Quasi-Dirac points in electron-energy spectra of crystals

Abstract

Specific properties, such as surface Fermi arcs, features of quantum oscillations and of various responses to a magnetic field, distinguish Dirac semimetals from ordinary materials. These properties are determined by Dirac points at which a contact of two electron-energy bands occurs and in the vicinity of which these bands disperse linearly in the quasimomentum. 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. 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. To illustrate the results, the spectrum of ZrTe₅ is considered. This spectrum contains a special quasi-Dirac point, similar to that in bismuth.

Introduction

In recent years much attention has been given to the so-called Dirac semimetals; see, e.g., reviews^1,2. In these 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 can exist in centrosymmetric crystals with a strong spin-orbit (SO) interaction. All the bands of such crystals are double degenerate in spin, and the Dirac points can have the following positions in the Brillouin zone: (i) Points with time reversal invariant momenta (TRIM) when some variable control parameter x is equal to a specific value x₀³. This case is realized, e.g., in the Bi_1−xSb_x alloys at the point L of their Brillouin zone, with the parameter x being the concentration of Sb, and x₀ ≈ 0.04⁴. (ii) Two symmetrically located points in a 3, 4, or 6-fold rotation axis when x lies in a certain interval of its values⁵. Such a pair of the points is created by the progressive inversion of the two bands in the axis, and the gap between the inverted bands in the middle of the axis can be considered as the parameter x. These pairs of the Dirac points were experimentally discovered in Na₃Bi⁶ and Cd₃As₂^7,8. (iii) TRIM on the Brillouin-zone surfaces of non-symmorphic crystals when the symmetry-enforced degeneracy of the bands occurs at such points⁵.

The Dirac points determine the special properties of the Dirac semimetals. In particular, the band inversion that produces a couple of the Dirac points also leads to the appearance of the surface Fermi arcs⁵. Such arcs were observed in Na₃Bi, using the angle-resolved photoemission spectroscopy⁹. Besides, the phase of quantum oscillations are widely used to detect the Dirac fermions (see, e.g., review article¹⁰) since this phase is noticeably affected by the Berry curvature produced by the Dirac points. The Dirac spectrum also manifests itself in the magneto-optical conductivity^11,12, the magnetic susceptibility and the magnetic torque^10,13,14,15, the magnetostriction¹⁶, and in the temperature correction to the quantum-oscillation frequency^17,18.

However, in contrast to the Weyl nodes, the Dirac points have zero Chern number¹, and so they are not protected by this topological invariant. Therefore, a small gap can appear in the Dirac spectrum under a little variation of the crystal potential. In particular, this gap appears when the above-mentioned parameter x slightly deviates from the value x₀, or if a uniaxial stress decreases the symmetry of the rotation axis, in which the two Dirac points lie. (The Dirac points cannot occur in the 2-fold symmetry axis⁵). Moreover, even without any external stress, if the magnetic field is not aligned with the axis of the two Dirac points, the magnetostriction of the crystal leads to its deformation. This deformation lifts the band degeneracy at the points. Thus, the existence of a small gap in the Dirac spectrum must be kept in mind when analyzing various experiments with the Dirac semimetals. Besides, even if the Dirac points are absent in a crystal, two bands can approach each other in a rotation axis without their crossing (see Results). If the gap between these bands is essentially smaller than the energy spacings separating them from the other bands in this region of the Brillouin zone, the electron energy spectrum can be described by a two-band k ⋅ p model. In crystals with strong SO interaction, this model reduces to the tilted Dirac spectrum with the gap. These situations for which the crossing of the two approaching bands is forbidden by the rotation symmetry are no less common than the case of the ordinary Dirac nodes. We will call all such gapped spectra the quasi-Dirac spectra and the point at which the direct gap reaches its minimal value will be named the quasi-Dirac (QD) point. Hence, the QD points are a generalization of the usual Dirac nodes.

In this article, we draw attention to the fact that the quasi-Dirac and Dirac spectra lead to almost identical physical properties of crystals. It is necessary to emphasize that this statement, which is intuitively evident for the case of a small gap as compared to the Fermi energy reckoned from the edge of one of the two close bands, is true even if the gap is essentially larger than this Fermi level, and therefore if the dispersion of the bands is not linear near the Fermi energy. We discuss in detail conditions under which this QD spectrum can appear in real crystals. The properties of the quasi-Dirac and Dirac points distinguish the charge carriers near these points from the carriers with other types of their spectra. It is also worth noting that for a QD point at a time reversal invariant momentum, the quasi-Dirac spectrum with a variable gap can describe the topological transition from an ordinary insulator to the topological one. As an example, we analyze the spectrum of ZrTe₅, in which a special quasi-Dirac point occurs. The specificity of this point is due to the layering of this material. Using published experimental data on the Shubnikov-de Haas oscillations in ZrTe₅, a simple (minimal) model of its spectrum is proposed, and the model parameters are estimated.

Results

The QD spectrum

Consider the dispersion of two charge-carrier bands c and v in the vicinity of a point where they approach each other. This dispersion follows from a 4 × 4 k ⋅ p Hamiltonian and can always be reduced to the form¹⁰:

$${\varepsilon }_{c,v}({{{\bf{p}}}})={\varepsilon }_{{{{\rm{D}}}}}+{{{\bf{a}}}}{{{\bf{p}}}}\pm \sqrt{{\Delta }^{2}+{v}_{x}^{2}{p}_{x}^{2}+{v}_{y}^{2}{p}_{y}^{2}+{v}_{z}^{2}{p}_{z}^{2}}.$$

(1)

Here the quasimomentum p is measured from the point (the QD point) where the direct gap in the spectrum is minimal, 2Δ is the value of this minimal gap, the constants v_i and a = (a_x, a_y, a_z) are the matrix elements of the velocity operator. At Δ = 0, formula (1) describes the case of the Dirac spectrum, and ε_D is the energy of the Dirac point. When a = 0, equation (1) is similar to the well-known Dirac equation for relativistic particles, with Δ playing the role of the mass term¹⁹. The parameter a specifies the so-called tilt of the Dirac spectrum. Such a tilt is absent for real relativistic particles, but for the pair of the Dirac points lying in the rotation axis, the vector a is aligned with this axis and differs from zero. At a nonzero a, the minimal gap in spectrum (1) is not direct, and it is equal to $2| {\Delta }_{\min }|$ (Fig. 1) where $| {\Delta }_{\min }| =| \Delta | (1-{\tilde{a}}^{2}),{\tilde{a}}^{2}={\tilde{a}}_{x}^{2}+{\tilde{a}}_{y}^{2}+{\tilde{a}}_{z}^{2}$, and ${\tilde{a}}_{i}\equiv {a}_{i}/{v}_{i}$. Below we always assume that the rotation axis coincides with the z axis, i.e., a = (0, 0, a_z), and we also imply that ${\tilde{a}}^{2} < 1$ (this condition means that the Dirac point is of the type I²⁰). The Fermi energy ε_F may have an arbitrary position relative to the edges ${\varepsilon }_{{{{\rm{D}}}}}\pm {\Delta }_{\min }$ of the bands, and if ${\varepsilon }_{{{{\rm{F}}}}}-{\varepsilon }_{{{{\rm{D}}}}} \sim \pm {\Delta }_{\min }$, the dispersion strongly deviates from the linear law.

