Advertisement

Journal of Low Temperature Physics

, Volume 189, Issue 5–6, pp 276–299 | Cite as

Lifshitz Transitions, Type-II Dirac and Weyl Fermions, Event Horizon and All That

  • G. E. Volovik
  • K. Zhang
Article

Abstract

The type-II Weyl and type-II Dirac points emerge in semimetals and also in relativistic systems. In particular, the type-II Weyl fermions may emerge behind the event horizon of black holes. In this case the horizon with Painlevé–Gullstrand metric serves as the surface of the Lifshitz transition. This relativistic analogy allows us to simulate the black hole horizon and Hawking radiation using the fermionic superfluid with supercritical velocity, and the Dirac and Weyl semimetals with the interface separating the type-I and type-II states. The difference between such type of the artificial event horizon and that which arises in acoustic metric is discussed. At the Lifshitz transition between type-I and type-II fermions the Dirac lines may also emerge, which are supported by the combined action of topology and symmetry. The type-II Weyl and Dirac points also emerge as the intermediate states of the topological Lifshitz transitions. Different configurations of the Fermi surfaces, involved in such Lifshitz transition, are discussed. 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. For the interacting electrons, the Lifshitz transitions may lead to the formation of the dispersionless (flat) band with zero energy and singular density of states, which opens the route to room-temperature superconductivity. Originally, the idea of the enhancement of \(T_\mathrm{c}\) due to flat band has been put forward by the nuclear physics community, and this also demonstrates the close connections between different areas of physics.

Notes

Acknowledgements

We thank Ivo Souza for pointing out mistake in the early version and Tero Heikkilä for discussion on the type-II Dirac lines. The work by GEV has been supported by the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (Grant Agreement No. 694248). The work by KZ has been supported in part by the National Natural Science Foundation of China (NSFC) under Grant Nos. 11674200, 11422433 and 11604392.

