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


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.



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.


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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

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