Spontaneous symmetry breaking is one of the fundamental ideas in physics. In condensed matter, the paradigm is the Ising model of magnetism. In the Ising model, spins spontaneously orient below the critical temperature, in one of two possible time-reversed orientations, say “up” or “down”. Without special preparation, when this happens, a system will form domains or both orientations, separated by domain boundaries. These boundaries are a simple example of a topological defect. Other examples are vortices in superfluids, and dislocations and disclinations in crystals. In general, domain walls form whenever there is a discrete broken symmetry, and so are not limited to Ising models.

These defects, which live in real space, are “old” examples of topology in physics. The more modern versions are defects in *momentum* space. Notable are Weyl points, which are linear crossings of a pair of non-degenerate bands, that can be viewed as monopoles of the Berry curvature. They are related to a variety of cool phenomena, like chiral edge states and Hall effects, and “teleportation” of electrons from one Weyl point to another.

What happens when we bring the two types of topology together? The magnetic order which forms a spontaneously broken symmetry state directly influences the formation and locations of Weyl points. This is because magnetic order locally acts like a Zeeman field on electrons, and modifies their band structure. So if Weyl points are present in a magnetic system, they generally re-orient in momentum space as one crosses a magnetic domain boundary. We can expect intimate coupling of the extended electronic states and domain walls. For example, chiral edge states may live on the latter, and the walls themselves may have distinct transport properties. Conversely, when one drives a current in the system, carried by the conduction electrons, it may “push” the domain boundaries in unusual ways.

Dr. Jianpeng Liu, a postdoc here, recently studied this with me in the context of Mn3Sn, an exciting material which shows a classic indication of band topology, the anomalous Hall effect, up to room temperature. It has interesting magnetic order, whose domains and domain boundaries we described theoretically, including their coupling to Weyl fermions. This material is being studied by a number of experimental groups – we learned about it thanks to Professors Satoru Nakatsuji and Yoshi-Chika Otani at the ISSP in Japan. We expect there will be a lot more to see experimentally in this interesting material! You can read our paper in PRL.