Observation of a linked-loop quantum state in a topological magnet

Quantum phases can be classified by topological invariants, which take on discrete values capturing global information about the quantum state^1–13. Over the past decades, these invariants have come to play a central role in describing matter, providing the foundation for understanding superfluids⁵, magnets^6,7, the quantum Hall effect^3,8, topological insulators^9,10, Weyl semimetals^11–13 and other phenomena. Here we report an unusual linking-number (knot theory) invariant associated with loops of electronic band crossings in a mirror-symmetric ferromagnet^14–20. Using state-of-the-art spectroscopic methods, we directly observe three intertwined degeneracy loops in the material’s three-torus, T³, bulk Brillouin zone. We find that each loop links each other loop twice. Through systematic spectroscopic investigation of this linked-loop quantum state, we explicitly draw its link diagram and conclude, in analogy with knot theory, that it exhibits the linking number (2, 2, 2), providing a direct determination of the invariant structure from the experimental data. We further predict and observe, on the surface of our samples, Seifert boundary states protected by the bulk linked loops, suggestive of a remarkable Seifert bulk–boundary correspondence. Our observation of a quantum loop link motivates the application of knot theory to the exploration of magnetic and superconducting quantum matter.