Fig. 1: Dispersion of the two energy bands along the rotation axis in the vicinity of the quasi-Dirac point.

These bands ε_c(p) and ε_v(p) (red solid lines) are described by Eq. (1). The quasi-Dirac point (p = 0) lies in the rotation axis z. The dashed lines corresponds to the case of the Dirac point (Δ = 0). The red circles mark the minimum of ε_c(p) and the maximum of ε_v(p). The minimal indirect gap $2{\Delta }_{\min }=2\Delta {(1-{\tilde{a}}_{z}^{2})}^{1/2}$ determined by these two points is less than 2Δ, the direct gap at p = 0. Here ${v}_{x}={v}_{y}\equiv {v}_{\perp },{v}_{z}/{v}_{\perp }=0.4,{\tilde{a}}_{z}=0.5$. The dotted lines show the Landau subbands l = 1, 2 at $2e\hslash {v}_{\perp }^{2}H/(c{\Delta }^{2})=2$. Inset: The cross section (ellipse) of the Fermi surface by the plane p_y = 0 at (ε_F − ε_D)/Δ = 2. The black dashed line marks the extremal cross section of the Fermi surface when the magnetic field lying in this plane is directed at the angle θ = π/4 to the z axis. This crosssection does not pass through the quasi-Dirac point p = 0 (black circle) at which the direct gap is minimal. The cross marks the center of the ellipse.

Formula (1) gives the strict definition of the QD spectrum. Now the question arises: What are conditions for existence of this spectrum in real crystals? Note that Eq. (1) does not contain quadratic in p_i terms in front of the square root and the terms ${p}_{i}^{n}$ with n > 2 under the radical. This means that the effect of other bands on dispersion (1) is negligible. This neglect is generally justified if (i) Δ is noticeably less than the energy spacing between ε_D and the remote bands at the point p = 0, and (ii) the strength of the spin-orbit interaction in the crystal, Δ_SO, is significantly larger than Δ. The latter condition usually ensures non-small values of all three v_i (at such v_i, the terms of higher orders in p_i are relatively small). The strength of the SO coupling is also important for the spectrum of the charge carriers in the magnetic field. (This spectrum depends both on dispersion (1) and on the g factor of electron orbits; see below.) The value of Δ_SO can be estimated from the band-structure calculations as a characteristic shift of the bands when the SO interaction is taken into account. If Δ itself has the spin-orbit origin, and Δ_SO ~ Δ, dispersion (1) usually occurs only in the planes perpendicular to a certain line in the Brillouin zone (see Discussion).

Consider now how the quasi-Dirac spectra appear in crystals. As was mentioned in the Introduction, they can occur if a small gap develops in the true Dirac spectrum. However, like the Dirac points, pairs of the QD points can exist in n-fold rotation axes when the control parameter x lies in its certain interval (Fig. 2). In a centrosymmetric crystal, all the electron states are doubly degenerate in spin. In the axes, these degenerate states are invariant under rotations through 2π/n angle. They are multiplied by factors u_c and ${u}_{c}^{* }$ for the band c and by u_v and ${u}_{v}^{* }$ for the band v when such a rotation occurs⁵. Here u_c and u_v are complex numbers (∣u_c∣ = ∣u_v∣ = 1), and the asterisk marks the complex conjugate value. For the Dirac point to occur in the axis, the pairs $({u}_{c},{u}_{c}^{* })$ and $({u}_{v},{u}_{v}^{* })$ have to be different, otherwise, the states of the bands c and v are “repulsed” from each other⁵. Since for a 2-fold rotation axis, the only possible pair is (i, − i), the Dirac points cannot appear in this case. The repulsion of the states can also occur in the 3, 4, and 6-fold axes if the pairs coincide for the bands c and v, i.e., the Dirac points cannot appear after the inversion of such bands. It turns out that in this case, the quasi-Dirac points arise; see Supplementary Note 1. In other words, the inversion of the bands in any rotation axis yields either the Dirac or quasi-Dirac points, and we expect the QD points are fairly common in crystals. In particular, in the 2-fold axis, only the pair of the quasi-Dirac points can occur.

Fig. 2: The two energy bands in the vicinity of the middle of a 2-fold rotation axis.

The rotation axis coincides with the z axis (with p_z = 0 being its middle). In the construction of the figure, dispersion relation (12) with the first set of the parameters from Table 2 has been used. The solid lines show the case of ZrTe₅ (Δ ≈ 0); the dotted line marks the position of the Fermi level in this material (ε_F ≈ 36.1 meV, mv_xv_y ≈ 1.06 eV). Even below this level, a deviation of the dispersion from the linear law is visible. The black and red dashed lines correspond to the bands when Δ/(mv_xv_y) = 0.1 and Δ/(mv_xv_y) = − 0.065 (i.e., before and after the band inversion), respectively. At $\Delta < {\Delta }_{{{{\rm{cr}}}}}=-{v}_{z}^{2}/({\alpha }_{c}+{\alpha }_{v})\approx -11.4$ meV, instead of the quasi-Dirac point with the gap 2∣Δ∣ at p_z = 0, the two quasi-Dirac points with the gap $2{[{\Delta }_{{{{\rm{cr}}}}}(2\Delta -{\Delta }_{{{{\rm{cr}}}}})]}^{1/2}$ exist at ${p}_{z}^{2}=2({\Delta }_{{{{\rm{cr}}}}}-\Delta )/({\alpha }_{c}+{\alpha }_{v})$.

The QD spectrum at nonzero magnetic fields

In the magnetic field H, the exact spectrum for the particles with dispersion relation (1) is well known for the case of true relativistic electrons when a = 0, Δ ≠ 0¹⁹. For the tilted dispersion when $0 < {\tilde{a}}^{2} < 1$, the spectrum was obtained at Δ = 0^21,22,23,24 and Δ ≠ 0²¹. It is important that this spectrum ε^(l)(p_n) can be described with the equation²¹,

$${S}_{c,v}({\varepsilon }^{(l)},{p}_{n})=\frac{2\pi \hslash eH}{c}l,$$

(2)

where e is the absolute value of the electron charge; l = 0, 1, 2, … ; S_c,v(ε^(l), p_n) is the area of the cross section of the constant-energy surface ε_c,v(p) = ε^(l) by the plane p_n = const.; the Landau levels (subbands) ε^(l)(p_n) in Eq. (2) are double degenerate in spin for all l ≠ 0, but the levels l = 0 for the bands c and v are nondegenerate. Here p_n is the component of the quasimomentum along the magnetic field. Note that in deriving formula (2), the Zeeman term describing the direct interaction of the electron spin with the magnetic field was disregarded.

