Abstract
Artificial quantum systems have emerged as platforms to realize topological matter in a well-controlled manner. So far, experiments have mostly explored non-interacting topological states, and the realization of many-body topological phases in solid-state platforms with atomic resolution has remained challenging. Here we construct topological quantum Heisenberg spin lattices by assembling spin chains and two-dimensional spin arrays from spin-1/2 Ti atoms on an insulating MgO film in a scanning tunnelling microscope. We engineer both topological and trivial phases of the quantum spin model and thereby realize first- and second-order topological quantum magnets. We probe the many-body excitations of the quantum magnets by single-atom electron spin resonance with an energy resolution better than 100 neV. Making use of the atomically localized magnetic field of the scanning tunnelling microscope tip, we visualize various many-body topological bound modes including topological edge states, topological defects and higher-order corner modes. Our results provide a bottom-up approach for the simulation of exotic quantum many-body phases of interacting spins.
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Data availability
The data that support the plots within this paper are available via the Figshare repository at https://doi.org/10.6084/m9.figshare.26379964. Additional data are available from the authors upon request.
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Acknowledgements
This work is supported by the National Key R&D Program of China (2022YFA1204100 (K.Y. and L.J.)), the Beijing Natural Science Foundation (Z230005 (K.Y., and P.F.)), the National Natural Science Foundation of China (12174433 (K.Y.), 52272170 (L.J.) and 62488201 (H.-J.G.)), and the CAS Project for Young Scientists in Basic Research (YSBR-003 (K.Y.)). J.L.L. acknowledges the computational resources provided by the Aalto Science-IT project, the financial support from the Academy of Finland Projects (nos. 331342 and 358088) and the Jane and Aatos Erkko Foundation.
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K.Y. and J.L.L. designed the experiment. H.W., P.F., J.C., L.J. and K.Y. carried out the STM measurements. J.L.L. developed the theoretical model. H.W., L.J., H.-J.G., K.Y. and J.L.L. performed the analysis and wrote the manuscript with help from all authors. All authors discussed the results and edited the manuscript.
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Extended data
Extended Data Fig. 1 Magnetic resonance imaging (MRI) of the dimerized 6-spin chain.
(a) STM image (1.3 nm × 5.6 nm) and binding sites for the dimerized 6-spin chain with topological configuration. Scale bar: 5 Å. (b) MRI of the chain showing spatially resolved spin resonance signal for a fixed frequency at f = 15.86, 17.18, 16.32 and 16.45 GHz, respectively (VDC = 50 mV, I = 50 pA, VRF = 16 mV, \({B}_{{\rm{ext}}}\) = 0.68 T).
Extended Data Fig. 2 Quantum coherence T2 of the end modes of a dimerized 6-spin chain and isolated Ti spins.
T2 was obtained by measuring the linewidth of the ESR peak at different VRF (VDC = 50 mV, I = 5–40 pA, VRF = 6–55 mV, \({B}_{{\rm{ext}}}\) = 0.68 T). Error bars are from the fitting uncertainties of T2 with a 95% confidence.
Extended Data Fig. 3 An 8-spin chain with topologically trivial configuration.
(a) STM image (1.5 nm × 7 nm) and binding sites of a dimerized 8-spin chain on MgO. Scale bar: 5 Å. (b) ESR spectra measured on each of the 8 spins as a function of setpoint current I, which is approximately proportional to \({B}_{{\rm{tip}}}\) (VDC = 50 mV, I = 20–220 pA, VRF = 4–25 mV, \({B}_{{\rm{ext}}}\) = 0.59 T). (c) ESR spectra measured along the dashed line in a (VDC = 50 mV, I = 60 pA, VRF = 20 mV). (d) Calculated average magnetization \(\left\langle {S}_{n}^{z}\right\rangle\) for its ground state \(\left|1\right\rangle\) and two excited states \(\left|2\right\rangle\) and \(\left|3\right\rangle\). (e) ESR spectra measured on a single Ti spin (VDC = 50 mV, I = 10–240 pA, VRF = 4–25 mV).
