$Z_3$ Scalar Singlet Dark Matter
  We consider the minimal scalar singlet dark matter stabilised by a $Z_3$
symmetry. Due to the cubic term in the scalar potential, semi-annihilations,
besides annihilations, contribute to the dark matter relic density. Unlike in
the $Z_2$ case, the dark matter spin independent direct detection cross section
is no more linked to the annihilation cross section. We study the extrema of
the potential and show that a too large cubic term would break the $Z_3$
symmetry spontaneously, implying a lower bound on the direct detection cross
section, and allowing the whole parameter space to be tested by XENON1T. In a
small region of the parameter space the model can avoid the instability of the
standard model vacuum up to the unification scale. If the semi-annihilations
are large, however, new physics will be needed at TeV scale because the model
becomes non-perturbative. The singlet dark matter mass cannot be lower than
53.8 GeV due to the constraint from Higgs boson decay into dark matter.