Formula (2) looks like the semiclassical quantization condition, which, for a band doubly degenerate in spin, reads²⁵:

$$S({\varepsilon }^{(l)},{p}_{n})=\frac{2\pi \hslash eH}{c}\left(l+\frac{1}{2}\pm \frac{g{m}_{* }}{4m}\right),$$

(3)

where g is the g factor of a semiclassical orbit in the magnetic field, m_∗ is the cyclotron mass of this orbit, and m is the free-electron mass. Equations (2) and (3) must coincide at least for large l when the semiclassical approximation is valid. This means that 0.5 ± gm_∗/4m are integer. A direct calculation^26,27 of the g factor for the dispersion law (1) does confirm this conclusion. Moreover, since Eq. (2) reveals the total coincidence of the semiclassical and exact spectra, the nondegeneracy of the level l = 0 and the double degeneracy of the Landau levels with l ≠ 0 give ∣g∣ = ∣2m/m_∗∣. Formula (2) also means that the Landau subbands ε^(l)(p_n) can be obtained from the semiclassical quantization condition for spinless particles²⁵,

$${S}_{c,v}({\varepsilon }^{(l)},{p}_{n})=\frac{2\pi \hslash eH}{c}\left(l+\gamma \right),$$

(4)

with constant γ = 0²⁸ different from the usual value γ = 1/2²⁵.

The coincidence of the semiclassical and exact spectra at all their quantum numbers is known for the two quantum systems²⁹. They are a harmonic oscillator (and hence, an electron with a parabolic dispersion) and an electron in the Coulomb field. Formula (2) demonstrates that quasiparticles with dispersion (1) give the third example of this total coincidence.

Consider the above result for the g factor in more detail. This will help us evaluate the scope of applicability of formula (2) to real situations. The area S in quantization condition (3) is defined by a dispersion relation, whereas the g factor of an electron orbit, e.g, in the band c, is determined by the following part $\hat{H}$ of the total electron Hamiltonian in the magnetic field H²⁶:

$$\hat{H}=\frac{e}{2mc}{{{\bf{H}}}}\left(2{{{{\bf{L}}}}}_{c}^{{{{\rm{intra}}}}}+{{{{\bf{L}}}}}_{c}^{{{{\rm{c}}}}\,{\mbox{-}}\,{{{\rm{v}}}}}+{{{{\bf{L}}}}}_{c}^{{{{\rm{c}}}}\,{\mbox{-}}\,{{{\rm{rem}}}}}+2{{{\bf{s}}}}\right).$$

(5)

The contribution of the first term $2{{{{\bf{HL}}}}}_{c}^{{{{\rm{intra}}}}}$ to the g factor is expressed via the Berry phase Φ_B of the orbit, g^intra = (2m/m_∗)(Φ_B/π), where ${{{{\bf{L}}}}}_{c}^{{{{\rm{intra}}}}}$ is the intraband orbital electron moment. In the second term ${{{{\bf{HL}}}}}_{c}^{{{{\rm{c}}}}\,{\mbox{-}}\,{{{\rm{v}}}}}$, the ${{{{\bf{L}}}}}_{c}^{{{{\rm{c}}}}\,{\mbox{-}}\,{{{\rm{v}}}}}$ is the part of the orbital electron moment associated with virtual electron transitions between the bands c and v. For H∥z,

$${{{{{\bf{L}}}}}}_c^{{{{\mathrm{c}}}}\mbox{-}{{{\mathrm{v}}}}}=\frac{\hbar m}{2i}\sum\limits_{\rho^{\prime\prime}=1,2} \frac{(v_x)_{c\rho,v\rho^{\prime\prime}}(v_y)_{v\rho^{\prime\prime},c\rho^{\prime}} -(v_y)_{c\rho,v\rho^{\prime\prime}}(v_x)_{v\rho^{\prime\prime},c\rho^{\prime}}}{\varepsilon_v({{{{{\bf{p}}}}}})-\varepsilon_c({{{{{\bf{p}}}}}})},$$

where ${({v}_{i})}_{c\rho ,v{\rho }^{{\prime\prime} }}$ are the matrix elements of the velocity operator between the double degenerate states (marked by $\rho ,{\rho }^{{\prime} },{\rho }^{{\prime\prime} }=1,2$) of the bands c and v. A similar expression corresponds to ${{{{\bf{L}}}}}_{c}^{{{{\rm{c}}}}\,{\mbox{-}}\,{{{\rm{rem}}}}}$ in the third term ${{{{\bf{HL}}}}}_{c}^{{{{\rm{c}}}}\,{\mbox{-}}\,{{{\rm{rem}}}}}$ which takes into account the virtual transitions between the c and remote bands. The ${{{{\bf{L}}}}}_{c}^{{{{\rm{c}}}}\,{\mbox{-}}\,{{{\rm{rem}}}}}$ is relatively small since it contains large denominators, ε_rem(p) − ε_c(p). This third term is usually of the order of the fourth term that describes the Zeeman interaction eHs/mc of the electron spin s with the magnetic field. In the case of dispersion (1), for which the remote bands are disregarded, the third term is absent, the fourth is neglected, and only the first two terms are actually taken into account in obtaining Eq. (2). It is necessary to emphasize that for various electron orbits in the vicinity of quasi-Dirac and Dirac points, the value of the Berry phase can be different. However, the total contribution of the first two terms of the Hamiltonian to the g factor is always equal to 2m/m_∗²⁷ [and hence, γ = 0 in Eq. (4)].

An estimate of the above contributions to the g factor shows that, like for Eq. (1), the necessary conditions for the applicability of formula (2) to real situations are (i) the relatively small Δ, and (ii) the strong SO interaction. The importance of the strong SO interaction becomes clear from the following reasoning: At $0 < ({\varepsilon }_{{{{\rm{F}}}}}-{\varepsilon }_{{{{\rm{D}}}}})-{\Delta }_{\min }\ll \Delta$, Eq. (1) leads to a parabolic dependence of ε_c on p near the minimum of the c band (Fig. 1). This dependence is typical for trivial (ordinary) charge carriers for which the g factor usually does not coincide with the specific value 2m/m_∗. As was emphasized above, the strength of the spin-orbit interaction, Δ_SO, is larger than Δ for the quasi-Dirac points. It is this condition that results in the specific value of the g factor. If Δ_SO decreases and becomes less than Δ, the value of the g factor begins to decrease, too. At small Δ_SO/Δ, we obtain the estimate g_orb ~ (Δ_SO/Δ) for the orbital part of the g factor²⁶ that is determined by the first three terms in Hamiltonian (5). Then, the total g ~ 2 + (Δ_SO/Δ). Thus, we arrive at the case of the trivial charge carriers only for sufficiently small Δ_SO.

