Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

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

Abstract

Quantum phases can be classified by topological invariants, which take on discrete values capturing global information about the quantum state1,2,3,4,5,6,7,8,9,10,11,12,13. 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 semimetals11,12,13 and other phenomena. Here we report an unusual linking-number (knot theory) invariant associated with loops of electronic band crossings in a mirror-symmetric ferromagnet14,15,16,17,18,19,20. 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.

This is a preview of subscription content, access via your institution

Access options

Buy article

Get time limited or full article access on ReadCube.

$32.00

All prices are NET prices.

Fig. 1: Signatures of linked node loops in Co2MnGa.
Fig. 2: Weyl loop trajectory in Co2MnGa.
Fig. 3: Linked Weyl loops in Co2MnGa.
Fig. 4: Linking number (2, 2, 2) in topological quantum matter.
Fig. 5: Seifert bulk–boundary correspondence.

Data availability

The datasets generated during and/or analysed during the current study are available in the Zenodo repository at https://doi.org/10.5281/zenodo.5793667Source data are provided with this paper.

References

  1. Buchanan, M. The unifying role of topology. Nat. Phys. 16, 818 (2020).

    CAS  Article  Google Scholar 

  2. Chaikin, P. M. & Lubensky, T. C. Principles of Condensed Matter Physics Ch. 9 (Cambridge Univ. Press, 1995).

  3. Haldane, F. D. M. Nobel lecture: Topological quantum matter. Rev. Mod. Phys. 89, 040502 (2017).

    ADS  MathSciNet  Article  Google Scholar 

  4. Wen, X.-G. Colloquium: Zoo of quantum-topological phases of matter. Rev. Mod. Phys. 89, 041004 (2017).

    ADS  MathSciNet  Article  Google Scholar 

  5. Volovik, G. E. The Universe in a Helium Droplet (Oxford Univ. Press, 2003).

  6. Zang, J., Cros, V. & Hoffmann, A. Topology in Magnetism (Springer, 2018).

  7. Nagaosa, N. & Tokura, Y. Topological properties and dynamics of magnetic skyrmions. Nat. Nanotechnol. 8, 899–911 (2013).

    ADS  CAS  PubMed  Article  Google Scholar 

  8. Avron, J. E., Osadchy, D. & Seiler, R. A topological look at the quantum Hall effect. Phys. Today 56, 38–42 (2003).

    CAS  Article  Google Scholar 

  9. Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    ADS  CAS  Article  Google Scholar 

  10. Bernevig, A. Topological Insulators and Topological Superconductors (Princeton Univ. Press, 2013).

  11. Hasan, M. Z., Xu, S.-Y., Belopolski, I. & Huang, S.-M. Discovery of Weyl fermion semimetals and topological Fermi arc states. Annu. Rev. Condens. Matter Phys. 8, 289–309 (2017).

    ADS  CAS  Article  Google Scholar 

  12. Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90, 015001 (2018).

    ADS  MathSciNet  CAS  Article  Google Scholar 

  13. Chiu, C.-K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).

    ADS  Article  Google Scholar 

  14. Belopolski, I. et al. Discovery of topological Weyl fermion lines and drumhead surface states in a room temperature magnet. Science 365, 1278–1281 (2019).

    ADS  CAS  PubMed  Article  Google Scholar 

  15. Chang, G. et al. Topological Hopf and chain link semimetal states and their application to Co2MnGa. Phys. Rev. Lett. 119, 156401 (2017).

    ADS  PubMed  Article  Google Scholar 

  16. Wu, Q., Soluyanov, A. A. & Bzdušek, T. Non-Abelian band topology in noninteracting metals. Science 365, 1273–1277 (2019).

    MathSciNet  CAS  PubMed  MATH  Article  Google Scholar 

  17. Yan, Z. et al. Nodal-link semimetals. Phys. Rev. B 96, 041103(R) (2017).

    ADS  Article  Google Scholar 

  18. Ezawa, M. Topological semimetals carrying arbitrary Hopf numbers: Fermi surface topologies of a Hopf link, Solomon’s knot, trefoil knot, and other linked nodal varieties. Phys. Rev. B 96, 041202 (2017).

