  We investigate the stability of the multipolar orderings in $f$-electron
material CeCoSi based on a self-consistent mean-field calculation for the
effective localized model. This material has two ordered phases in the
temperature-pressure phase diagram: the antiferromagnetic phase and the
nonmagnetic phase, the latter of which has been suggested to be an
antiferroquadrupolar phase. Meanwhile, the origin of the antiferroquadrupolar
phase has been unclear, since a quadrupole degree of freedom is present only
between the ground-state level and highly separated excited-state level under a
tetragonal crystalline electric field (CEF), whose energy scale is much larger
than the transition temperature. To understand the sequence of the phase
transition from the paramagnetic phase, antiferroquadrupolar phase, and
antiferromagnetic phase when decreasing the temperature, we examine the
important interaction parameters in the effective localized model. We clarify
that the $3z^2-r^2$-type of the antiferroquadrupolar interactions can
renormalize the CEF level splitting, which assists a quadrupolar ordering even
in a tetragonal system without orbital degeneracy. Moreover, the stability of
the antiferroquadrupolar and antiferromagnetic states in a magnetic field and
the behavior of the magnetic/quadrupolar susceptibility are also presented for
the information to identify the unknown order parameter in the nonmagnetic
ordered phase.