The above necessary conditions can be formulated as follows:

$$\eta =\frac{{\varepsilon }_{0}}{{E}_{0}}\ll 1.$$

(6)

Here ${\varepsilon }_{0} \sim \max (| {\varepsilon }_{{{{\rm{F}}}}}-{\varepsilon }_{{{{\rm{D}}}}}| ,\Delta )$ is the characteristic scale of the energy spectrum under study, the scale E₀ is of the order of the smallest value of the two energies: $m{v}_{\perp }^{2}$ and a gap between ε_D and the closest remote band at the point p = 0, v_⊥ is the typical charge-carrier velocity in the plane perpendicular to the magnetic field; see, e.g, Supplementary Eq. (33). If the parameter η is not too small, additional terms that take into account the remote bands should be introduced into dispersion relation (1). Beside this modification of Eq. (1), the last two terms in Eq. (5) produce a correction Δg to the above universal value of the g factor in semiclassical condition (3), g = 2(m/m_∗) + Δg. This correction adds  ± Δgm_∗/4m to γ = 0 in formula (4) and leads to a splitting of the Landau levels with l ≥ 1. The splitting Δgm_∗/2m is proportional to the parameter η since $| {m}_{* }| /m\approx | {\varepsilon }_{{{{\rm{F}}}}}-{\varepsilon }_{{{{\rm{D}}}}}| /(m{v}_{\perp }^{2}) \sim \eta$. Strictly speaking, this estimate of the splitting is valid for low magnetic fields when many Landau levels lie under the Fermi surface, and the semiclassical approximation is accurate. However, the splitting is usually observed only for the lowest Landau levels^30,31,32, i.e., for strong H. In this case, the correction to the g factor can increase even more since the lowest Landau levels for the modified dispersion (1) are no longer described by the semiclassical formula.

To distinguish the Dirac electrons from the trivial charge carriers in crystals, a number of the physical effects are commonly used (see Introduction). Consider now these effects in the case of the quasi-Dirac spectra.

Quantum-oscillation phenomena

The quasi-Dirac spectra can be analyzed with the quantum-oscillation phenomena, e.g., with the de Haas-van Alphen and Shubnikov-de Haas effects. In Supplementary Note 2, formulas for the quantum-oscillation frequencies F_i and the cyclotron masses m_∗,i are presented in the case of charge carriers with dispersion (1). The subscript i = x, y, z means that the appropriate quantity corresponds to the magnetic field directed along the ith axis. For simplicity, consider the case a = 0. Then, Supplementary formulas (26)–(28) give

$$\frac{2e\hslash {F}_{i}}{c| {m}_{* ,i}| }=\frac{{({\varepsilon }_{{{{\rm{F}}}}}-{\varepsilon }_{{{{\rm{D}}}}})}^{2}-{\Delta }^{2}}{| {\varepsilon }_{{{{\rm{F}}}}}-{\varepsilon }_{{{{\rm{D}}}}}| }\equiv {\tilde{\varepsilon }}_{{{{\rm{F}}}}},$$

(7)

$$\frac{{F}_{i}}{{F}_{j}}=\frac{{m}_{* ,i}}{{m}_{* ,j}}\equiv {\epsilon }_{ij},$$

(8)

where ϵ_ij = v_i/v_j. Besides, when the magnetic field lies in the i − j plane at the angle θ to the i axis, the θ dependence of the frequency F has the form:

$$F(\theta )=\frac{{F}_{i}}{{({\cos }^{2}\theta +{\epsilon }_{ij}^{2}{\sin }^{2}\theta )}^{1/2}}.$$

(9)

As in the case of Dirac points^10,33, relationship (7) shows that the ratio F_i/∣m_∗,i∣ is the same for the three directions of the magnetic field (in fact, it is the same for all directions of H). This key statement is true in the general case a ≠ 0 if Δ is replaced by ${\Delta }_{\min }$ (Supplementary Note 2). Note that for trivial electrons with a parabolic dispersion, the ratio F_i/∣m_∗,i∣ is also independent of the direction of the magnetic field. In this case, Eqs. (8), (9) remain valid with ${\epsilon }_{ij}={({m}_{j}/{m}_{i})}^{1/2}$ where m_i are the effective masses of the parabolic spectrum. However, if ε_F and F_i for such electrons change, e.g., due to the doping of the sample by impurities, their cyclotron masses remain unchanged. On the other hand, for the Dirac and quasi-Dirac points, one has m_∗,i ∝ ε_F − ε_D. (For QD points, at $| {\varepsilon }_{{{{\rm{F}}}}}-{\varepsilon }_{{{{\rm{D}}}}}| \to {\Delta }_{\min }$, the decreasing ∣m_∗,i∣ reaches a constant proportional to Δ.) Interestingly, without any doping of the sample, the ε_F dependence of the cyclotron masses can be revealed by measuring the temperature correction to the F_i^17,18.

The constant γ (the g factor) in the semiclassical quantization condition determines the phase of the quantum oscillations and can be found in experiments²⁵. For example, the first harmonic of the magnetization produced by the quasi-Dirac point is proportional to ${\sum }_{\pm }\sin [2\pi ({F}_{i}/H-{\phi }_{\pm })-\pi /4]$ where ϕ_± are the phases specified by Eq. (3),

$${\phi }_{\pm }=\frac{1}{2}\pm \frac{g{m}_{* }}{4m}.$$

Since ∣gm_∗∣/4m = 1/2 for the QD points, the phases ϕ_± coincide with γ introduced in Eq. (4), ϕ_± = γ = 0 (the phase values 0 and 1 are equivalent). We emphasize that according to Eq. (2), the above specific value of γ (of the g factor) is valid for any cross section of the Fermi surface surrounding the quasi-Dirac point. In particular, if the maximal cross section of the Fermi surface does not pass through the Dirac point (H is not perpendicular to a ≠ 0, Fig. 1), or if there is a nonzero gap 2Δ in the spectrum, the Berry phase of the orbit differs from π. However, quantum-oscillations experiments should give γ = 0 in both these cases. In other words, the phase of the quantum oscillation, γ, rather than its constituent, the Berry phase, is robust with respect to small crystal-potential perturbations generating the gap. Therefore, the measurements of the phase of the oscillations is the most direct way to distinguish the charge carriers near the QD points from the carriers with a dispersion different from Eq. (1). However, these measurements does not differentiate between the Dirac and quasi-Dirac points.

Although the value of γ is the hallmark of the quasi-Dirac points, in a number of the experiments with Cd₃As₂^31,32 and ZrTe₅^12,34,35, the measured γ deviates from the zero value. The ratio F_i/m_∗,i for these materials also slightly depends on the direction of H^12,32, and hence F_i/F_j does not coincide with m_∗,i/m_∗,j. All these discrepancies between the theoretical and experimental results indicate that the dispersion of the charge carriers, at least along one of the axes, deviates from Eq. (1), i.e., the parameter η in Eq. ((6)) is not small enough. In this case, a modification of the dispersion is required to describe experimental data. Below we will discuss this issue in more detail, using ZrTe₅ as an example.