Extended Data Fig. 4 Dimerized 6-spin chains without and with the second-nearest-neighbor coupling.
STM images and ESR spectra of dimerized 6-spin chains without (a–c) and with (d–f) second-nearest-neighbor coupling (~1 GHz). Scale bars in (a, d): 5 Å. For the spin chain in (d), the second-nearest-neighbor coupling is between S1 and S3 as well as between S4 and S6. (b, e) ESR spectra measured on spins S1, S2 and S3 as a function of setpoint current I, which is approximately proportional to \({B}_{{\rm{tip}}}\) (VDC = 50 mV, I = 10–142 pA, VRF = 2.5–10 mV, \({B}_{{\rm{ext}}}\) = 0.68 T). (c, f) Zoom-in ESR spectra showing the singlet-triplet splitting (VDC = 50 mV, I = 10 pA, VRF = 10-20 mV).
Extended Data Fig. 5 ESR spectra of S1 to S4 and S6 to S9 of the 9-spin chain.
(a) STM image (1.75 nm × 8.3 nm) and binding sites of the dimerized 9-spin chain on MgO with a topological defect in the middle. Scale bar: 5 Å. (b) ESR spectra measured on each of the 8 spins as a function of setpoint current I, which is approximately proportional to \({B}_{{\rm{tip}}}\) (VDC = 50 mV, I = 20–155 pA, VRF = 6–30 mV, \({B}_{{\rm{ext}}}\) = 0.68 T).
Extended Data Fig. 6 ESR spectra of all the spins of the dimerized 4×4 spin lattice.
(a) STM image (4 nm × 4 nm) of the 4×4 spin lattice. Scale bar: 5 Å. (b) ESR spectra measured on each of the 16 spins as a function of setpoint current I, which is approximately proportional to \({B}_{{\rm{tip}}}\) (VDC = 50 mV, I = 10–90 pA, VRF = 12–32 mV, \({B}_{{\rm{ext}}}\) = 0.77 T). Red arrows indicate the corner modes. (c) ESR spectra measured on a single Ti spin (VDC = 50 mV, I = 40–125 pA, VRF = 10–16 mV). (d) Energy level diagram showing the lowest 64 eigenenergies (as a function of \({B}_{{\rm{ext}}}\) and \({B}_{{\rm{tip}}}\)) of the dimerized 4×4 spin lattice. (e) Energy level diagram of a 2×2 spin lattice, which is approximate to the low-energy spectrum of the dimerized 4×4 spin lattice. (f) Calculated average magnetization \(\left\langle {S}_{n}^{z}\right\rangle\) of different sites for the ground state |1〉.
Extended Data Fig. 7 Dimerized 6-spin chains with varied coupling between the end spins and the bulk spins.
(a) STM image and binding sites for spin chains with Jend-bulk = 10, 6, 3.6, 0.9 GHz, respectively. The bulk coupling is J(1+δ)/J(1-δ) = 25/6 GHz. Scale bars: 5 Å. (b) ESR spectra measured on the end spins as a function of setpoint current I (VDC = 50 mV, I = 10−200 pA, VRF = 5−28 mV, \({B}_{{\rm{ext}}}\) = 0.68 T). (c) Zoom-in of the ESR spectra measured at low setpoint currents (VDC = 50 mV, I = 5−10 pA, VRF = 21−28 mV), showing transitions I and II, the frequency difference of which gives the singlet-triplet splittings.
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Supplementary Figs. 1–16 and Sections 1 and 2.
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Wang, H., Fan, P., Chen, J. et al. Construction of topological quantum magnets from atomic spins on surfaces. Nat. Nanotechnol. (2024). https://doi.org/10.1038/s41565-024-01775-2
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DOI: https://doi.org/10.1038/s41565-024-01775-2
