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

TL;DR

This paper reports the experimental observation of linked quantum loops in a ferromagnet's electronic structure, revealing a topological invariant linked to knot theory, and suggests a new approach to understanding exotic quantum matter properties.

Contribution

It provides the first direct experimental observation of linked loop quantum states and their topological invariants using advanced spectroscopic techniques.

Findings

Observation of three intertwined degeneracy loops in the Brillouin zone

Determination of linking number (2,2,2) from experimental data

Prediction and observation of Seifert boundary states

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.