Thermodynamic quantities dependent on magnetic field

As an example of such thermodynamic quantities, consider magnetic susceptibility χ, and compare χ produced by the Dirac and quasi-Dirac points. The magnetic susceptibility of the QD points was calculated for weak^36,37,38 and strong²¹ magnetic fields. The specific case of the Dirac point was analyzed in a number of papers^{10,21,37,38,39}.

The main results can be summarized as follows: The magnetic susceptibility for the Dirac point is diamagnetic and diverges logarithmically when the Fermi level ε_F approaches the Dirac energy, $\chi \propto \ln | {\varepsilon }_{{{{\rm{F}}}}}-{\varepsilon }_{{{{\rm{D}}}}}|$. This divergence is cut off at $| {\varepsilon }_{{{{\rm{F}}}}}-{\varepsilon }_{{{{\rm{D}}}}}| \sim \max (T,\Delta {\varepsilon }_{H})$ where Δε_H = eℏH/(m_∗c) is the spacing between the Landau levels, and T is the temperature. For the case of the quasi-Dirac point, χ is the same, but this cut-off occurs at $| {\varepsilon }_{{{{\rm{F}}}}}-{\varepsilon }_{{{{\rm{D}}}}}| \sim \max (T,\Delta {\varepsilon }_{H},{\Delta }_{\min })$ where $2{\Delta }_{\min }$ is the minimal indirect gap for dispersion (1), Fig. 1. Thus, the difference between the Dirac and the quasi-Dirac cases can manifest itself only at $T < {\Delta }_{\min }$ in the sufficiently low magnetic fields $\Delta {\varepsilon }_{H} < {\Delta }_{\min }$ and for the Fermi level lying in the gap, $| {\varepsilon }_{{{{\rm{F}}}}}-{\varepsilon }_{{{{\rm{D}}}}}| < {\Delta }_{\min }$, or near it.

The similarity of the thermodynamic quantities for the Dirac and quasi-Dirac points can be understood from the following considerations: The characteristic feature of the Dirac spectrum is that the lowest Landau subband (l = 0) is independent of H and coincides with the dispersion law of the charge carriers along the direction of the magnetic field¹³. It is clear from Eq. (2) that if there is a nonzero gap in the spectrum, the lowest Landau subband (l = 0) still is independent of H. That is why the thermodynamic quantities practically do not “feel” the gap. In contrast, for the trivial electrons, one has γ = 1/2, S_c,v(ε⁽⁰⁾, p_n) = πℏeH/c, and the lowest Landau subband ε⁽⁰⁾(p_n) depends on the magnetic field.

Fermi arcs, chiral anomaly, and magneto-optical conductivity

In a crystal with two Dirac points lying in a rotation axis, the Fermi arcs can be observed on its surface⁹. The existence of these arcs is due to the nonzero topological invariant ν_2D defined in the plane which is perpendicular to the axis and passes through its middle Γ ^5,40. This invariant is determined by the parities of the electron bands at TRIM in this plane of a crystal. When the inversion of the bands occurs at Γ, the invariant changes. This change does not depend on whether the Dirac or the quasi-Dirac points appear in the axis after the inversion. Therefore, the arcs are expected to be observed in both these cases.

It is clear that the chiral anomaly⁴¹ cannot occur for the quasi-Dirac points with a nonzero gap. However, the negative longitudinal magnetoresistance, which accompanies this anomaly, can be observed for the quasi-Dirac spectra⁴². Moreover, the negative magnetoresistance can arise even in crystals with trivial charge carriers if in magnetic fields, these carriers are redistributed between their Fermi-surface pockets with different mobilities¹⁶.

Measurements of the magneto-optical conductivity make it possible to find the gap in electron spectra^{11,12,43,44,45,46}, and therefore, to distinguish between the quasi-Dirac and Dirac points. (The appropriate formulas in the case of the QD spectrum are presented in Supplementary Note 3). However, if ∣ε_F − ε_D∣ ≫ Δ, a reliable detection of a nonzero Δ is obviously not an easy task.

Spectrum of ZrTe₅

To illustrate the above results, consider ZrTe₅. Crystals of this material have an orthorhombic layered structure, in which the layers stack along the b axis. As a result, ZrTe₅ shows a strong anisotropy. The Dirac point can occur in the center Γ of the Brillouin zone^47,48, i.e., the case (i) mentioned in Introduction is realized in ZrTe₅. In absence of the magnetic field, the Hamiltonian of the charge carriers near the Γ point has the form¹¹:

$${\hat{H}}_{\Gamma }={\hat{H}}_{{{{\rm{diag}}}}}+{v}_{x}{p}_{x}{\tau }_{x}{\sigma }_{z}+{v}_{y}{p}_{y}{\tau }_{y}{\sigma }_{0}+{v}_{z}{p}_{z}{\tau }_{x}{\sigma }_{x},$$

(10)

where ${\hat{H}}_{{{{\rm{diag}}}}}=\Delta {\tau }_{z}{\sigma }_{0}$; σ_i and τ_i are the Pauli matrices, whereas τ₀ and σ₀ are the identity matrices (the τ and σ mark the band and spin indices, respectively); v_x, v_y, v_z are real constants with the dimension of a velocity; 2Δ is the gap in the spectrum (to obtain the true Dirac point, a small deformation of ZrTe₅ is required^34,48); the energy is measured from the middle of the gap (i.e., ε_D = 0 below). The z axis coincides with the b axis, whereas x and y are along the a and c axes of the crystal, respectively. The diagonalization of Hamiltonian (10) yields dispersion (1) with a = 0.

Let us check whether experimental results on the quantum oscillations in this material confirm the existence of the quasi-Dirac point. The Shubnikov - de Haas oscillations in ZrTe₅ were investigated in a number of works (see, e.g., papers^{12,34,35,49,50}). The most full data were presented by Yuan et al.¹². In Table 1, the frequencies of the oscillations and the cyclotron masses are written for the magnetic fields directed along the principal axes of the crystal¹². In this Table, the ratios ${\tilde{\varepsilon }}_{{{{\rm{F}}}}}$ calculated with Eq. (7) are also given. Although this ratio for the quasi-Dirac point should be independent of the orientation of H within the accuracy of the experiment, the discrepancy between the obtained values of ${\tilde{\varepsilon }}_{{{{\rm{F}}}}}$ is about 30%, which slightly exceeds the possible experimental error.

