  We investigate antiferro quadrupole orders in a diamond lattice under
magnetic fields by Monte Carlo simulations for two types of classical effective
models. One is an XY model with Z_3 anisotropy, and the other is a
two-component phi^4 model with a third-order anisotropy. We confirm that the
universality class of the zero-field transition is that for the
three-dimensional XY model. Magnetic field corresponds to a Z_3 field in the
effective model, and under this field, we find that collinear and canted
antiferro-quadrupole orders compete. Each phase is characterized by symmetry
breaking in the sector of (sublattice Z_2)x(reflection Z_2 for the order
parameter). When Z_3 anisotropy and magnetic field vary, it turns out that this
system is a good playground for various multicritical points; bicritical and
tetracritical points emerge in a finite field. Another important finding is
about the scaling of parasitic ferro quadrupole order at the zero-field
critical point. This is the secondary order parameter induced by the primary
antiferro order, and its critical exponent beta'=0.815 clearly differs from the
expected value that is twice the value for the primary order parameter. The
corresponding correlation length exponent is also different, nu'=0.597(12). We
also discuss relation of the present effective quadrupole models with the
3-state Potts model as well as implication to undertanding of orbital orders in
Pr-based 1-2-20 compounds.
