Characterizing topological electronic states in crystals is a formidable goal in condensed matter physics. Such states offer substantial breakthroughs toward a deeper understanding of matter and potential applications ranging from electronics to quantum computing. Their unique electronic features – robust boundary states and quantized bulk electromagnetic responses – derive from topological properties of the wave functions in reciprocal space as a refinement of Bloch band theory. Direct evidence of the band topology traditionally relies on the macroscopic response of the electrons to external electromagnetic fields in ultra-clean samples.

In this seminar, I will introduce an alternative approach to identify topological systems. I will present two experiments (from a theoretical physicist perspective) that image the local response of the electrons to a boundary in the local density of states (LDOS). The first experiment focuses on graphene — a 2D semimetal — imaged by scanning tunneling microscopy. The second experiment emulates a 1D insulator with dielectric resonators inside a microwave cavity. Remarkably, both systems exhibit wavefront dislocations in the LDOS near the boundary. I will show that the dislocation charge is a real-space measure of the reciprocal-space band topology.