    ADS  Article  Google Scholar 

  19. Chang, P.-Y. & Yee, C.-H. Weyl-link semimetals. Phys. Rev. B 96, 081114 (2017).

    ADS  Article  Google Scholar 

  20. Zhong, C. et al. Three-dimensional pentagon carbon with a genesis of emergent fermions. Nat. Commun. 8, 15641 (2017).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  21. Bian, G. et al. Topological nodal-line fermions in spin-orbit metal PbTaSe2. Nat. Commun. 7, 10556 (2016).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  22. Watanabe, H., Po, H. C. & Vishwanath, A. Structure and topology of band structures in the 1651 magnetic space groups. Sci. Adv. 4, eaat8685 (2018).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  23. Wang, Y. & Nandkishore, R. Topological surface superconductivity in doped Weyl loop materials. Phys. Rev. B 95, 060506 (2017).

    ADS  Article  Google Scholar 

  24. Stenull, O., Kane, C. L. & Lubensky, T. C. Topological phonons and Weyl lines in three dimensions. Phys. Rev. Lett. 117, 068001 (2016).

    ADS  PubMed  Article  CAS  Google Scholar 

  25. Nandkishore, R. Weyl and Dirac loop superconductors. Phys. Rev. B 93, 020506(R) (2016).

    ADS  Article  CAS  Google Scholar 

  26. Sun, X.-Q., Lian, B. & Zhang, S.-C. Double helix nodal line superconductor. Phys. Rev. Lett. 119, 147001 (2017).

    ADS  PubMed  Article  Google Scholar 

  27. Lian, B., Vafa, C., Vafa, F. & Zhang, S.-C. Chern-Simons theory and Wilson loops in the Brillouin zone. Phys. Rev. B 95, 094512 (2017).

    ADS  Article  Google Scholar 

  28. Seifert, H. Über das Geschlecht von Knoten. Math. Ann. 110, 571–592 (1935).

    MathSciNet  MATH  Article  Google Scholar 

  29. Bergholtz, E. J., Budich, J. C. & Kunt, F. K. Exceptional topology of non-Hermitian systems. Rev. Mod. Phys. 93, 015005 (2021).

    ADS  MathSciNet  Article  Google Scholar 

  30. Li, L., Lee, C. H. & Gong, J. Emergence and full 3D-imaging of nodal boundary Seifert surfaces in 4D topological matter. Commun. Phys. 2, 135 (2019).

    Article  Google Scholar 

  31. Carlström, J., Stålhammar, M., Budich, J. C. & Bergholtz, E. J. Knotted non-Hermitian metals. Phys. Rev. B 99, 161115 (2019).

    ADS  Article  Google Scholar 

  32. Zhang, X. et al. Tidal surface states as fingerprints of non-Hermitian nodal knot metals. Commun. Phys. 4, 47 (2021).

    Article  Google Scholar 

  33. Sakai, A. et al. Iron-based binary ferromagnets for transverse thermoelectric conversion. Nature 581, 53–57 (2020).

    ADS  CAS  PubMed  Article  Google Scholar 

  34. Sakai, A. et al. Giant anomalous Nernst effect and quantum-critical scaling in a ferromagnetic semimetal. Nat. Phys. 14, 1119–1124 (2018).

    CAS  Article  Google Scholar 

  35. Guin, S. N. et al. Anomalous Nernst effect beyond the magnetization scaling relation in the ferromagnetic Heusler compound Co2MnGa. NPG Asia Mater. 11, 16 (2019).

    ADS  CAS  Article  Google Scholar 

  36. Park, G.-H. et al. Thickness dependence of the anomalous Nernst effect and the Mott relation of Weyl semimetal Co2MnGa thin films. Phys. Rev. B 101, 060406 (2020).

