  We present combined spin model and first principles electronic structure
calculations to study the weak ferromagnetism in bulk Mn$_3$Z (Z=Sn, Ge, Ga)
compounds. The spin model parameters were determined from a spin-cluster
expansion technique based on the relativistic disordered local moment formalism
implemented in the screened Korringa--Kohn--Rostoker method. We describe the
magnetic ground state of the system within a three-sublattice model and
investigate the formation of the weak ferromagnetic states in terms of the
relevant model parameters. First, we give a group-theoretical argument how the
point-group symmetry of the lattice leads to the formation of weak
ferromagnetic states. Then we study the ground states of the classical spin
model and derive analytical expressions for the weak ferromagnetic distortions
by recovering the main results of the group-theoretical analysis. As a third
approach we obtain the weak ferromagnetic ground states from self-consistent
density functional calculations and compare our results with previous first
principles calculations and with available experimental data. In particular, we
demonstrate that the orbital moments follow a decomposition predicted by group
theory. For a deeper understanding of the formation of weak ferromagnetism we
selectively trace the effect of the spin-orbit coupling at the Mn and Z sites.
In addition, for the case of Mn$_3$Ga, we gain information on the role of the
induced moment of Ga from constrained local density functional calculations.

Computational Physics;Strongly Correlated Electrons