Table 1 The frequencies and the cyclotron masses of the quantum oscillations in ZrTe₅

Yuan et al.¹² also measured the angular dependences of the oscillation frequency when the magnetic field was rotated in the z-x and z-y planes. These dependences are well approximated with formula (9) (Fig. 3), but the obtained values of ϵ_zy = F_z/F_y and ϵ_zx = F_z/F_x are about 25% less than m_∗,z/m_∗,y and m_∗,z/m_∗,x, respectively. On the other hand, the parameter of the anisotropy in the x-y plane, ϵ_yx = v_y/v_x, is practically the same when it is calculated as F_y/F_x or m_∗,y/m_∗,x. Using Supplementary Eq. (22) for F_z at ${\tilde{a}}_{z}=0$, the values of ϵ_yx and the Fermi energy $| {\varepsilon }_{{{{\rm{F}}}}}| \approx {\tilde{\varepsilon }}_{{{{\rm{F}}}}}=36.1$ meV from Table 1, we obtain the velocities v_x ≈ 5.6 × 10⁵ ms⁻¹ and v_y ≈ 3.3 × 10⁵ ms⁻¹. (We take ${\tilde{\varepsilon }}_{{{{\rm{F}}}}}\approx | {\varepsilon }_{{{{\rm{F}}}}}|$ since Δ is small in ZrTe₅^{11,12,43,44,45,46}.) Then, with v_z/v_x = ϵ_zx = 0.096 found in Fig. 3, we arrive at the estimate v_z ≈ 5.4 × 10⁴ ms⁻¹. Note that $\sqrt{{v}_{x}{v}_{y}}\approx 4.3\times 1{0}^{5}$ ms⁻¹ is close to the values (4.8 − 5) × 10⁵ ms⁻¹ obtained from magneto-optical measurements^{11,43,44,45,46} and to 4.1 × 10⁵ ms⁻¹ found in the recent Shubnikov-de Haas experiment⁵⁰.

Fig. 3: The angular dependences of the quantum-oscillations frequency for ZrTe₅.

The θ is the angle between the magnetic field H and the z axis. The experimental data¹² for the frequency F of the oscillations are shown for H lying in the z − x (cycles) and z − y (squares) planes. For dispersion (1), the dashed lines depict dependence (9) with ϵ_zx = 0.096 and ϵ_zy = 0.164. The red solid lines correspond to dispersion (12) and are calculated with Eqs. (13)–(16) and the first set of the parameters from Table 2. Inset: The red lines are the same as in the main panel. To clarify the role of the terms ${\alpha }_{i}{p}_{z}^{2}$ in dispersion (12), the dashed lines are calculated with the same formulas as the red lines, but setting α_c = α_v = 0 in the first set of the parameters [i.e., in fact, with Eq. (9) and ϵ_zx ≈ 0.065, ϵ_zy ≈ 0.11].

The above analysis of the frequencies and masses seems to indicate some deviation of the electron spectrum along the z axis in ZrTe₅ from the quasi-Dirac form. A more distinct support of this conclusion follows from the measurements of the phase of the Shubnikov - de Haas oscillations^12,34,35. In particular, Yuan et al.¹² found that at H∥z, this phase corresponds to the Dirac spectrum, whereas the phase changes by about 1/2 when the direction of H becomes almost perpendicular to the z axis.

The deviation of the dispersion from Eq. (1) may be due to the layered structure of ZrTe₅. This layering leads to a relatively small velocity v_z found above. To take into account the above-mentioned deviation, it is worth noting the following: The symmetry of the point Γ admits the terms of the form ${\alpha }_{i}{p}_{i}^{2}$ in the diagonal part of the Hamiltonian, ${\hat{H}}_{{{{\rm{diag}}}}}$, where i = x, y, z, and the constants α_i are of the order of 1/m. However, the terms ${\alpha }_{x}{p}_{x}^{2},{\alpha }_{y}{p}_{y}^{2}$ are relatively small as compared to v_xp_x, v_yp_y and can be omitted. This is not the case for the quadratic term ${\alpha }_{z}{p}_{z}^{2}$ due to the relatively small v_z. Indeed, typical values of p_z are determined by the relation v_zp_z ~ ε_F, whereas the quadratic term at such p_z is of the order of $(1/m){({\varepsilon }_{{{{\rm{F}}}}}/{v}_{z})}^{2}\approx 2.2{\varepsilon }_{{{{\rm{F}}}}}$ for the values of v_z and ε_F obtained above. Therefore, the quadratic terms along the z axis are important, and instead of ${\hat{H}}_{{{{\rm{diag}}}}}=\Delta {\tau }_{z}{\sigma }_{0}$, we will use the following expression:

$${\hat{H}}_{{{{\rm{diag}}}}}=\left(\Delta +\frac{{\alpha }_{c}+{\alpha }_{v}}{2}{p}_{z}^{2}\right){\tau }_{z}{\sigma }_{0}+\frac{({\alpha }_{c}-{\alpha }_{v})}{2}{p}_{z}^{2}{\tau }_{0}{\sigma }_{0},$$

(11)

where α_c and α_v are some constants. Then, dispersion relation (1) is replaced by

$${\varepsilon }_{c,v}({{{\bf{p}}}})= \, \frac{{\alpha }_{c}-{\alpha }_{v}}{2}{p}_{z}^{2}\\ \pm \sqrt{{(\Delta +\frac{{\alpha }_{c}+{\alpha }_{v}}{2}{p}_{z}^{2})}^{2}+{v}_{x}^{2}{p}_{x}^{2}+{v}_{y}^{2}{p}_{y}^{2}+{v}_{z}^{2}{p}_{z}^{2}}.$$

(12)

In fact, formula (12) takes into account the effect of the remote bands on the dispersion of the c and v bands along the z axis, and the quasi-Dirac spectrum occurs only in the x − y plane. Curiously, if Δ is negative and $\Delta < {\Delta }_{{{{\rm{cr}}}}}=-{v}_{z}^{2}/({\alpha }_{c}+{\alpha }_{v})$, dispersion (12) predicts the splitting of the quasi-Dirac point at p_z = 0 into two QD points lying in the two-fold rotation axis z (Fig. 2). It is necessary to note that the energy-band dispersion similar to Eq. (12) has already been proposed previously^43,46. However, the term ${v}_{z}^{2}{p}_{z}^{2}$ was disregarded by Martino et al.⁴³, and it was implied^43,46 that α_c = α_v although this restriction is not dictated by the symmetry of ZrTe₅.

Interestingly, the Hamiltonian described by Eqs. (10)–(12) is equivalent to the Hamiltonian of McClure⁵¹ for the electrons located near the point L of the Brillouin zone of Bi. In bismuth, the role of the b axis plays the axis “2” directed along the length of the electron Fermi-surface pocket. This equivalence of the Hamiltonians means that results obtained with McClure model for Bi can be extended to the case of ZrTe₅. In particular, experimental data on frequencies and cyclotron masses of pure bismuth and its alloys with Sb were successfully described with McClure model⁴. We derive similar formulas in the case of dispersion relation (12) and find the parameters of this relation (see Methods). In Table 2, the values of these parameters are presented for Δ ≈ 0^11,12, Δ = 2.5 meV⁴⁴ and Δ = −5 meV⁴⁵ obtained in the magneto-optical experiments. Dependences F(θ) calculated with these formulas practically coincide with those given by Eq. (9), Fig. 3. There is a tiny discrepancy between them only at $\cos \theta \sim {v}_{z}/{v}_{x}$.

