Abstract: I shall discuss the quantum critical and incommensurate magnetism of three distinct topologically non-trivial semi-metals that we have studied using neutron and x-ray scattering techniques. In CeRu4Sn6 there is no phase transition down to T=0.2 K and the magnetic excitation spectrum is gapless and critical through much of the Brillouin zone. The observation of quantum criticality in a stoichiometric intermetallic without tuning suggests the possibility of a quantum critical phase. NdAlSi does have a magnetic phase transition to incommensurate magnetic order at T=7.2 K followed by a transition to commensurate ferrimagnetism at T=3.3 K.

I shall discuss the relation of these magnetic structures to topologically protected Weyl points in the electronic band structure of NdAlSi. Finally, I discuss the magnetic order and excitations in Mn3X. While both Mn3Sn and Mn3Ge display strong anomalous Hall and Nernst effects at room temperature, their low temperature magnetism is distinct and indicative of itinerant magnetism. By comparing these three distinct materials I aim to illustrate the rich fundamental science and the potential for applications that is to be found when combining topologically non-trivial electronic bands with magnetism. This work was supported as part of the Institute for Quantum Matter, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award No. DE-SC0019331.