JETP Letters

, Volume 105, Issue 8, pp 519–525 | Cite as

Lifshitz transitions via the type-II dirac and type-II Weyl points

  • K. ZhangEmail author
  • G. E. Volovik
Methods of Theoretical Physics


The type-II Weyl and type-II Dirac points emerge in semimetals and in relativistic systems. In particular, the type-II Weyl fermions may emerge behind the event horizon of black holes. The type-II Weyl and Dirac points also emerge as the intermediate states of the topological Lifshitz transitions. In one case, the type-II Weyl point connects the Fermi pockets, and the Lifshitz transition corresponds to the transfer of the Berry flux between the Fermi pockets. In the other case, the type-II Weyl point connects the outer and inner Fermi surfaces. At the Lifshitz transition, the Weyl point is released from both Fermi surfaces. They loose their Berry flux, which guarantees the global stability, and without the topological support, the inner surface disappears after shrinking to a point at the second Lifshitz transition. These examples reveal the complexity and universality of topological Lifshitz transitions, which originate from the ubiquitous interplay of a variety of topological characters of the momentum-space manifolds.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. Weyl, I. Z. Phys. 56, 330 (1929).ADSCrossRefGoogle Scholar
  2. 2.
    H. B. Nielsen and M. Ninomiya, Nucl. Phys. B 185, 20 (1981), Nucl. Phys. B 193,173 (1981).ADSCrossRefGoogle Scholar
  3. 3.
    G. E. Volovik, The Universe in a Helium Droplet (Clarendon, Oxford, 2003).zbMATHGoogle Scholar
  4. 4.
    J. von Neumann and E. Wigner, Phys. Z. 30, 467 (1929).Google Scholar
  5. 5.
    S. P. Novikov, Sov. Math. Dokl. 23, 298 (1981).Google Scholar
  6. 6.
    C. D. Froggatt and H. B. Nielsen, Origin of Symmetry (World Scientific, Singapore, 1991).CrossRefGoogle Scholar
  7. 7.
    P. Horava, Phys. Rev. Lett. 95, 016405 (2005).ADSCrossRefGoogle Scholar
  8. 8.
    B. Simon, Phys. Rev. Lett. 51, 2167 (1983).ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    G. E. Volovik, JETP Lett. 46, 98 (1987).ADSGoogle Scholar
  10. 10.
    T. D. C. Bevan, A. J. Manninen, J. B. Cook, J. R. Hook, H. E. Hall, T. Vachaspati, and G. E. Volovik, Nature 386, 689 (1997).ADSCrossRefGoogle Scholar
  11. 11.
    M. Krusius, T. Vachaspati, and G. E. Volovik, condmat/9802005.Google Scholar
  12. 12.
    G. E. Volovik, Physica B 255, 86 (1998); condmat/9802091.ADSCrossRefGoogle Scholar
  13. 13.
    C. Herring, Phys. Rev. 52, 365373 (1937).Google Scholar
  14. 14.
    A. A. Abrikosov and S. D. Beneslavskii, JETP 32, 699 (1971).ADSGoogle Scholar
  15. 15.
    A. A. Abrikosov, J. Low Temp. Phys. 5, 141 (1972).ADSCrossRefGoogle Scholar
  16. 16.
    H. B. Nielsen and M. Ninomiya, Phys. Lett. B 130, 389 (1983).ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    A. A. Burkov and L. Balents, Phys. Rev. Lett. 107, 127205 (2011).ADSCrossRefGoogle Scholar
  18. 18.
    A. A. Burkov, M. D. Hook, and L. Balents, Phys. Rev. B 84, 235126 (2011).ADSCrossRefGoogle Scholar
  19. 19.
    H. Weng, Ch. Fang, Zh. Fang, B. A. Bernevig, and X. Dai, Phys. Rev. X 5, 011029 (2015).Google Scholar
  20. 20.
    Sh.-M. Huang, S.-Y. Xu, I. Belopolski, Ch.-Ch. Lee, G. Chang, B. K. Wang, N. Alidoust, G. Bian, M. Neupane, Ch. Zhang, Sh. Jia, A. Bansil, H. Lin, and M. Z. Hasan, Nat. Commun. 6, 7373 (2015).CrossRefGoogle Scholar