References

  1. 1.
    H. Weyl, Elektron und Gravitation. I. Z. Phys. 56, 330–352 (1929)ADSCrossRefMATHGoogle Scholar
  2. 2.
    H.B. Nielsen, M. Ninomiya, Absence of neutrinos on a lattice. I—Proof by homotopy theory. Nucl. Phys. B 185, 20 (1981)ADSCrossRefMathSciNetGoogle Scholar
  3. 3.
    H.B. Nielsen, M. Ninomiya, Absence of neutrinos on a lattice. II—Intuitive homotopy proof. Nucl. Phys. B 193, 173 (1981)ADSCrossRefGoogle Scholar
  4. 4.
    G.E. Volovik, The Universe in a Helium Droplet (Clarendon Press, Oxford, 2003)MATHGoogle Scholar
  5. 5.
    J. von Neumann, E. Wigner, Über das Verhalten von Eigenwerten bei adiabatischen Prozessen. Phys. Z. 30, 467 (1929)MATHGoogle Scholar
  6. 6.
    S.P. Novikov, Magnetic Bloch functions and vector bundles. Typical dispersion laws and their quantum numbers. Sov. Math., Dokl. 23, 298–303 (1981)MATHGoogle Scholar
  7. 7.
    B. Simon, Holonomy, the quantum adiabatic theorem, and Berry’s phase. Phys. Rev. Lett. 51, 2167 (1983)ADSCrossRefMathSciNetGoogle Scholar
  8. 8.
    G.E. Volovik, Zeros in the fermion spectrum in superfluid systems as diabolical points. JETP Lett. 46, 98–102 (1987)ADSGoogle Scholar
  9. 9.
    T.D.C. Bevan, A.J. Manninen, J.B. Cook, J.R. Hook, H.E. Hall, T. Vachaspati, G.E. Volovik, Momentum creation by vortices in superfluid \(^{3}\)He as a model of primordial baryogenesis. Nature 386, 689–692 (1997)ADSCrossRefGoogle Scholar
  10. 10.
    M. Krusius, T. Vachaspati and G.E. Volovik, Flow instability in 3He-A as analog of generation of hypermagnetic field in early Universe, cond-mat/9802005 (1998)Google Scholar
  11. 11.
    G.E. Volovik, Axial anomaly in 3He-A: simulation of baryogenesis and generation of primordial magnetic field in Manchester and Helsinki. Phys. B 255, 86–107 (1998). cond-mat/9802091ADSCrossRefGoogle Scholar
  12. 12.
    C. Herring, Accidental degeneracy in the energy bands of crystals. Phys. Rev. 52, 365–373 (1937)ADSCrossRefGoogle Scholar
  13. 13.
    A.A. Abrikosov, S.D. Beneslavskii, Possible existence of substances intermediate between metals and dielectrics. JETP 32, 699–798 (1971)ADSGoogle Scholar
  14. 14.
    A.A. Abrikosov, Some properties of gapless semiconductors of the second kind. J. Low Temp. Phys. 5, 141–154 (1972)ADSCrossRefGoogle Scholar
  15. 15.
    H.B. Nielsen, M. Ninomiya, The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal. Phys. Lett. B 130, 389–396 (1983)ADSCrossRefMathSciNetGoogle Scholar
  16. 16.
    A.A. Burkov, L. Balents, Weyl semimetal in a topological insulator multilayer. Phys. Rev. Lett. 107, 127205 (2011)ADSCrossRefGoogle Scholar
  17. 17.
    A.A. Burkov, M.D. Hook, L. Balents, Topological nodal semimetals. Phys. Rev. B 84, 235126 (2011)ADSCrossRefGoogle Scholar
  18. 18.
    H. Weng, C. Fang, Z. Fang, B.A. Bernevig, X. Dai, Weyl semimetal phase in noncentrosymmetric transition-metal monophosphides. Phys. Rev. X 5, 011029 (2015)Google Scholar
  19. 19.
    S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang, B.K. Wang, N. Alidoust, G. Bian, M. Neupane, C. Zhang, S. Jia, A. Bansil, H. Lin, M.Z. Hasan, A Weyl fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class. Nat. Commun. 6, 7373 (2015)CrossRefGoogle Scholar
  20. 20.
    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, H. Ding, Experimental discovery of Weyl semimetal TaAs. Phys. Rev. X 5, 031013 (2015)Google Scholar
  21. 21.
    X. Su-Yang, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D.S. Sanchez, B.K. Wang, A. Bansil, F. Chou, P.P. Shibayev, H. Lin, S. Jia, M. Zahid Hasan, Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015)ADSCrossRefGoogle Scholar