Table 2 The parameters of the spectrum for the charge carriers in ZrTe₅

Discussion

It was predicted^47,48 that in ZrTe₅, the temperature expansion increases the parameter Δ, and the band inversion occurs at Δ = 0. This inversion corresponds to the transition from the strong topological insulator (STI), for which Δ < 0, to the weak topological insulator (WTI) characterized by a positive gap 2Δ. The existence of this transition and the “initial” state of ZrTe₅ at low temperatures are widely discussed in the literature; see, e.g., recent papers^44,45. Unlike spectrum (1), which does not depend on the sign of Δ, dispersion (12) makes it possible to distinguish between the STI and WTI phases, using the bulk properties of ZrTe₅. Indeed, as it was mentioned above, the critical value 2Δ_cr of the negative gap exists. At Δ < Δ_cr, the quasi-Dirac point Γ splits into the two QD points ζ, which gradually shift along the z axis with increasing ∣Δ∣ (Fig. 2). Therefore, if two gaps is observed in magneto-optical measurements, this is indicative of the STI phase in ZrTe₅⁴⁶. The first gap is still equal to 2∣Δ∣, whereas the second gap $2| {\Delta }_{\zeta }| =2\sqrt{{\Delta }_{{{{\rm{cr}}}}}(2\Delta -{\Delta }_{{{{\rm{cr}}}}})}$ corresponds to the two quasi-Dirac points. The observation of the two gaps at low temperatures was indeed reported for ZrTe₅^45,46. In particular, Jiang et al.⁴⁶ obtained 2∣Δ∣ ≈ 15 meV and 2∣Δ_ζ∣ ≈ 11.2 meV. These data lead to ∣Δ_cr∣ ≈ 2.5 meV. At v_z ≈ 5 × 10⁴ ms⁻¹⁴⁶ (which is comparable with v_z from Table 2), the formula for ∣Δ_cr∣ gives m(α_c + α_v) ≈ 5.7. This value is approximately an order of magnitude larger than those shown in Table 2. Although the doping of ZrTe₅ depends on the method of growing its single crystals⁴³, the parameters α_c and α_v are determined by the remote bands. Hence, the large discrepancy between the values of (α_c + α_v) can hardly be explained by a difference in these methods for different experiments. To resolve this contradiction between the magneto-optical⁴⁶ and oscillation¹² data, it would be useful to measure the three frequencies F_i and the cyclotron masses m_∗,i of the Shubnikov-de Haas oscillations for the samples exhibiting the two gaps in magneto-optical experiments.

Keeping in mind the similarity of the electron Hamiltonians for Bi and ZrTe₅, let us point out a correspondence of certain results for these materials. (i) As was mentioned above, the phase of the Shubnikov-de Haas oscillations sharply changes in ZrTe₅ when the magnetic-field direction approaches the y axis^12,34,35. A similar change of the g factor for the electron orbits was observed in bismuth when H was almost perpendicular to the axis “2”⁵². This angular dependence of the g factor in Bi was quantitatively described within McClure model²⁷. (ii) For ZrTe₅, the magnetic-field⁵³ and temperature⁵⁴ dependences of the magnetization were experimentally investigated at H∥z. For Bi, such dependences were measured^55,56 and analyzed within McClure model⁵⁷ many years ago. Thus, the approaches of the papers^27,57 can be useful to obtain additional information on the electron spectrum of ZrTe₅.

There is a similarity in the manifestations of the quasi-Dirac points in crystals with the strong spin-orbit interaction and of nodal lines in semimetals, for which this interaction is weak. Such lines mainly occur in neglect of the SO coupling^1,2,10. The coupling usually lifts the degeneracy of the bands, and the spectrum takes on the quasi-Dirac form (with small Δ = Δ_SO) in any plane perpendicular to the line. As in the case of the quasi-Dirac point, the g factor of an electron orbit surrounding the line in such a plane is the sum: g = g_intra + g_inter. Here g_inter and g_intra = (2m/m_∗)(Φ_B/π) are associated with the interband electron orbital moment and with the Berry phase Φ_B of the orbit, respectively. The total g factor again has the universal value 2m/m_∗, and γ = 0 in quantization condition (4)⁵⁸. This result justifies the concept of the nodal lines since it ensures stability of their physical properties with respect to the lifting of the degeneracy. Moreover, when the radius of the orbit (i.e., ∣ε_F − ε_D∣) increases, one has Φ_B → π and g_inter/g ~ (Δ/∣ε_F − ε_D∣) → 0. Thus, already near the nodal line, the constant γ = 0 can be represented as 1/2 − Φ_B/(2π) with Φ_B = π. It is important that in crystals with the weak SO interaction, these values of Φ_B and γ do not depend on the size and shape of the electron orbit²⁸. This result explains why the properties of nodal-line semimetals (e.g., the drumhead surface states) remain valid far away from the line. Note that for the quasi-Dirac points in crystals with the strong SO interaction, the same universal value γ = 0 occurs only near these points where Eq. (1) accurately describes the electron spectrum. The same statement is true for the Weyl points⁵⁹.

Conclusions

The Landau levels for charge carriers located near the quasi-Dirac and Dirac points are very similar. This similarity leads to a practical coincidence of the physical phenomena determined by these points. In the case of ZrTe₅, the published experimental data on the Shubnikov-de Haas effect indicate that the spectrum of this 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 formula (1). Using ZrTe₅ as an example, we show how such a deviation can be taken into account to describe experimental data.

Methods

Determination of the parameters describing the charge carriers in ZrTe₅

The frequencies F_x, F_y and the cyclotron masses m_∗,x, m_∗,y can be calculated analytically in the case of dispersion relation (12). In particular, we obtain,

$$\frac{2\pi e\hslash {F}_{x}}{c}=\frac{4{({\varepsilon }_{{{{\rm{F}}}}}^{2}-{\Delta }^{2})}^{3/4}{(2R)}^{3/2}}{3{v}_{y}{({\alpha }_{c}{\alpha }_{v})}^{1/4}}[(1-{k}^{2})K(k)+(2{k}^{2}-1)E(k)],$$

(13)

$${m}_{* ,x}= \frac{2{\varepsilon }_{{{{\rm{F}}}}}}{\pi {v}_{y}{({\alpha }_{c}{\alpha }_{v})}^{1/4}{({\varepsilon }_{{{{\rm{F}}}}}^{2}-{\Delta }^{2})}^{1/4}{(2R)}^{1/2}}\\ \left[K(k)-\frac{({\alpha }_{c}-{\alpha }_{v}){({\varepsilon }_{{{{\rm{F}}}}}^{2}-{\Delta }^{2})}^{1/2}R}{{\varepsilon }_{{{{\rm{F}}}}}{({\alpha }_{c}{\alpha }_{v})}^{1/2}}[K(k)-(1-{k}^{2})E(k)]\right],$$

