Log-periodic quantum oscillations in topological or Dirac materials

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  1. 1.

    L. Schubnikow and W. J. De Haas, A new phenomenon in the change of resistance in a magnetic field of single crystals of bismuth, Nature126(3179), 500 (1930)

    ADS  Article  Google Scholar 

  2. 2.

    D. Schoenberg, Magnetic Oscillations in Metals, Cambridge University Press, 1984

    Google Scholar 

  3. 3.

    Y. Imry, Introduction to Mesoscopic Physics, Oxford University Press, 1997

    Google Scholar 

  4. 4.

    W. J. de Haas and P. M. van Alphen, The dependence of the susceptibility of diamagnetic metals upon the field, Proc. Netherlands Roy. Acad. Sci. 33(1106), 170 (1930)

    MATH  Google Scholar 

  5. 5.

    R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, Observation of h/e Aharonov–Bohm oscillations in normal-metal rings, Phys. Rev. Lett. 54(25), 2696 (1985)

    ADS  Article  Google Scholar 

  6. 6.

    B. L. Al’Tshuler, A. G. Aronov, and B. Z. Spivak, The Aaronov-Bohm effect in disordered conductors, JETP Lett. 33(2), 94 (1981)

    ADS  Google Scholar 

  7. 7.

    W. Gao, N. Hao, F. W. Zheng, W. Ning, M. Wu, X. Zhu, G. Zheng, J. Zhang, J. Lu, H. Zhang, C. Xi, J. Yang, H. Du, P. Zhang, Y. Zhang, and M. Tian, Extremely large magnetoresistance in a topological semimetal candidate pyrite PtBi2, Phys. Rev. Lett. 118(25), 256601 (2017)

    ADS  Article  Google Scholar 

  8. 8.

    Y. Zhao, H. Liu, C. Zhang, H. Wang, J. Wang, Z. Lin, Y. Xing, H. Lu, J. Liu, Y. Wang, S. M. Brombosz, Z. Xiao, S. Jia, X. C. Xie, and J. Wang, Anisotropic fermi surface and quantum limit transport in high mobility threedimensional Dirac semimetal Cd3As2, Phys. Rev.. 5(3), 031037 (2015)

    Article  Google Scholar 

  9. 9.

    A. B. Fowler, F. F. Fang, W. E. Howard, and P. J. Stiles, Magneto-oscillatory conductance in silicon surfaces, Phys. Rev. Lett. 16(20), 901 (1966)

    ADS  Article  Google Scholar 

  10. 10.

    Z. Xiang, Y. Kasahara, T. Asaba, B. Lawson, C. Tinsman, L. Chen, K. Sugimoto, S. Kawaguchi, Y. Sato, G. Li, S. Yao, Y. L. Chen, F. Iga, J. Singleton, Y. Matsuda, and L. Li, Quantum oscillations of electrical resistivity in an insulator, Scienc. 362(6410), 65 (2018)

    ADS  Article  Google Scholar 

  11. 11.

    M. Tian, J. Wang, Q. Zhang, N. Kumar, T. E. Mallouk, and M. H. W. Chan, Superconductivity and quantum oscillations in crystalline Bi nanowire, Nano Lett. 9(9), 3196 (2009)

    ADS  Article  Google Scholar 

  12. 12.

    J. Wang, X. C. Ma, L. Lu, A. Z. Jin, C. Z. Gu, X. C. Xie, J. F. Jia, X. Chen, and Q. K. Xue, Anomalous magnetoresistance oscillations and enhanced superconductivity in single-crystal Pb nanobelts, Appl. Phys. Lett. 92(23), 233119 (2008)

    ADS  Article  Google Scholar 

  13. 13.

    H. Wang, H. Liu, Y. Li, Y. Liu, J. Wang, J. Liu, Ji-Yan Dai, Y. Wang, L. Li, J. Yan, D. Mandrus, X. C. Xie, and J. Wang, Discovery of log-periodic oscillations in ultraquantum topological materials, Sci. Adv.4(11), eaau5096 (2018)

    Google Scholar 

  14. 14.

    H. Weng, X. Dai, and Z. Fang, Transition-metal pentatelluride ZrTe5 and HfTe5: A paradigm for large-gap quantum spin Hall insulators, Phys. Rev. X4(1), 011002 (2014)

    Google Scholar 

  15. 15.

    Q. Li, D. E. Kharzeev, C. Zhang, Y. Huang, I. Pletikosić, A. V. Fedorov, R. D. Zhong, J. A. Schneeloch, G. D. Gu, and T. Valla, Chiral magnetic effect in ZrTe5, Nat. Phys. 12(6), 550 (2016)

    Article  Google Scholar 

  16. 16.

    R. Y. Chen, Z. G. Chen, X. Y. Song, J. A. Schneeloch, G. D. Gu, F. Wang, and N. L. Wang, Magnetoinfrared spectroscopy of Landau levels and Zeeman splitting of threedimensional massless Dirac fermions in ZrTe5, Phys. Rev. Lett. 115(17), 176404 (2015)

    ADS  Article  Google Scholar 

  17. 17.