  22. 22.
    L. Ling, Z. Wang, D. Ye, L. Ran, F. Liang, J.D. Joannopoulos, M. Soljacic, Experimental observation of Weyl points. Science 349, 622–624 (2015)ADSCrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    M.Z. Hasan, S.-Y. Xu, I. Belopolski, S.-M. Huang, Discovery of Weyl fermion semimetals and topological Fermi arc states (2017), arXiv:1702.07310
  24. 24.
    G.E. Volovik, V.A. Konyshev, Properties of the superfluid systems with multiple zeros in fermion spectrum. JETP Lett. 47, 250–254 (1988)ADSGoogle Scholar
  25. 25.
    V. Pardo, W.E. Pickett, Half-metallic semi-Dirac-point generated by quantum confinement in TiO\(_2\)/VO\(_2\) nanostructures. Phys. Rev. Lett. 102, 166803 (2009)ADSCrossRefGoogle Scholar
  26. 26.
    S. Banerjee, W.E. Pickett, Phenomenology of a semi-Dirac semi-Weyl semimetal. Phys. Rev. B 86, 075124 (2012)ADSCrossRefGoogle Scholar
  27. 27.
    A.A. Soluyanov, D. Gresch, Z. Wang, Q.S. Wu, M. Troyer, X. Dai, B.A. Bernevig, Type-II Weyl semimetals. Nature 527, 495–498 (2015)ADSCrossRefGoogle Scholar
  28. 28.
    X. Yong, F. Zhang, C. Zhang, Structured Weyl points in spin-orbit coupled fermionic superfluids. Phys. Rev. Lett. 115, 265304 (2015)ADSCrossRefGoogle Scholar
  29. 29.
    T.-R. Chang, S.-Y. Xu, G. Chang, C.-C. Lee, S.-M. Huang, B.K. Wang, G. Bian, H. Zheng, D.S. Sanchez, I. Belopolski, N. Alidoust, M. Neupane, A. Bansil, H.-T. Jeng, H. Lin, M. Zahid Hasan, Prediction of an arc-tunable Weyl Fermion metallic state in Mo\(_{x}\)W\(_{1-x}\)Te\(_{2}\). Nat. Commun. 7, 10639 (2016)ADSCrossRefGoogle Scholar
  30. 30.
    G. Autes, D. Gresch, A.A. Soluyanov, M. Troyer, O.V. Yazyev, Robust type-II Weyl semimetal phase in transition metal diphosphides XP\(_{2}\) (X = Mo, W). Phys. Rev. Lett. 117, 066402 (2016)ADSCrossRefGoogle Scholar
  31. 31.
    S.-Y. Xu, N. Alidoust, G. Chang, H. Lu, B. Singh, I. Belopolski, D.S. Sanchez, X. Zhang, G. Bian, H. Zheng, M.-A. Husanu, Y. Bian, S.-M. Huang, C.-H. Hsu, T.-R. Chang, H.-T. Jeng, A. Bansil, V.N. Strocov, H. Lin, S. Jia, M.Z. Hasan, Discovery (theoretical and experimental) of Lorentz-violating Weyl fermion semimetal state in LaAlGe materials. Sci. Adv. 3(6), e1603266 (2017)CrossRefADSGoogle Scholar
  32. 32.
    J. Jiang, Z.K. Liu, Y. Sun, H.F. Yang, R. Rajamathi, Y.P. Qi, L.X. Yang, C. Chen, H. Peng, C.-C. Hwang, S.Z. Sun, S.-K. Mo, I. Vobornik, J. Fujii, S.S.P. Parkin, C. Felser, B.H. Yan, Y.L. Chen, Observation of the type-II Weyl semimetal phase in MoTe\(_{2}\). Nat. Commun. 8, 13973 (2017)ADSCrossRefGoogle Scholar
  33. 33.
    I.M. Lifshitz, Anomalies of electron characteristics of a metal in the high pressure region. Sov. Phys. JETP 11, 1130 (1960)Google Scholar
  34. 34.
    G.E. Volovik, Topological Lifshitz transitions. Fizika Nizkikh Temperatur 43, 57–67 (2017). arXiv:1606.08318; Exotic Lifshitz transitions in topological materials, doi: 10.3367/UFNr.2017.01.038218, doi: 10.3367/UFNe.2017.01.038218, arXiv:1701.06435
  35. 35.
    P. Huhtala, G.E. Volovik, Fermionic microstates within Painlevé-Gullstrand black hole. ZhETF 121, 995–1003 (2002). JETP 94, 853-861; gr-qc/0111055Google Scholar
  36. 36.
    F.R. Klinkhamer, G.E. Volovik, Emergent CPT violation from the splitting of Fermi points. Int. J. Mod. Phys. A 20, 2795–2812 (2005). hep-th/0403037ADSCrossRefMATHGoogle Scholar
  37. 37.
    G.E. Volovik, Quantum phase transitions from topology in momentum space, in Quantum Analogues: From Phase Transitions to Black Holes and Cosmology, ed. by W.G. Unruh, R. Schützhold (Springer Lecture Notes in Physics, 2007), pp. 31–73Google Scholar