    ADS  CAS  Article  Google Scholar 

  37. Markou, A. et al. Hard magnet topological semimetals in xPt3 compounds with the harmony of Berry curvature. Commun. Phys. 4, 104 (2021).

    CAS  Article  Google Scholar 

  38. Strocov, V. N. et al. High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectroscopies. J. Synchrotron Radiat. 17, 631–643 (2010).

    CAS  PubMed  PubMed Central  Article  Google Scholar 

  39. Strocov, V. N. et al. Soft-X-ray ARPES facility at the ADRESS beamline of the SLS: concepts, technical realisation and scientific applications. J. Synchrotron Radiat. 21, 32–44 (2014).

    CAS  PubMed  Article  Google Scholar 

  40. Wu, L. et al. Quantized Faraday and Kerr rotation and axion electrodynamics of a 3D topological insulator. Science 354, 1124–1127 (2016).

    ADS  MathSciNet  CAS  PubMed  MATH  Article  Google Scholar 

  41. Zahid Hasan, M. et al. Weyl, Dirac and high-fold chiral fermions in topological quantum matter. Nat. Rev. Mater. 6, 784–803 (2021).

    ADS  Article  CAS  Google Scholar 

  42. Yin, J.-X., Pan, S. H. & Zahid Hasan, M. Probing topological quantum matter with scanning tunnelling microscopy. Nat. Rev. Phys. 3, 249–263 (2021).

    Article  Google Scholar 

  43. Webster, P. J. Magnetic and chemical order in Heusler alloys containing cobalt and manganese. J. Phys. Chem. Solids 32, 1221–1231 (1971).

    ADS  CAS  Article  Google Scholar 

  44. Ido, H. & Yasuda, S. Magnetic properties of Co-Heusler and related mixed alloys. J. Phys. 49, C8-141–C8-142 (1988).

    Google Scholar 

  45. Strocov, V. N. et al. Three-dimensional electron realm in VSe2 by soft-X-ray photoelectron spectroscopy: origin of charge-density waves. Phys. Rev. Lett. 109, 086401 (2012).

    ADS  PubMed  Article  CAS  Google Scholar 

  46. Kresse, G. & Furthmueller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996).

    ADS  CAS  Article  Google Scholar 

  47. Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59, 1758–1775 (1999).

    ADS  CAS  Article  Google Scholar 

  48. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).

    ADS  CAS  PubMed  Article  Google Scholar 

  49. Belopolski, I. et al. Signatures of Weyl fermion annihilation in a correlated kagome magnet. Phys. Rev. Lett. 127, 256403 (2021). 

  50. Salomaa, M. M. & Volovik, G. E. Quantized vortices in superfluid 3He. Rev. Mod. Phys. 59, 533–613 (1987).

    ADS  CAS  Article  Google Scholar 

  51. Nakahara, M. Geometry, Topology and Physics (Institute of Physics, 2003).

  52. Bansil, A., Lin, H. & Das, T. Colloquium: Topological band theory. Rev. Mod. Phys. 88, 021004 (2016).

    ADS  Article  Google Scholar 

  53. Kane, C. L. in Topological Insulators Ch. 1 (Elsevier, 2013).

  54. Tokura, Y., Yasuda, K. & Tsukazaki, A. Magnetic topological insulators. Nat. Rev. Phys. 1, 126–143 (2019).

    Article  Google Scholar 

  55. Hasan, M. Z., Xu, S.-Y. & Bian, G. Topological insulators, topological superconductors and Weyl fermion semimetals: discoveries, perspectives and outlooks. Phys. Scr. 2015, T164 (2015).

    Google Scholar 

  56. Yin, J.-X. et al. Quantum-limit Chern topological magnetism in TbMn6Sn6. Nature 583, 533–536 (2020).

  57. Chen, W., Lu, H.-Z. & Hou, J.-M. Topological semimetals with a double-helix nodal link. Phys. Rev. B 96, 041102 (2017).

