Visualizing higher-fold topology in chiral crystals

Novel topological phases of matter are fruitful platforms for the discovery of unconventional electromagnetic phenomena. Higher-fold topology is one example, where the low-energy description goes beyond Standard Model analogs. Despite intensive experimental studies, conclusive evidence remains elusive for the \textit{multi-gap topological nature of higher-fold chiral fermions}. In this Letter, we leverage a combination of fine-tuned chemical engineering and photoemission spectroscopy with photon energy contrast to discover the higher-fold topology of a chiral crystal. We identify all bulk branches of a higher-fold chiral fermion for the first time, critically important for allowing us to explore unique Fermi arc surface states in multiple inter-band gaps, which exhibit an emergent ladder structure. Through designer chemical gating of the samples in combination with our measurements, we uncover an unprecedented multi-gap bulk boundary correspondence. Our demonstration of multi-gap electronic topology will propel future research on unconventional topological responses.