The urgent imperative to discover and understand topological quantum matter (TQM) is based both on the fundamental significance of its rich new physics and on the potential for unprecedented applications e.g. topological quantum computing. Specific topological ordered states, such as topological superconductors, ferromagnetic topological insulators, and topological Kondo insulators, are now the focus of physics research. Their microscopic quantum states have proven very difficult to address experimentally, one reason being the lack of instrumentation designed to deal with their novel measurement challenges, especially in high-resolution visualization of TQM. However, new opportunities for this field, based on discovery of viable new materials and on the development of new visualization techniques, have emerged very recently. Therefore, we propose to develop and utilize a suite of new spectroscopic imaging scanning tunnelling microscope instruments capable of visualizing topological quantum matter at millikelvin temperatures. The objective is to achieve the extremely high energy-resolution required to access the energy gaps and exotic electronic states of TQM. Using these unique instruments, we propose a specific sequence of key experiments on direct visualization and quantitative understanding of topological quantum matter. These include direct measurement of the momentum-space structure of the energy gaps of topological superconductors, ferromagnetic topological insulators, and topological Kondo insulators. We plan atomic scale visualization of the Cooper-pair condensate, the ferromagnetic state, and the heavy-fermion hybridization to explore how the real-space structure of these ordered states influences their topological characteristics. Finally, we propose to visualize scattering interference of topological quasiparticles to detect the spectra of extremely exotic Majorana/Jackiw-Rebbi/Heavy-Dirac states that are predicted for these three topological orders.
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