  38. 38.
    G.E. Volovik, M.A. Zubkov, Emergent Weyl spinors in multi-fermion systems. Nucl. Phys. B 881, 514 (2014)ADSCrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    C.D. Froggatt, H.B. Nielsen, Origin of Symmetry (World Scientific, Singapore, 1991)CrossRefGoogle Scholar
  40. 40.
    P. Hořava, Stability of Fermi surfaces and \(K\)-theory. Phys. Rev. Lett. 95, 016405 (2005)ADSCrossRefGoogle Scholar
  41. 41.
    D. Li, B. Rosenstein, B.Y. Shapiro, I. Shapiro, Effect of the type-I to type-II Weyl semimetal topological transition on superconductivity. Phys. Rev. B 95, 094513 (2017)ADSCrossRefGoogle Scholar
  42. 42.
    L. Kimme, T. Hyart, Existence of zero-energy impurity states in different classes of topological insulators and superconductors and their relation to topological phase transitions. Phys. Rev. B 93, 035134 (2016)ADSCrossRefGoogle Scholar
  43. 43.
    T.T. Heikkilä, G.E. Volovik, Nexus and Dirac lines in topological materials. New J. Phys. 17, 093019 (2015)ADSCrossRefGoogle Scholar
  44. 44.
    L.H. Kauffman, Knots and Physics (World Scientific, Singapore, 2001)CrossRefMATHGoogle Scholar
  45. 45.
    R. Bi, Z. Yan, L. Lu, Z. Wang, Nodal-knot semimetals (2017), arXiv:1704.06849
  46. 46.
    P. Painlevé, La mécanique classique et la théorie de la relativité. C. R. Hebd. Acad. Sci. (Paris) 173, 677–680 (1921)ADSMATHGoogle Scholar
  47. 47.
    A. Gullstrand, Allgemeine Lösung des statischen Einkörperproblems in der Einsteinschen Gravitationstheorie. Arkiv. Mat. Astron. Fys. 16, 1–15 (1922)MATHGoogle Scholar
  48. 48.
    W.G. Unruh, Experimental Black-Hole Evaporation. Phys. Rev. Lett. 46, 1351 (1981)ADSCrossRefGoogle Scholar
  49. 49.
    W.G. Unruh, Sonic analogue of black holes and the effects of high frequencies on black hole evaporation. Phys. Rev. D 51, 2827–2838 (1995)ADSCrossRefMathSciNetGoogle Scholar
  50. 50.
    P. Kraus, F. Wilczek, Some applications of a simple stationary line element for the Schwarzschild geometry. Mod. Phys. Lett. A 9, 3713–3719 (1994)ADSCrossRefMATHMathSciNetGoogle Scholar
  51. 51.
    C. Doran, New form of the Kerr solution. Phys. Rev. D 61, 067503 (2000)ADSCrossRefMathSciNetGoogle Scholar
  52. 52.
    A. Kostelecky, N. Russell, Data Tables for Lorentz and CPT Violation. Rev. Mod. Phys. 83, 11 (2011)ADSCrossRefGoogle Scholar
  53. 53.
    G. Rubtsov, P. Satunin, S. Sibiryakov, Constraints on violation of Lorentz invariance from atmospheric showers initiated by multi-TeV photons, arXiv:1611.10125
  54. 54.
    G.E. Volovik, Topological invariants for Standard Model: from semi-metal to topological insulator. JETP Lett. 91, 55–61 (2010). arXiv:0912.0502 ADSCrossRefGoogle Scholar
  55. 55.
    M. Visser, Acoustic black holes: horizons, ergospheres, and Hawking radiation. Class. Quant. Grav. 15, 1767–1791 (1998)ADSCrossRefMATHMathSciNetGoogle Scholar
  56. 56.
    G.E. Volovik, Simulation of Panleve–Gullstrand black hole in thin \(^3\)He-A film. JETP Lett. 69, 705–713 (1999)ADSCrossRefGoogle Scholar
  57. 57.
    M.K. Parikh, F. Wilczek, Hawking radiation as tunneling. Phys. Rev. Lett. 85, 5042 (2000)ADSCrossRefMATHMathSciNetGoogle Scholar
  58. 58.
    D. Gosalbez-Martinez, I. Souza, D. Vanderbilt, Chiral degeneracies and Fermi-surface Chern numbers in bcc Fe. Phys. Rev. B 92, 085138 (2015)ADSCrossRefGoogle Scholar
  59. 59.
    V.A. Khodel, V.R. Shaginyan, Superfluidity in system with fermion condensate. JETP Lett. 51, 553 (1990)ADSGoogle Scholar
  60. 60.
