Heller, E. J. Bound-state eigenfunctions of classically chaotic Hamiltonian systems: scars of periodic orbits. Phys. Rev. Lett. 53, 1515â1518 (1984).
Stöckmann, H.-J. Quantum Chaos: An Introduction (American Association of Physics Teachers, 2000).
Gutzwiller, M. C. Chaos in Classical and Quantum Mechanics Vol. 1 (Springer Science & Business Media, 2013).
Heller, E., Crommie, M., Lutz, C. & Eigler, D. Scattering and absorption of surface electron waves in quantum corrals. Nature 369, 464â466 (1994).
Crook, R. et al. Imaging fractal conductance fluctuations and scarred wave functions in a quantum billiard. Phys. Rev. Lett. 91, 246803 (2003).
Martins, F. et al. Imaging electron wave functions inside open quantum rings. Phys. Rev. Lett. 99, 136807 (2007).
Burke, A. et al. Periodic scarred states in open quantum dots as evidence of quantum Darwinism. Phys. Rev. Lett. 104, 176801 (2010).
Aoki, N. et al. Direct imaging of electron states in open quantum dots. Phys. Rev. Lett. 108, 136804 (2012).
Cabosart, D. et al. Recurrent quantum scars in a mesoscopic graphene ring. Nano Lett. 17, 1344â1349 (2017).
Ge, Z. et al. Imaging quantum interference in stadium-shaped monolayer and bilayer graphene quantum dots. Nano Lett. 21, 8993â8998 (2021).
Lee, J. et al. Imaging electrostatically confined Dirac fermions in graphene quantum dots. Nat. Phys. 12, 1032â1036 (2016).
Ge, Z. et al. Visualization and manipulation of bilayer graphene quantum dots with broken rotational symmetry and nontrivial topology. Nano Lett. 20, 8682â8688 (2020).
Huang, L., Lai, Y.-C., Ferry, D. K., Goodnick, S. M. & Akis, R. Relativistic quantum scars. Phys. Rev. Lett. 103, 054101 (2009).
Huang, L., Xu, H.-Y., Grebogi, C. & Lai, Y.-C. Relativistic quantum chaos. Phys. Rep. 753, 1â128 (2018).
Luukko, P. J. et al. Strong quantum scarring by local impurities. Sci. Rep. 6, 37656 (2016).
Keski-Rahkonen, J., Luukko, P. J., Kaplan, L., Heller, E. & Räsänen, E. Controllable quantum scars in semiconductor quantum dots. Phys. Rev. B 96, 094204 (2017).
Keski-Rahkonen, J., Ruhanen, A., Heller, E. & Räsänen, E. Quantum Lissajous scars. Phys. Rev. Lett. 123, 214101 (2019).
Xu, H., Huang, L., Lai, Y.-C. & Grebogi, C. Chiral scars in chaotic dirac fermion systems. Phys. Rev. Lett. 110, 064102 (2013).
Song, M.-Y., Li, Z.-Y., Xu, H.-Y., Huang, L. & Lai, Y.-C. Quantization of massive Dirac billiards and unification of nonrelativistic and relativistic chiral quantum scars. Phys. Rev. Res. 1, 033008 (2019).
Keski-Rahkonen, J., Graf, A. & Heller, E. Antiscarring in chaotic quantum wells. Preprint at https://arxiv.org/abs/2403.18081 (2024).
Berry, M. Quantum chaology, not quantum chaos. Phys. Scr. 40, 335 (1989).
Einstein, A. Zum quantensatz von Sommerfeld und Epstein. Verh. Dtsch. Phys. Ges. 19, 82â92 (1917).
Stone, A. D. Einsteinâs unknown insight and the problem of quantizing chaos. Phys. Today 58, 37 (2005).
Pilatowsky-Cameo, S. et al. Ubiquitous quantum scarring does not prevent ergodicity. Nat. Commun. 12, 852 (2021).
Hummel, Q., Richter, K. & Schlagheck, P. Genuine many-body quantum scars along unstable modes in BoseâHubbard systems. Phys. Rev. Lett. 130, 250402 (2023).
Evrard, B., Pizzi, A., Mistakidis, S. I. & Dag, C. B. Quantum scars and regular eigenstates in a chaotic spinor condensate. Phys. Rev. Lett. 132, 020401 (2024).
