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Direct visualization of relativistic quantum scars in graphene quantum dots

Nature volume 635, pages 841–846 (2024)Cite this article

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Abstract

Quantum scars refer to eigenstates with enhanced probability density along unstable classical periodic orbits. First predicted 40 years ago1, scars are special eigenstates that counterintuitively defy ergodicity in quantum systems whose classical counterpart is chaotic2,3. Despite the importance and long history of scars, their direct visualization in quantum systems remains an open field4,5,6,7,8,9,10. Here we demonstrate that, by using an in situ graphene quantum dot (GQD) creation and a wavefunction mapping technique11,12, quantum scars are imaged for Dirac electrons with nanometre spatial resolution and millielectronvolt energy resolution with a scanning tunnelling microscope. Specifically, we find enhanced probability densities in the form of lemniscate ∞-shaped and streak-like patterns within our stadium-shaped GQDs. Both features show equal energy interval recurrence, consistent with predictions for relativistic quantum scars13,14. By combining classical and quantum simulations, we demonstrate that the observed patterns correspond to two unstable periodic orbits that exist in our stadium-shaped GQD, thus proving that they are both quantum scars. In addition to providing unequivocal visual evidence of quantum scarring, our work offers insight into the quantum–classical correspondence in relativistic chaotic quantum systems and paves the way to experimental investigation of other recently proposed scarring species such as perturbation-induced scars15,16,17, chiral scars18,19 and antiscarring20.

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Fig. 1: STM characterization of an in situ-created stadium-shaped GQD.
Fig. 2: VG and VS dependence of dI/dVS maps.
Fig. 3: Equal energy interval recurrence of the ∞-shaped and streak-like dI/dVS patterns.
Fig. 4: Classical and quantum dynamics simulations for a stadium-shaped GQD.

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Data availability

Source data that support the findings of this study are available on Zenodo at https://doi.org/10.5281/zenodo.13751637 (ref. 53).

Code availability

All codes used in this article are available from the corresponding authors upon request.

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Acknowledgements

We thank M. Crommie and A. Zettl for discussions about quantum chaos at the initial stages of the project; D. Liu for assistance with STM cryogen refilling at the beginning of the project; and the Hummingbird Computational Cluster team at UC Santa Cruz for providing computational resources for the numerical tight-binding calculations performed in this work. J.V.J. and Z.G. acknowledge support from the National Science Foundation under award DMR-1753367. J.V.J. acknowledges support from the Army Research Office under contract W911NF-17-1-0473 and the Gordon and Betty Moore Foundation award 10.37807/GBMF11569. V.I.F. and S.S. acknowledge support from the Research Collaborations grant 1185409051 under the International Science Partnerships Fund. V.I.F. acknowledges support from Lloyd Register Foundation Nanotechnology Grant, EPSRC grants EP/V007033/1, EP/S030719/1 and EP/N010345/1. K.W. and T.T. acknowledge support from the JSPS KAKENHI (grant numbers 21H05233 and 23H02052) and the World Premier International Research Center Initiative (WPI), MEXT, Japan. A.M.G. acknowledges support from the Harvard Quantum Initiative. J.K.-R. acknowledges support from the Emil Aaltonen Foundation and the Oskar Huttunen Foundation.

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Author notes

  1. Zhehao Ge

    Present address: Department of Physics, University of California, Berkeley, Berkeley, CA, USA

Authors and Affiliations

  1. Department of Physics, University of California, Santa Cruz, Santa Cruz, CA, USA

    Zhehao Ge, Peter Polizogopoulos, Ryan Van Haren, David Lederman & Jairo Velasco Jr

  2. Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA

    Anton M. Graf

  3. Department of Physics, Harvard University, Cambridge, MA, USA

    Joonas Keski-Rahkonen & Eric J. Heller

  4. Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA, USA

    Joonas Keski-Rahkonen & Eric J. Heller

  5. Department of Physics and Astronomy, University of Manchester, Manchester, UK

    Sergey Slizovskiy & Vladimir I. Fal’ko

  6. National Graphene Institute, University of Manchester, Manchester, UK

    Sergey Slizovskiy & Vladimir I. Fal’ko

  7. Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Japan

    Takashi Taniguchi

  8. Research Center for Electronic and Optical Materials, National Institute for Materials Science, Tsukuba, Japan

    Kenji Watanabe

  9. Henry Royce Institute for Advanced Materials, Manchester, UK

    Vladimir I. Fal’ko

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Contributions

J.V.J. and Z.G. conceived of the work and designed the research strategy. Z.G. fabricated the samples and performed data analysis under the supervision of J.V.J. Z.G. carried out tunnelling spectroscopy measurements with assistance from P.P. and under the supervision of J.V.J. A.M.G. and J.K.-R. performed quantum dynamics simulations under the supervision of E.J.H. Z.G. performed classical dynamics and tight-binding simulations with input from S.S. under the supervision of V.I.F. and J.V.J. K.W. and T.T. provided hBN crystals. R.V.H. and D.L. provided instrument support. Z.G., J.V.J., J.K.-R., A.M.G. and E.J.H. wrote the paper. All authors discussed the paper and commented on the paper.

Corresponding authors

Correspondence to Zhehao Ge or Jairo Velasco Jr.

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Extended data figures and tables

Extended Data Fig. 1 Raw dI/dVS (VS,d) data used to acquire Fig. 1e,f.

a-b, Experimentally measured dI/dVS (VS,d) at VG = −19 V for the same stadium-shaped GQD shown in Fig. 1 across its center along the horizontal (a) and vertical (b) directions. The inset shows the schematic of the measurement direction. The setpoint used to acquire the dI/dVS (VS,d) data was I = 1 nA, VS = −500 mV with a 10 mV ac modulation. Panels a and b reproduced from ref. 50 under a Creative Commons licence CC BY 4.0.

Extended Data Fig. 2 Extended dI/dVS map data at VG = −19 V from VS = −30 mV to −6 mV.

a-m, dI/dVS maps measured at VG = −19 V with different applied VS for the same in-situ created GQD shown in Fig. 3. The applied VS is shown at the top right corner of each dI/dVS map. A 2 mV ac modulation was used to acquire the dI/dVS maps. The red star marks the dI/dVS maps that show a ∞-shaped pattern with enhanced dI/dVS intensity. Panels a and g–l adapted from ref. 50 under a Creative Commons licence CC BY 4.0. Panel d reproduced from ref. 50 under a Creative Commons licence CC BY 4.0.

Extended Data Fig. 3 Extended dI/dVS map data at VG = −19 V from VS = 6 mV to 36 mV.

a-n, dI/dVS maps measured at VG = −19 V with different applied VS for the same in-situ created GQD shown in Fig. 3. The applied VS is shown at the top right corner of each dI/dVS map. A 2 mV ac modulation was used to acquire the dI/dVS maps. The green star marks the dI/dVS maps that show a streak-like pattern with enhanced dI/dVS intensity. Panels a–f, k and n adapted from ref. 50 under a Creative Commons licence CC BY 4.0.

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Ge, Z., Graf, A.M., Keski-Rahkonen, J. et al. Direct visualization of relativistic quantum scars in graphene quantum dots. Nature 635, 841–846 (2024). https://doi.org/10.1038/s41586-024-08190-6

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