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.
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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.5793667. Source data are provided with this paper.
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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.
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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 k ⋅ p 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.
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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 (hν = 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 θ ≡ ky/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.
a–c, 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 (a–c), 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 X1 − X2 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, X1 − X2 and c, X3 − X1 obtained at photon energy hν = 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 hν = 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.
a–c, 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.
a–d, 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.
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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
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DOI: https://doi.org/10.1038/s41586-022-04512-8
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