(14)

where K(k) and E(k) are the complete elliptic integrals of the first and second kinds, respectively,

$${k}^{2}= \frac{0.5B+R}{2R},\quad R=\sqrt{1+\frac{{B}^{2}}{4}},\\ B= -\frac{{\varepsilon }_{{{{\rm{F}}}}}({\alpha }_{c}-{\alpha }_{v})+\Delta ({\alpha }_{c}+{\alpha }_{v})+{v}_{z}^{2}}{\sqrt{{\alpha }_{c}{\alpha }_{v}({\varepsilon }_{{{{\rm{F}}}}}^{2}-{\Delta }^{2})}}.$$

(15)

Formulas (13)–(15) agree with the expressions derived in the case of bismuth⁵¹. Formulas for F_y and m_∗,y are obtained by the replacement of v_y by v_x in Eqs. (13) and (14). This replacement shows that the relation F_x/F_y = m_∗,x/m_∗,y = v_x/v_y remains true in the case of dispersion (12), cf. Eq. (8). As to F_z and m_∗z, they are the same for dispersion relations (1) and (12) and are described by Supplementary formulas (22) and (23) with ${\tilde{a}}_{z}=0$. Therefore, using these two formulas and the equality F_x/F_y = v_x/v_y, one can find ε_F, v_x, v_y if Δ is known. In Table 2, we present the values of these parameters for Δ ≈ 0^11,12, Δ = 2.5 meV⁴⁴, and Δ = − 5 meV⁴⁵.

If the magnetic field is at the angle θ to the z axis, formulas for F(θ) and m_∗(θ) at H lying in the x − z and y − z planes are obtained from Eqs. (13)–(15) by the following substitutions:

$$\begin{array}{r}{\alpha }_{c,v}\to {\alpha }_{c,v}{\sin }^{2}\theta ,\quad {v}_{z}^{2}\to {v}_{z}^{2}{\sin }^{2}\theta +{v}_{x}^{2}{\cos }^{2}\theta ,\\ {\alpha }_{c,v}\to {\alpha }_{c,v}{\sin }^{2}\theta ,\quad {v}_{z}^{2}\to {v}_{z}^{2}{\sin }^{2}\theta +{v}_{y}^{2}{\cos }^{2}\theta ,\end{array}$$

(16)

respectively. When α_c and α_v are relatively small, i. e., when the parameter $\eta =\sqrt{{\alpha }_{c}{\alpha }_{v}}{\varepsilon }_{{{{\rm{F}}}}}/{v}_{z}^{2}\ll 1$, we obtain the following expressions from Eqs. (13)–(16):

$$B \, \approx -\frac{{v}_{z}^{2}+{v}_{x}^{2}{\cot }^{2}\theta }{\sqrt{{\alpha }_{c}{\alpha }_{v}({\varepsilon }_{{{{\rm{F}}}}}^{2}-{\Delta }^{2})}},\quad | B| \gg 1,\quad 2R\approx | B| +\frac{2}{| B| },\\ {k}^{2} \, \approx \, \frac{1}{{B}^{2}}\ll 1,\quad K(k)\approx \frac{\pi }{2}(1+\frac{{k}^{2}}{4}),\quad E(k)\approx \frac{\pi }{2}(1-\frac{{k}^{2}}{4}),\\ \frac{2\pi e\hslash F(\theta )}{c} \, \approx \, \frac{4{({\varepsilon }_{{{{\rm{F}}}}}^{2}-{\Delta }^{2})}^{3/4}| B{| }^{3/2}}{3{v}_{y}{({\alpha }_{c}{\alpha }_{v})}^{1/4}\sin \theta }\frac{3\pi }{4{B}^{2}}\approx \frac{\pi ({\varepsilon }_{{{{\rm{F}}}}}^{2}-{\Delta }^{2})}{{v}_{y}\sqrt{{v}_{z}^{2}{\sin }^{2}\theta +{v}_{x}^{2}{\cos }^{2}\theta }},\\ {m}_{* }(\theta ) \, \approx \, \frac{{\varepsilon }_{{{{\rm{F}}}}}}{{v}_{y}\sqrt{{v}_{z}^{2}{\sin }^{2}\theta +{v}_{x}^{2}{\cos }^{2}\theta }}.$$

The last two formulas reduce to Supplementary Eqs. (22), (23) at θ = 0 and to Eq. (9) at θ ≠ 0.

If α_c and α_v increase, the parameter η can become of the order of unity. This means that the dispersion of the bands along the z axis noticeably deviates from the quasi-Dirac form. Nevertheless, we still have $F(\theta )\approx F(0)/\cos \theta$ at the angles θ ≲ 1 due to the large value of v_x/v_z. However, when θ is close to π/2 ($\cot \theta \sim {v}_{z}/{v}_{x}$), the dependence F(θ) in this region of the angles becomes sensitive to the value of α_c/α_v. If α_c/α_v is large, the parameter k² remains small at θ → π/2 [see formula for the parameter B in Eqs. (15)], and F(θ) tends to the universal form given by Eq. (9) (the dashed lines in Fig. 4). If α_c/α_v decreases, F(θ) deviates from dependence (9). Therefore, a precise measurement of F(θ) in the region $\cos \theta \lesssim {v}_{z}/{v}_{x}$ enables one to find the value of α_c/α_v.

Fig. 4: Another representation of the angular dependences shown in the main panel of Fig. 3.

Here F and θ are the frequency of the oscillations and the angle between the magnetic field H and the z axis, respectively. The experimental data¹² are shown for H lying in the z − x (cycles) and z − y (squares) planes. The red solid lines are calculated with Eqs. (13)–(16) and the first set of the parameters from Table 2 (the second and third sets provide the same accuracy of the fit to the data). The dashed lines are described by Eq. (9) with ϵ_zx = 0.096, ϵ_zy = 0.164.

For ZrTe₅, the dependences F(θ) were measured for the magnetic field lying in the z − x and z − y planes¹². These dependences are really close to $F(0)/\cos \theta$ at $(1/\cos \theta )\lesssim 3$, but they deviate from this simple dependence when θ tends to π/2 (Fig. 4). There is also a slight deviation of the experimental data from the universal form given by Eq. (9).

In order to determine the values of v_z, α_c, and α_v for the ZrTe₅, it is convenient to use formula (13) and an expression for $2\pi e\hslash {F}_{x}{\varepsilon }_{{{{\rm{F}}}}}/[c{m}_{* ,x}({\varepsilon }_{{{{\rm{F}}}}}^{2}-{\Delta }^{2})]$ that is obtained as the ratio of Eqs. (13) and (14). We also impose the requirement that the calculated dependences F(θ) provide the best fit to the experimental data¹² in the interval $3 < 1/\cos \theta < 10$. With these three conditions, we find the values of the parameters v_z, α_c, and α_v presented in Table 2. Interestingly, for F_z, m_∗,z, F_x, m_∗,x, ϵ_yx from Table 1, the first two conditions can be satisfied only at α_c/α_v≥ 0.81, 0.95, 1.02, if Δ = − 5, 0, 2.5 meV, respectively.