Probing topological quantum matter with scanning tunnelling microscopy

The search for topological phases of matter is evolving towards strongly interacting systems, including magnets and superconductors, where exotic effects emerge from the quantum-level interplay between geometry, correlation and topology. Over the past decade or so, scanning tunnelling microscopy has become a powerful tool to probe and discover emergent topological matter, because of its unprecedented spatial resolution, high-precision electronic detection and magnetic tunability. Scanning tunnelling microscopy can be used to probe various topological phenomena, as well as complement results from other techniques. We discuss some of these proof-of-principle methodologies applied to probe topology, with particular attention to studies performed under a tunable vector magnetic field, which is a relatively new direction of recent focus. We then project the future possibilities for atomic-resolution tunnelling methods in providing new insights into topological matter.