  58. Muro, T. et al. Soft X-ray ARPES for three-dimensional crystals in the micrometre region. J. Synchr. Radiat. 28, 1631–1638 (2021). 

    CAS  Article  Google Scholar 

  59. Strocov, V. N. et al. Soft-X-ray ARPES at the Swiss Light Source: from 3D materials to buried interfaces and impurities. Synchrotron Radiat. News 27, 31–40 (2014).

    Article  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

Authors

Contributions

M.Z.H. supervised the project. I.B., G. Chang and T.A.C. initiated the project. I.B., T.A.C., Z.-J.C. and M.Z.H. acquired and analysed ARPES spectra with help from X.P.Y., D.M., J.-X.Y., M. Litskevich, N.S. and S.S.Z. ARPES measurements were supported by N.B.M.S., A.C., C.P., B.T., M. Leandersson, J.A. and V.N.S. G. Chang performed the first-principles calculations. I.B. wrote down the kp model with help from G. Chang and S.-M.H. I.B. developed the linking number theory with help from C.H. G. Cheng and N.Y. performed the scanning transmission electron microscopy measurements. K.M., C.S. and C.F. synthesized and characterized the single crystals. I.B. wrote the manuscript with contributions from all authors.

Corresponding authors

Correspondence to Ilya Belopolski or M. Zahid Hasan.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature thanks Sergey Borisenko and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 Topological invariants in physics.

a, An example of an order parameter winding in real space: a magnetic vortex2,5,6,7,50,51. In this case, the order parameter is the local magnetization m(x), confined to a magnetic easy plane in real space (x, y). It may happen that m(x) winds around a point in real space, forming a magnetic vortex characterized by a winding number topological invariant, in this example given by w = 1. b, An example of a quantum wavefunction winding in momentum space: the one-dimensional topological insulator (Su-Schrieffer-Heeger model)3,4,8,9,10,11,12,13,52,53,54,55,56. This phase is described by Bloch Hamiltonian h(k) = d(k) σ, where k is the one-dimensional crystal momentum, σ refers to the Pauli matrices and d(k) is a two-component object confined to the (dx, dy) plane. The normalized quantity \(\widehat{{\bf{d}}}(k)\equiv {\bf{d}}(k)/|{\bf{d}}(k)|\) (orange arrow) moves around the unit circle (dotted blue) as k varies. The topological invariant is related to how many times \(\widehat{{\bf{d}}}(k)\) winds around the origin as k scans through the one-dimensional Brillouin zone. c, Node loops linking in momentum space17,18,19,20,57: a three-dimensional electronic structure may exhibit multiple node loops (cyan and purple), characterized by kn(θ), where n indexes the loops and θ parametrizes the loop trajectory in momentum space. The loops may link one another, encoding a linking number topological invariant. This example shows a Hopf link. (See also Supplementary Information.).

Extended Data Fig. 2 Crystal structure and Brillouin zone of Co2MnGa.

a, Conventional unit cell with representative crystallographic mirror plane M (orange). b, The primitive unit cell (grey) includes one formula unit. c, Brillouin zone, with conventional reciprocal lattice basis vectors (black). Brillouin zone edges color-coded to correspond to the mirror planes: magenta M1 plane, (001); red M2 plane, (010), orange M3 plane, (100). d, Slice through Γ in an extended zone scheme.

Extended Data Fig. 3 Energy dispersion of the Weyl loop.

a, Crossing point energies EB and b, crossing point momenta (kx, ky) systematically extracted from cone dispersions observed in the ARPES spectra (magenta squares), same dataset as Fig. 2c ( = 544 eV), with fit of the Weyl loop momentum trajectory and energy dispersion (cyan, see main text). The crossing point energies are parametrized by a polar angle θ defined by tan θky/kx. c, Weyl loop trajectory from DFT, with dotted lines indicating the DFT energy-momentum slices shown in Fig. 2b. The binding energy axes in (b) and (c) are collapsed58.