    Z. Fan, Q. F. Liang, Y. B. Chen, S. H. Yao, and J. Zhou, Transition between strong and weak topological insulator in ZrTe5 and HfTe5, Sci. Rep. 7(1), 45667 (2017)

    ADS  Article  Google Scholar 

  18. 18.

    B. I. Halperin, Possible states for a three-dimensional electron gas in a strong magnetic field, Jpn. J. Appl. Phys. 26, 1913 (1987)

    Article  Google Scholar 

  19. 19.

    Y. Liu, X. Yuan, C. Zhang, Z. Jin, A. Narayan, C. Luo, Z. Chen, L. Yang, J. Zou, X. Wu, S. Sanvito, Z. Xia, L. Li, Z. Wang, and F. Xiu, Zeeman splitting and dynamical mass generation in Dirac semimetal ZrTe5, Nat. Commun. 7(1), 12516 (2016)

    ADS  Article  Google Scholar 

  20. 20.

    B. Fauqué, D. LeBoeuf, B. Vignolle, M. Nardone, C. Proust, and K. Behnia, Two phase transitions induced by a magnetic field in graphite, Phys. Rev. Lett. 110(26), 266601 (2013)

    ADS  Article  Google Scholar 

  21. 21.

    D. Sornette, Discrete-scale invariance and complex dimensions, Phys. Rep. 297(5), 239 (1998)

    ADS  MathSciNet  Article  Google Scholar 

  22. 22.

    L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd Ed., Perganon Press, 1977

    Google Scholar 

  23. 23.

    T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola, H. C. Nägerl, and R. Grimm, Evidence for Efimov quantum states in an ultracold gas of caesium atoms, Natur. 440(7082), 315 (2006)

    ADS  Article  Google Scholar 

  24. 24.

    B. Huang, L. A. Sidorenkov, R. Grimm, and J. M. Hutson, Observation of the second triatomic resonance in Efimov’s scenario, Phys. Rev. Lett. 112(19), 190401 (2014)

    ADS  Article  Google Scholar 

  25. 25.

    R. Pires, J. Ulmanis, S. Häfner, M. Repp, A. Arias, E. D. Kuhnle, and M. Weidemüller, Observation of Efimov resonances in a mixture with extreme mass imbalance, Phys. Rev. Lett. 112(25), 250404 (2014)

    ADS  Article  Google Scholar 

  26. 26.

    S. K. Tung, K. Jiménez-García, J. Johansen, C. V. Parker, and C. Chin, Geometric scaling of Efimov states in a 6Li-133Cs mixture, Phys. Rev. Lett. 113(24), 240402 (2014)

    ADS  Article  Google Scholar 

  27. 27.

    M. Kunitski, S. Zeller, J. Voigtsberger, A. Kalinin, L. Ph. H. Schmidt, M. Schöffler, A. Czasch, W. Schöllkopf, R. E. Grisenti, T. Jahnke, D. Blume, and R. Dörner, Observation of the Efimov state of the helium trimer, Scienc. 348(6234), 551 (2015)

    ADS  Article  Google Scholar 

  28. 28.

    Y. B. Zeldovich and V. S. Popov, Electronic structure of superheavy atoms, Sov. Phys. Usp. 14(6), 673 (1972)

    ADS  Article  Google Scholar 

  29. 29.

    W. Greiner, B. Muller, and J. Rafelski, Quantum Electrodynamics of Strong Fields, Springer Science & Business Media, 2013

    Google Scholar 

  30. 30.

    D. Kennedy and C. Norman, So much more to know, Science309(5731), 78b (2005)

    Article  Google Scholar 

  31. 31.

    H. Liu, H. Jiang, Z. Wang, R. Joynt, and X. C. Xie, Discrete scale invariance in topological semimetals, arXiv: 1807.02459 (2018)

    Google Scholar 

  32. 32.

    P. Zhang and H. Zhai, Efimov effect in the Dirac Semimetals, Front. Phys. 13(5), 137204 (2018)

    Article  Google Scholar 

  33. 33.

    H. Wang, Y. Liu, Y. Liu, C. Xi, J. Wang, J. Liu, Y. Wang, L. Li, S. P. Lau, M. Tian, J. Yan, D. Mandrus, J.-Y. Dai, H. Liu, X. C. Xie, and J. Wang, Log-periodic quantum magneto-oscillations and discrete scale invariance in topological material HfTe 5, arXiv: 1810.03109 (2018)

    Google Scholar 

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We thank X. C. Xie for theoretical contribution. This work was financially supported by the National Key Research and Development Program of China (Grant Nos. 2018YFA0305604 and 2017YFA0303300), the National Natural Science Foundation of China (Grant No. 11774008), and the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB28000000).

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Correspondence to Jian Wang.

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Wang, H., Liu, Y., Liu, H. et al. Log-periodic quantum oscillations in topological or Dirac materials. Front. Phys. 14, 23201 (2019). https://doi.org/10.1007/s11467-018-0878-8

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