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

Ilya Belopolski (), Guoqing Chang, Tyler A. Cochran, Zi-Jia Cheng, Xian P. Yang, Cole Hugelmeyer, Kaustuv Manna, Jia-Xin Yin, Guangming Cheng, Daniel Multer, Maksim Litskevich, Nana Shumiya, Songtian S. Zhang, Chandra Shekhar, Niels B. M. Schröter, Alla Chikina, Craig Polley, Balasubramanian Thiagarajan, Mats Leandersson, Johan Adell, Shin-Ming Huang, Nan Yao, Vladimir N. Strocov, Claudia Felser and M. Zahid Hasan ()
Additional contact information
Ilya Belopolski: Princeton University
Guoqing Chang: Nanyang Technological University
Tyler A. Cochran: Princeton University
Zi-Jia Cheng: Princeton University
Xian P. Yang: Princeton University
Cole Hugelmeyer: Princeton University
Kaustuv Manna: Max Planck Institute for Chemical Physics of Solids
Jia-Xin Yin: Princeton University
Guangming Cheng: Princeton University
Daniel Multer: Princeton University
Maksim Litskevich: Princeton University
Nana Shumiya: Princeton University
Songtian S. Zhang: Princeton University
Chandra Shekhar: Max Planck Institute for Chemical Physics of Solids
Niels B. M. Schröter: Swiss Light Source, Paul Scherrer Institut
Alla Chikina: Swiss Light Source, Paul Scherrer Institut
Craig Polley: Lund University
Balasubramanian Thiagarajan: Lund University
Mats Leandersson: Lund University
Johan Adell: Lund University
Shin-Ming Huang: National Sun Yat-sen University
Nan Yao: Princeton University
Vladimir N. Strocov: Swiss Light Source, Paul Scherrer Institut
Claudia Felser: Max Planck Institute for Chemical Physics of Solids
M. Zahid Hasan: Princeton University

Nature, 2022, vol. 604, issue 7907, 647-652

Abstract: Abstract Quantum phases can be classified by topological invariants, which take on discrete values capturing global information about the quantum state1–13. Over the past decades, these invariants have come to play a central role in describing matter, providing the foundation for understanding superfluids5, magnets6,7, the quantum Hall effect3,8, topological insulators9,10, Weyl semimetals11–13 and other phenomena. Here we report an unusual linking-number (knot theory) invariant associated with loops of electronic band crossings in a mirror-symmetric ferromagnet14–20. 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.

Date: 2022
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DOI: 10.1038/s41586-022-04512-8

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