  21. 21.
    B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, Phys. Rev. X 5, 031013 (2015).Google Scholar
  22. 22.
    S.-Y. Xu, I. Belopolski, N. Alidoust, et al., Science 349, 613 (2015).ADSCrossRefGoogle Scholar
  23. 23.
    L. Lu, Zh. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljacic, Science 349, 622 (2015).ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    M. Z. Hasan, S.-Y. Xu, I. Belopolski, and Sh.-M. Huang, arXiv:1702.07310.Google Scholar
  25. 25.
    G. E. Volovik and V. A. Konyshev, JETP Lett. 47, 250 (1988).ADSGoogle Scholar
  26. 26.
    V. Pardo and W. E. Pickett, Phys. Rev. Lett. 102, 166803 (2009).ADSCrossRefGoogle Scholar
  27. 27.
    S. Banerjee and W. E. Pickett, Phys. Rev. B 86, 075124 (2012).ADSCrossRefGoogle Scholar
  28. 28.
    A. A. Soluyanov, D. Gresch, Zh. Wang, Q. Sh. Wu, M. Troyer, X. Dai, and B. A. Bernevig, Nature 527, 495 (2015).ADSCrossRefGoogle Scholar
  29. 29.
    Y. Xu, F. Zhang, and Ch. Zhang, Phys. Rev. Lett. 115, 265304 (2015).ADSCrossRefGoogle Scholar
  30. 30.
    T.-R. Chang, S.-Y. Xu, G. Chang, et al., Nat. Commun. 7, 10639 (2016).ADSCrossRefGoogle Scholar
  31. 31.
    G. Autes, D. Gresch, A. A. Soluyanov, M. Troyer, and O. V. Yazyev, arXiv:1603.04624.Google Scholar
  32. 32.
    S.-Y. Xu, N. Alidoust, G. Chang, et al., arXiv:1603.07318.Google Scholar
  33. 33.
    J. Jiang, Z. K. Liu, Y. Sun, et al., arXiv:1604.00139.Google Scholar
  34. 34.
    T. E. O’Brien, M. Diez, and C. W. J. Beenakker, arXiv:1604.01028.Google Scholar
  35. 35.
    I. M. Lifshitz, Sov. Phys. JETP 11, 1130 (1960).Google Scholar
  36. 36.
    G. E. Volovik, J. Low Temp. Phys. 43, 47 (2017); arXiv:1606.08318; arXiv:1701.06435.CrossRefGoogle Scholar
  37. 37.
    P. Huhtala and G. E. Volovik, J. Exp. Theor. Phys. 94, 853 (2002); gr-qc/0111055.ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    G. E. Volovik and M. A. Zubkov, Nucl. Phys. B 881, 514 (2014).ADSCrossRefGoogle Scholar
  39. 39.
    F. R. Klinkhamer and G. E. Volovik, Int. J. Mod. Phys. A 20, 2795 (2005); hep-th/0403037.ADSCrossRefGoogle Scholar
  40. 40.
    G. E. Volovik, Lect. Notes Phys. 718, 31 (2007).ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    D. Gosalbez-Martinez, I. Souza, and D. Vanderbilt, Phys. Rev. B 92, 085138 (2015); arXiv:1505.07727.ADSCrossRefGoogle Scholar
  42. 42.
    J. W. McClure, Phys. Rev. 108, 612 (1957).ADSCrossRefGoogle Scholar
  43. 43.
    G. P. Mikitik and Yu. V. Sharlai, Phys. Rev. B 73, 235112 (2006).ADSCrossRefGoogle Scholar
  44. 44.
    G. P. Mikitik and Yu. V. Sharlai, Low Temp. Phys. 34, 794 (2008).ADSCrossRefGoogle Scholar
  45. 45.
    T. T. Heikkilä and G. E. Volovik, New J. Phys. 17, 093019 (2015).ADSCrossRefGoogle Scholar
  46. 46.
    J. Nissinen and G. E. Volovik, arXiv:1702.04624.Google Scholar
  47. 47.
    K. Zhang and G. E. Volovik, arXiv:1604.00849, JETP Lett. 105 (2017, in press).Google Scholar

Copyright information

© Pleiades Publishing, Inc. 2017

Authors and Affiliations

  1. 1.State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser SpectroscopyShanxi UniversityTaiyuanPeople’s Republic of China
  2. 2.Low Temperature LaboratoryAalto UniversityAaltoFinland
  3. 3.Landau Institute for Theoretical PhysicsRussian Academy of SciencesMoscowRussia

Personalised recommendations