Bernien, H. et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature 551, 579â584 (2017).
Heller, E. J. The Semiclassical Way to Dynamics and Spectroscopy (Princeton Univ. Press, 2018).
Zelditch, S. Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55, 919â941 (1987).
Bohigas, O., Giannoni, M.-J. & Schmit, C. Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys. Rev. Lett. 52, 1 (1984).
Sridhar, S. Experimental observation of scarred eigenfunctions of chaotic microwave cavities. Phys. Rev. Lett. 67, 785 (1991).
Stein, J. & Stöckmann, H.-J. Experimental determination of billiard wave functions. Phys. Rev. Lett. 68, 2867 (1992).
Chinnery, P. A. & Humphrey, V. F. Experimental visualization of acoustic resonances within a stadium-shaped cavity. Phys. Rev. E 53, 272 (1996).
Kudrolli, A., Abraham, M. C. & Gollub, J. P. Scarred patterns in surface waves. Phys. Rev. E 63, 026208 (2001).
Manoharan, H., Lutz, C. & Eigler, D. Quantum mirages formed by coherent projection of electronic structure. Nature 403, 512â515 (2000).
Crommie, M. F., Lutz, C. P. & Eigler, D. M. Confinement of electrons to quantum corrals on a metal surface. Science 262, 218â220 (1993).
Ghahari, F. et al. An on/off Berry phase switch in circular graphene resonators. Science 356, 845â849 (2017).
Behn, W. A. et al. Measuring and tuning the potential landscape of electrostatically defined quantum dots in graphene. Nano Lett. 21, 5013â5020 (2021).
Ge, Z. et al. Giant orbital magnetic moments and paramagnetic shift in artificial relativistic atoms and molecules. Nat. Nanotechnol. 18, 250â256 (2023).
Zhao, Y. et al. Creating and probing electron whispering-gallery modes in graphene. Science 348, 672â675 (2015).
Gutiérrez, C., Brown, L., Kim, C.-J., Park, J. & Pasupathy, A. N. Klein tunnelling and electron trapping in nanometre-scale graphene quantum dots. Nat. Phys. 12, 1069â1075 (2016).
Zheng, Q., Zhuang, Y.-C., Sun, Q.-F. & He, L. Coexistence of electron whispering-gallery modes and atomic collapse states in graphene/WSe2 heterostructure quantum dots. Nat. Commun. 13, 1597 (2022).
Akis, R., Ferry, D. & Bird, J. Wave function scarring effects in open stadium shaped quantum dots. Phys. Rev. Lett. 79, 123 (1997).
Novoselov, K. S. et al. Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197â200 (2005).
Katsnelson, M. I., Novoselov, K. S. & Geim, A. K. Chiral tunnelling and the Klein paradox in graphene. Nat. Phys. 2, 620â625 (2006).
Berry, M. V. & Mondragon, R. Neutrino billiards: time-reversal symmetry-breaking without magnetic fields. Proc. R. Soc. Lond. A 412, 53â74 (1987).
Chen, S. et al. Electron optics with pn junctions in ballistic graphene. Science 353, 1522â1525 (2016).
Cao, H. & Wiersig, J. Dielectric microcavities: model systems for wave chaos and non-Hermitian physics. Rev. Mod. Phys. 87, 61â111 (2015).
Young, A. F. & Kim, P. Quantum interference and Klein tunnelling in graphene heterojunctions. Nat. Phys. 5, 222â226 (2009).
Ge, Z. Wavefunction Mapping and Magnetic Field Response of Electrostatically Defined Graphene Quantum Dots. PhD thesis, Univ. California, Santa Cruz (2023).
Zomer, P., Dash, S., Tombros, N. & Van Wees, B. A transfer technique for high mobility graphene devices on commercially available hexagonal boron nitride. Appl. Phys. Lett. 99, 232104 (2011).
Goossens, A. et al. Mechanical cleaning of graphene. Appl. Phys. Lett. 100, 073110 (2012).
Ge, Z. et al. Source data for âDirect visualization of relativistic quantum scarsâ. Zenodo. https://doi.org/10.5281/zenodo.13751637 (2024).