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One of the signatures of the face-centered cubic (fcc) antiferromagnet as atypical example of a geometrically frustrated system is the large ground statedegeneracy of the classical nearest neighbor and next-nearest neighborHeisenberg (isotropic) model on this lattice. In particular, collinear statesare degenerate with non-collinear and non-coplanar ones: this degeneracy isaccidental and is expected to be lifted by anisotropic exchange interactions.In this work, we derive the most general nearest and next-nearest neighborexchange model allowed by the space-group symmetry of the noncentrosymmetrichalf-Heusler compounds, which includes three anisotropic terms: the so-calledKitaev, Gamma and Dzyaloshinskii-Moriya interactions -- most notably, thelatter is allowed by the breaking of inversion symmetry in these materials andhas not been previously been studied in the context of the fcc lattice. Wecompute the resulting phase diagram and show how the different terms lift theground state degeneracy of the isotropic model, and lay emphasis on findingregimes where multi-q (non-collinear/non-coplanar) states are selected byanisotropy. We then discuss the role of quantum fluctuations and the couplingto a magnetic field in the ground state selection, and show that these effectscan stabilize non-coplanar (triple-q) states. These results suggest that somehalf-Heusler antiferromagnets might host rare non-collinear/non-coplanarorders, which may in turn explain the unusual transport properties detected inthese semimetals.

Anisotropic exchange and non-collinear antiferromagnets on a noncentrosymmetric fcc structure as in the half-Heuslers