Discovery of a topological quantum link

Quantum phases can be classified by topological invariants, which take on discrete values capturing global information about the quantum state. Over the past decades, these invariants have come to play a key role in describing matter, providing the foundation for understanding superfluids, magnets, the quantum Hall effect, topological insulators and Weyl semimetals. We introduce a remarkable linking number (knot theory) invariant associated with loops of electronic band crossings in the mirror-symmetric ferromagnet Co₂MnGa [1-4]. Using state-of-the-art soft X-ray and vacuum ultraviolet ARPES, we observe three intertwined degeneracy loops in the bulk Brillouin zone three-torus, T³. We find that each loop links each other loop twice. Through systematic investigation of this linked loop quantum state, we explicitly draw its link diagram and conclude, in analogy with knot theory, that it exhibits linking number (2,2,2), providing a direct experimental determination of the topological invariant. On the sample surface, we further predict and observe Seifert boundary states protected by the bulk linked loops, suggestive of a Seifert bulk-boundary correspondence. Our observation of a quantum loop link motivates the application of knot theory to the exploration of quantum matter.

1. I. Belopolski et al. Nature 604, 647 (2022).

2. I. Belopolski et al. Phys. Rev. Lett. 127, 256403 (2021).

3. I. Belopolski et al. Science 365, 6459 (2019).

4. M. Z. Hasan, G. Chang, I. Belopolski et al. Nat. Rev. Mat. 6, 784 (2021).