    G.E. Volovik, A new class of normal Fermi liquids. JETP Lett. 53, 222 (1991)ADSGoogle Scholar
  61. 61.
    P. Nozieres, Properties of Fermi liquids with a finite range interaction. J. Phys. (Fr.) 2, 443 (1992)ADSCrossRefGoogle Scholar
  62. 62.
    S.T. Belyaev, On the nature of the first excited states of even–even spherical nuclei. JETP 12, 968–976 (1961)Google Scholar
  63. 63.
    T.T. Heikkilä, G.E. Volovik, Flat bands as a route to high-temperature superconductivity in graphite, in: Basic Physics of Functionalized Graphite (Springer 2016, pp. 123–143), arXiv:1504.05824
  64. 64.
    A.A. Shashkin, V.T. Dolgopolov, J.W. Clark, V.R. Shaginyan, M.V. Zverev, V.A. Khodel, Merging of Landau levels in a strongly-interacting two-dimensional electron system in silicon. Phys. Rev. Lett. 112, 186402 (2014)ADSCrossRefGoogle Scholar
  65. 65.
    D. Yudin, D. Hirschmeier, H. Hafermann, O. Eriksson, A.I. Lichtenstein, M.I. Katsnelson, Fermi condensation near van Hove singularities within the Hubbard model on the triangular lattice. Phys. Rev. Lett. 112, 070403 (2014)ADSCrossRefGoogle Scholar
  66. 66.
    G.E. Volovik, On Fermi condensate: near the saddle point and within the vortex core. JETP Lett. 59, 830 (1994)ADSGoogle Scholar
  67. 67.
    A.P. Drozdov, M.I. Eremets, I.A. Troyan, Conventional superconductivity at 190 K at high pressures, IEEE/CSC & ESAS SUPERCONDUCTIVITY NEWS FORUM (global edition), January 2015, arXiv:1412.0460
  68. 68.
    A.P. Drozdov, M.I. Eremets, I.A. Troyan, V. Ksenofontov, S.I. Shylin, Conventional superconductivity at 203 K at high pressures. Nature 525, 73 (2015)ADSCrossRefGoogle Scholar
  69. 69.
    Y. Quan, W.E. Pickett, Impact of van Hove singularities in the strongly coupled high temperature superconductor H\(_3\)S. Phys. Rev. B 93, 104526 (2016)ADSCrossRefGoogle Scholar
  70. 70.
    A. Bianconi, T. Jarlborg, Superconductivity above the lowest Earth temperature in pressurized sulfur hydride. EPL 112, 37001 (2015)ADSCrossRefGoogle Scholar
  71. 71.
    A. Bianconi, T. Jarlborg, Lifshitz transitions and zero point lattice fluctuations in sulfur hydride showing near room temperature superconductivity. Novel Superconducting Materials 1, 15 (2015)ADSCrossRefGoogle Scholar
  72. 72.
    T. Jarlborg, A. Bianconi, Breakdown of the Migdal approximation at Lifshitz transitions with a giant zero-point motion in H\(_3\)S superconductor. Sci. Rep. 6, 24816 (2016)ADSCrossRefGoogle Scholar
  73. 73.
    A. Bussmann-Holder, J. Kohler, M.-H. Whangbo, A. Bianconi, A. Simon, High temperature superconductivity in sulfur hydride under ultrahigh pressure: A complex superconducting phase beyond conventional BCS. Nov. Supercond. Mater. 2, 37–42 (2016)Google Scholar
  74. 74.
    J.W. McClure, Band structure of graphite and de Haas-van Alphen effect. Phys. Rev. 108, 612–618 (1957)ADSCrossRefGoogle Scholar
  75. 75.
    G.P. Mikitik, Y.V. Sharlai, Band-contact lines in the electron energy spectrum of graphite. Phys. Rev. B 73, 235112 (2006)ADSCrossRefGoogle Scholar
  76. 76.
    G.P. Mikitik, Y.V. Sharlai, The Berry phase in graphene and graphite multilayers. Low Temp. Phys. 34, 780–794 (2008)CrossRefGoogle Scholar
  77. 77.
    T. Hyart, T.T. Heikkilä, Momentum-space structure of surface states in a topological semimetal with a nexus point of Dirac lines. Phys. Rev. B 93, 235147 (2016)ADSCrossRefGoogle Scholar
  78. 78.
    J. Nissinen, G.E. Volovik, Type-III and IV interacting Weyl points. JETP Lett. 105, 447–452 (2017)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

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

Personalised recommendations