Extended Data Fig. 4 Link ‘depth’ of the Weyl loops.

ac, Distance between the extrema of the Weyl loops and the bulk Brillouin zone W points for the M1, M2 and M3 Weyl loops. We estimate s1 = 0.32 ± 0.1 Å−1, s2 = 0.27 ± 0.1 Å−1 and s3 = 0.29 ± 0.1 Å−1. d, The link depth captures how far in momentum space one would need to slide the Weyl loops in order to unlink them, providing a measure of the stability of the link. Based on the loop Fermi surfaces (ac), we estimate d12 = 0.58 ± 0.14 Å−1, d23 = 0.55 ± 0.14 Å−1 and d31 = 0.60 ± 0.14 Å−1. The average gives a typical link depth extracted from ARPES, davg = 0.58 ± 0.08 Å−1. e, Energy-momentum slice along the high-symmetry path X1X2 from DFT, passing through two linked Weyl loops. We obtain dDFT = 0.68 Å−1.

Extended Data Fig. 5 Supplementary measurement of the link depth.

a, M1, M2 and M3 Weyl loops, with trajectories obtained from the analytical model (see main text), showing that M1 links M2 twice and M3 twice. Energy-momentum photoemission cuts along the high-symmetry paths b, X1X2 and c, X3X1 obtained at photon energy  = 642 eV. We observe d12 = 0.56 ± 0.1 Å−1 and d31 = 0.61 ± 0.1 Å−1, consistent with Extended Data Fig. 4. For both cuts, exactly one branch of each Weyl cone exhibits appreciable photoemission cross-section, as expected from the mirror-symmetric measurement geometry59. d, Fermi surface acquired at  = 642 eV, exhibiting an in-plane Weyl loop contour, M1. We further observe spectral weight emanating along kx and ky from the center of M1, corresponding to the linearly dispersive branches in (b, c), again suggesting that M1 is linked by M2 and M3.

Extended Data Fig. 6 Unsymmetrized Fermi surfaces.

ac, Left: photoemission spectra displayed in Fig. 1d–f, without symmetrization. Right: the same spectra, with the experimentally-determined Weyl loop trajectory overlaid across multiple Brillouin zones. The irrelevant Γ pocket is consistently observed in all unsymmetrized spectra. Signatures of Weyl loops are observed around all X points.

Extended Data Fig. 7 SX-ARPES systematics.

ad, Photoemission energy-momentum cuts through the Weyl loop, used to extract Fig. 2c.

Extended Data Fig. 8 Unsymmetrized energy-momentum cuts.

Photoemission spectra displayed in Fig. 4a, without symmetrization.

Extended Data Fig. 9 Linked Weyl loop Fermi surface.

Constant-energy slice of the pockets (navy) making up the linked Weyl loops obtained by ab initio calculation, at binding energy EB = −10 meV below the experimental Fermi level. Plotted a, in an extended zone scheme (only two loops shown for simplicity) and b, the reduced Brillouin zone (all three loops shown). The Fermi surface pockets touch at a set of discrete points, where the Weyl loop disperses through this particular EB. For reference, the full Weyl loop trajectories are indicated, collapsed in energy (magenta around X1, red around X2, orange around X3). The Weyl loop Fermi surface pockets form a linked structure.

Extended Data Fig. 10 Measured Fermi surfaces in an extended zone scheme.

The Brillouin zone corresponds to Γ(066) in the primitive reciprocal basis.

Supplementary information

Supplementary Information

This file contains Supplementary text, figures and equations.

Peer Review File

Source data

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Belopolski, I., Chang, G., Cochran, T.A. et al. Observation of a linked-loop quantum state in a topological magnet. Nature 604, 647–652 (2022). https://doi.org/10.1038/s41586-022-04512-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41586-022-04512-8

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing