Quasi-Dirac points in electron-energy spectra of crystals

In topological Dirac semimetals, two electron energy bands contact at discrete (Dirac) points of the Brillouin zone and disperse linearly in all directions around these nodes. The Dirac points determine specific properties of these semimetals, such as the presence of the surface Fermi arcs, the phase shift of the quantum oscillations, features of the magnetic susceptibility, the magnetostriction,  and of other responses to the magnetic field.  These properties distinguish the Dirac semimetals from ordinary materials.  This work shows that almost the same properties are inherent in a wider class of materials in which the Dirac spectrum can have a noticeable gap comparable with the Fermi energy (see Figure). In other words, the degeneracy of the bands at the point and their linear dispersion are not necessary for the existence of these properties. The only sufficient condition is the following: In the vicinity of such a quasi-Dirac point, the two close bands are well described by a two-band model that takes into account the strong spin-orbit interaction. In the article, the spectra described by this model are called the quasi-Dirac spectra. The true Dirac spectrum is the special case of this model. 

Figure. Dispersion of the Dirac and quasi-Dirac electrons along the rotation axis in the vicinity of the Dirac point and quasi-Dirac point, respectively. Both the spectra are slightly tilted along the axis. For the quasi-Dirac electrons, the Fermi energy measured from the edge of the band can be less than the gap, and then, their dispersion significantly deviates from the linear dependence.

 Conditions under which the quasi-Dirac spectra can appear in real crystals are thoroughly discussed in the paper.  Neglecting  the other bands in the two-band model means the following:  (i) the energy gap 2Δ between the two bands under study is noticeably less than the energy spacing between these two bands and the remaining remote bands,(ii) the strength of the spin-orbit interaction in the crystal is larger than 2Δ. These are the necessary conditions for the quasi-Dirac spectrum to occur. An analysis carried out in the article also shows that similar to the case of the true Dirac points, the quasi-Dirac points can have the following positions in the Brillouin zone: (i) Points with time reversal invariant momenta. (In particular, doped topological and conventional insulators with a narrow gap fall into this class.)  (ii) Two symmetrically located points in a rotation axis of a crystal. (Such a pair of the points is created by the progressive inversion of the two bands in the axis.)
The exact spectrum in the magnetic field H (i.e., the Landau levels) for charge carriers located near the quasi-Dirac point was obtained in the article [1] published long before the discovery of the Dirac semimetals. The important point of Ref. [1] is that the exact Landau levels for all their quantum numbers coincide with the levels obtained from the Onsager-Lifshitz semiclassical quantization rule.  It follows from this “semiclassical representation” of the exact spectrum that the Landau levels for the charge carriers located near the quasi-Dirac and true Dirac points are very similar. In particular, the lowest Landau level for the Dirac electrons does not depend on H, and this characteristic feature persists when a gap appears in the spectrum.  It this similarity of the Landau levels that leads to a practical coincidence of the physical properties determined by the Dirac and quasi-Dirac points.
One of the questions thoroughly discussed in the article is the following: How can the quasi-Dirac spectra be detected with the quantum-oscillation phenomena, e.g., with the de Haas-van Alphen and Shubnikov-de Haas effects? Measurements of the phase φ of the quantum oscillations are often used to distinguish the Dirac electrons from the conventional charge carriers. This φ is determined by the so-called Berry phase ΦB of electron orbit and an interband part Linter of the electron orbital moment. (If one considers a semiclassical electron as a wave packet, this Linter can be interpreted as the moment associated with self-rotation of the wave packet around its center of mass.) For the Dirac electrons with tilted spectrum, the Berry phase, in general, is not equal to a quantized value, and the term associated with Linter  does not vanish in φ.  These results are usually used to explain deviations of the measured φ from a quantized value when such deviations are really observed in experiments with the Dirac semimetals. However, the exact spectrum for the quasi-Dirac electrons in the magnetic field shows that although the two contributions to φ can have various values for these electrons, the total phase of the quantum oscillations always takes on the universal value. In other words, this phase is the distinctive property of the quasi- Dirac and true Dirac charge carriers, but  the measurements of the phase do not distinguish between them. Therefore, the main cause of the difference of the experimental phase from the universal value is due to deviations of the real spectrum of a material from the two-band model.
Electron spectrum of ZrTe5
To illustrate the obtained results, the spectrum of ZrTe5 is considered in the article. Analyzing the published experimental data [2] on the Shubnikov-de Haas effect, we conclude that the spectrum of this layering material has the quasi-Dirac form only in the plane of its layers. In the direction perpendicular to them, the real charge-carrier dispersion deviates from the quasi-Dirac dispersion. The example of ZrTe5 also shows how such a deviation can be taken into account to describe experimental data. As a result, a simple model for the spectrum of ZrTe5 is presented. This model generalizes the spectra proposed in papers [3,4], and the model parameters are estimated, using the data [2]. Interestingly, the presented model is equivalent to the model of McClure [5] for the electrons in Bi. This equivalence means that results obtained for Bi can be extended to the case of ZrTe5. 
[1] Mikitik, G.P. and  Sharlai, Yu.V. Low Temp. Phys. 22, 585-592 (1996).     The pdf file of this paper is available  here
[2] Yuan, X. et al. NPG Asia Materials 8, e235 (2016).
[3] Martino, E. et al. Phys. Rev. Lett. 122, 217402 (2019).
[4] Jiang, Y. et al. Phys. Rev. Lett. 125, 046403 (2020).
[5] McClure, J.W.    J. Low Temp. Phys. 25, 527-540 (1976).