  One of the signatures of the face-centered cubic (fcc) antiferromagnet as a
typical example of a geometrically frustrated system is the large ground state
degeneracy of the classical nearest neighbor and next-nearest neighbor
Heisenberg (isotropic) model on this lattice. In particular, collinear states
are degenerate with non-collinear and non-coplanar ones: this degeneracy is
accidental and is expected to be lifted by anisotropic exchange interactions.
In this work, we derive the most general nearest and next-nearest neighbor
exchange model allowed by the space-group symmetry of the noncentrosymmetric
half-Heusler compounds, which includes three anisotropic terms: the so-called
Kitaev, Gamma and Dzyaloshinskii-Moriya interactions -- most notably, the
latter is allowed by the breaking of inversion symmetry in these materials and
has not been previously been studied in the context of the fcc lattice. We
compute the resulting phase diagram and show how the different terms lift the
ground state degeneracy of the isotropic model, and lay emphasis on finding
regimes where multi-q (non-collinear/non-coplanar) states are selected by
anisotropy. We then discuss the role of quantum fluctuations and the coupling
to a magnetic field in the ground state selection, and show that these effects
can stabilize non-coplanar (triple-q) states. These results suggest that some
half-Heusler antiferromagnets might host rare non-collinear/non-coplanar
orders, which may in turn explain the unusual transport properties detected in
these semimetals.
