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.
Original language | English (US) |
---|---|
Pages (from-to) | 647-652 |
Number of pages | 6 |
Journal | Nature |
Volume | 604 |
Issue number | 7907 |
DOIs | |
State | Published - Apr 28 2022 |
All Science Journal Classification (ASJC) codes
- General
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In: Nature, Vol. 604, No. 7907, 28.04.2022, p. 647-652.
Research output: Contribution to journal › Article › peer-review
TY - JOUR
T1 - Observation of a linked-loop quantum state in a topological magnet
AU - Belopolski, Ilya
AU - Chang, Guoqing
AU - Cochran, Tyler A.
AU - Cheng, Zi Jia
AU - Yang, Xian P.
AU - Hugelmeyer, Cole
AU - Manna, Kaustuv
AU - Yin, Jia Xin
AU - Cheng, Guangming
AU - Multer, Daniel
AU - Litskevich, Maksim
AU - Shumiya, Nana
AU - Zhang, Songtian S.
AU - Shekhar, Chandra
AU - Schröter, Niels B.M.
AU - Chikina, Alla
AU - Polley, Craig
AU - Thiagarajan, Balasubramanian
AU - Leandersson, Mats
AU - Adell, Johan
AU - Huang, Shin Ming
AU - Yao, Nan
AU - Strocov, Vladimir N.
AU - Felser, Claudia
AU - Hasan, M. Zahid
N1 - Funding Information: I.B. thanks N. Lvov and Z. Szabó for discussions on linking numbers. We thank D. Lu and M. Hashimoto at Beamline 5-2 of the Stanford Synchrotron Radiation Lightsource at the SLAC National Accelerator Laboratory, CA, USA for support. I.B. and D.M. thank T. Muro for experimental support during preliminary ARPES measurements carried out at BL25SU of SPring-8 in Hyogo, Japan. I.B. thanks B. Lian for discussions on the topological magneto-electric effect. I.B., T.A.C., X.P.Y. and D.M. thank J. McChesney and F. Rodolakis for experimental support during preliminary ARPES measurements carried out at BL29 of the Advanced Photon Source in Illinois, USA. I.B. acknowledges discussions with B. Belopolski on Savitzky–Golay analysis. G. Chang acknowledges the support of the National Research Foundation, Singapore under its NRF Fellowship Award (NRF-NRFF13-2021-0010) and the Nanyang Assistant Professorship grant from Nanyang Technological University. T.A.C. acknowledges support by the National Science Foundation Graduate Research Fellowship Program under grant number DGE-1656466. A.C. acknowledges funding from the Swiss National Science Foundation under grant number 200021-165529. We acknowledge synchrotron radiation beamtime at the ADRESS beamline of the Swiss Light Source of the Paul Scherrer Institut in Villigen, Switzerland under proposals 20170898, 20190740 and 20191674. S.-M.H. acknowledges funding by the MOST-AFOSR Taiwan program on Topological and Nanostructured Materials under grant no. 110-2124-M-110-002-MY3. We further acknowledge use of Princeton’s Imaging and Analysis Center, which is partially supported by the Princeton Center for Complex Materials, a National Science Foundation Materials Research Science and Engineering Center (DMR-2011750). This research used resources of the Advanced Photon Source, a US Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under contract number DE-AC02-06CH11357. We acknowledge beamtime at BL25SU of SPring-8 under proposal 2017A1669 and at BL29 of the Advanced Photon Source under proposals 54992 and 60811. K.M. and C.F. acknowledge financial support from the European Research Council Advanced Grant no. 742068 “TOP-MAT”. C.F. acknowledges the DFG through SFB 1143 (project ID. 247310070) and the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter ct.qmat (EXC2147, project ID. 39085490). M.Z.H. acknowledges support from the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Science Center and Princeton University. M.Z.H. acknowledges visiting scientist support at Berkeley Lab (Lawrence Berkeley National Laboratory) during the early phases of this work. Work at Princeton University was supported by the Gordon and Betty Moore Foundation (grant numbers GBMF4547 and GBMF9461; M.Z.H.). The ARPES and theoretical work were supported by the US DOE under the Basic Energy Sciences programme (grant number DOE/BES DE-FG-02-05ER46200; M.Z.H.). Use of the Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, is supported by the US DOE, Office of Science, Office of Basic Energy Sciences, under contract number DE-AC02-76SF00515. We acknowledge MAX IV Laboratory for time on the BLOCH Beamline under proposal 20210268. Research conducted at MAX IV, a Swedish national user facility, is supported by the Swedish Research council under contract 2018-07152, the