Recently the quantum spin Hall effect was theoretically predicted and experimentally realized in quantum wells based on the binary semiconductor HgTe (refs 1-3). The quantum spin Hall state and topological insulators are new states of quantum matter interesting for both fundamental condensed-matter physics and material science. Many Heusler compounds with C1(b) structure are ternary semiconductors that are structurally and electronically related to the binary semiconductors. The diversity of Heusler materials opens wide possibilities for tuning the bandgap and setting the desired band inversion by choosing compounds with appropriate hybridization strength (by the lattice parameter) and magnitude of spin-orbit coupling (by the atomic charge). Based on first-principle calculations we demonstrate that around 50 Heusler compounds show band inversion similar to that of HgTe. The topological state in these zero-gap semiconductors can be created by applying strain or by designing an appropriate quantum-well structure, similar to the case of HgTe. Many of these ternary zero-gap semiconductors (LnAuPb, LnPdBi, LnPtSb and LnPtBi) contain the rare-earth element Ln, which can realize additional properties ranging from superconductivity (for example LaPtBi; ref. 12) to magnetism (for example GdPtBi; ref. 13) and heavy fermion behaviour (for example YbPtBi; ref. 14). These properties can open new research directions in realizing the quantized anomalous Hall effect and topological superconductors.

A large anomalous Hall effect is observed in the triangular antiferromagnet Mn3Ge arising from an intrinsic electronic Berry phase.

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The anomalous Hall effect is investigated theoretically for the non-collinear antiferromagnetic order of the hexagonal compounds Mn3Ge and Mn3Sn using various planar triangular magnetic configurations as well as unexpected non-planar configurations. The former give rise to anomalous Hall conductivities (AHC) that are found to be extremely anisotropic. For the planar cases the AHC is connected with Weyl-points in the energy-band structure, which are described in detail. If this case were observable in Mn3Ge, a large AHC of about σzx ≈ 900 (Ωcm) −1 should be expected. However, in Mn3Ge it is the non-planar configuration that is energetically favored, in which case it gives rise to an AHC of σxy ≈ 100 (Ωcm) −1 . The non-planar configuration allows a quantitative evaluation of the topological Hall effect (THE) that is seen to determine this value of σxy to a large extent. For Mn3Sn it is the planar configurations that are predicted to be observable. In this case the AHC can be as large as σyz ≈ 250 (Ωcm) −1 .

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