry with the ordering wavenumber at the L points in the face-centered cubic lattice Brillouin zone as observed experimentally, we construct a Landau free energy and analyze the resulting free energy. We find that the unidentified high-temperature ordered phase under the magnetic field ${\bf H} \parallel [001]$ reported earlier arises from the rotation of the quadrupole moments of f electrons on the Pr site. We also discuss the phase diagram for other magnetic-field directions and possible double-${\bf q}$ quadrupolar orders in this system.