Swedish Governmental Agency for Innovation Systems under contract 2018-04969, and Formas under contract 2019-02496. Materials characterization and the study of topological quantum properties were supported by the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Science Center and Princeton University. Funding Information: I.B. thanks N.?Lvov and Z.?Szab? for discussions on linking numbers. We thank D.?Lu and M.?Hashimoto at Beamline 5-2 of the Stanford Synchrotron Radiation Lightsource at the SLAC National Accelerator Laboratory, CA, USA for support. I.B. and D.M. thank T.?Muro for experimental support during preliminary ARPES measurements carried out at BL25SU of SPring-8 in Hyogo, Japan. I.B. thanks B.?Lian for discussions on the topological magneto-electric effect. I.B., T.A.C., X.P.Y. and D.M. thank J.?McChesney and F.?Rodolakis for experimental support during preliminary ARPES measurements carried out at BL29 of the Advanced Photon Source in Illinois, USA. I.B. acknowledges discussions with?B. Belopolski on Savitzky?Golay analysis. G.?Chang acknowledges the support of the National Research Foundation, Singapore under its NRF Fellowship Award (NRF-NRFF13-2021-0010) and the Nanyang Assistant Professorship grant from Nanyang Technological University. T.A.C. acknowledges support by the National Science Foundation Graduate Research Fellowship Program under grant number DGE-1656466. A.C. acknowledges funding from the Swiss National Science Foundation under grant number 200021-165529. We acknowledge synchrotron radiation beamtime at the ADRESS beamline of the Swiss Light Source of the Paul Scherrer Institut in Villigen, Switzerland under proposals 20170898, 20190740 and 20191674. S.-M.H. acknowledges funding by the MOST-AFOSR Taiwan program on Topological and Nanostructured Materials under grant no. 110-2124-M-110-002-MY3.?We further acknowledge use of Princeton?s Imaging and Analysis Center, which is partially supported by the Princeton Center for Complex Materials, a National Science Foundation Materials Research Science and Engineering Center (DMR-2011750). This research used resources of the Advanced Photon Source, a US Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under contract number DE-AC02-06CH11357. We acknowledge beamtime at BL25SU of SPring-8 under proposal 2017A1669 and at BL29 of the Advanced Photon Source under proposals 54992 and 60811. K.M. and C.F. acknowledge financial support from the European Research Council Advanced Grant no. 742068 ?TOP-MAT?. C.F. acknowledges the DFG through SFB 1143 (project ID. 247310070) and the W?rzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter ct.qmat (EXC2147, project ID. 39085490).?M.Z.H. acknowledges support from the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Science Center and Princeton University. M.Z.H. acknowledges visiting scientist support at Berkeley Lab (Lawrence Berkeley National Laboratory) during the early phases of this work. Work at Princeton University was supported by the Gordon and Betty Moore Foundation (grant numbers GBMF4547 and GBMF9461; M.Z.H.). The ARPES and theoretical work were supported by the US DOE under the Basic Energy Sciences programme (grant number DOE/BES DE-FG-02-05ER46200; M.Z.H.). Use of the Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, is supported by the US DOE, Office of Science, Office of Basic Energy Sciences, under contract number DE-AC02-76SF00515. We acknowledge MAX IV Laboratory for time on the BLOCH Beamline under proposal 20210268. Research conducted at MAX IV, a Swedish national user facility, is supported by the Swedish Research council under contract 2018-07152, the Swedish Governmental Agency for Innovation Systems under contract 2018-04969, and Formas under contract 2019-02496.?Materials characterization and the study of topological quantum properties were supported by the US?Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Science Center and Princeton University. Publisher Copyright: © 2022, The Author(s), under exclusive licence to Springer Nature Limited.
PY - 2022/4/28
Y1 - 2022/4/28
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85128917922&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85128917922&partnerID=8YFLogxK
U2 - 10.1038/s41586-022-04512-8
DO - 10.1038/s41586-022-04512-8
M3 - Article
C2 - 35478239
AN - SCOPUS:85128917922
SN - 0028-0836
VL - 604
SP - 647
EP - 652
JO - Nature
JF - Nature
IS - 7907
ER -