  Anisotropic antiferromagnets in an external magnetic field show a rich
variety of different ground states meeting in transition lines and
multicritical points. We study the dependence of the ground states of these
systems in the three dimensional space on physical parameters as exchange,
single ion and cubic anisotropy.
  One identifies four different ground states: the paramagnetic (PM), the
antiferromagnetic (AF), the spin flop (SF) and the biconical (BC) ground state.
In the case of absence of a cubic anisotropy the transition lines separating
the different ground states can be calculated analytically, otherwise they have
to be calculated numerically. We also considered the behavior of the staggered
magnetization which characterizes the different ground states. From its
behavior the order of the transition from one state to the other is determined.
But also the order of the transition changes along the transition lines when
including the cubic anisotropy, especially at the reeentrant region where a
transition from SF to BC and back to SF by increasing the external field $H$
occurs. Multicritical points are founded which are assumed to be tricritical or
critical endpoints. The results obtained may be relevant for other systems
since the antiferromagnetic model can be mapped to a lattice gas model where
the biconical ground state is interpreted as supersolid phase. Recent
renormalization group calculations show that such a phase would indicate the
existence of a tetracritical point.
