What's knot to like? Observation of a linked loop quantum state

Abstract

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 central role in describing matter, providing the foundation for understanding superfluids, magnets, the quantum Hall effect, topological insulators, Weyl semimetals and other phenomena. Here we report a remarkable linking number (knot theory) invariant associated with loops of electronic band crossings in a mirror-symmetric ferromagnet. Using state-of-the-art spectroscopic methods, we directly observe three intertwined degeneracy loops in the material's bulk Brillouin zone three-torus, ${\mathbb{T}}^3$. 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 linking number $(2,2,2)$, providing a direct determination of the invariant structure from the experimental data. On the surface of our samples, we further predict and observe 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 exotic properties of quantum matter.

arXiv commentSee popular summary at https://research.princeton.edu/news/electrons-crystal-exhibit-linked-and-knotted-quantum-twists

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• Observation of Weyl point pair annihilation in a gyromagnetic photonic crystal
• Realization of a Hopf insulator in circuit systems
• Intrinsic Magnetic Topological Materials
• Synthetic Topological Vacua of Yang-Mills Fields in Bose-Einstein Condensates
• A topological Hund nodal line antiferromagnet
• Unremovable linked nodal structures protected by crystalline symmetries in stacked bilayer graphene with Kekulé texture
• Nodal-line semimetals and their variance
• Realization of Hopf-link structure in phonon spectra: Symmetry guidance and High-throughput investigation
• Quantum oscillation in Hopf-link semimetals
• Emergence of Nodal-Knot Transitions by Disorder