Dirac and Weyl semimetals are in the center of attention of many researchers because of their unique properties such as the chiral anomaly, negative magnetoresistance, and Fermi arc surface states [1]. These materials allow the so-called massless modes satisfying the Dirac and Weyl equations [2, 3]. Three-dimensional Dirac semimetals are of fundamental interest because they provide a convenient platform for the study of the 3 + 1 Dirac vacuum. In contrast to a Weyl semimetal, the linear spectrum in a Dirac semimetal is due to the crossing of degenerate bands (see, e.g., [4, 5]). The appearance of the Dirac point because of such crossing is usually protected by the symmetries of the space group of the crystal lattice (the most well-known example is Dirac points in graphene [6, 7]) rather than a nonzero Chern number as in Weyl semimetals or by the nonequivalence of representations of the symmetry group of the Brillouin zone to which crossing bands belong.

Dirac systems with a tilted spectrum are of separate interest. Such a spectrum arises at the overlapping of electron and hole Fermi pockets [8–12]. In this work, we study the possibility of the band overlapping near the Dirac point and the tilt of the spectrum induced by this overlapping. The physical reasons for band overlapping is discussed below. Here, we list the most general conditions. In the general case, such overlapping is possible when overlapping bands belong to equivalent representations of the symmetry group. Thus, we study the situation where bands overlap with each other and simultaneously cross each other at some points of the Brillouin zone. The possibility of such stable crossing points was predicted in the pioneering work by Herring [13], according to which such isolated band crossing points can appear in crystals without an inversion center in the plane of symmetry of the Brillouin zone or in a plane perpendicular to the second-order axis. The crossing of bands belonging to equivalent representations in crystals with an inversion center is also possible but on closed curves in the Brillouin zone rather than at isolated points. In this work, we consider the case of isolated crossing points.

To demonstrate that bands belonging to different but equivalent representations can overlap with each other, it is necessary to show that the overlap integral

$$S_{k}^{{\alpha \beta }} = \int {{d}^{3}}{\mathbf{r}}\psi _{k}^{{\left( \alpha \right)\text{*}}}\psi _{k}^{{\left( \beta \right)}}$$
(1)

is nonzero. Here, k is the quantum number characterizing the band (in our case, it is the wave vector), and a pair of upper superscripts \(\left( {\alpha ,\beta } \right)\) specifies a representation of the group to which the corresponding bands belong; i.e.,

$$\psi _{k}^{{\left( {\alpha ,\beta } \right)}} = \sum\limits_q G_{{kq}}^{{\left( {\alpha ,\beta } \right)}}\psi _{q}^{{\left( {\alpha ,\beta } \right)}}.$$
(2)

Substituting Eq. (2) into Eq. (1) and summing both sides over all elements of the group, we obtain

$$gS_{k}^{{\alpha \beta }} = \sum\limits_{qq'} \int {{d}^{3}}{\mathbf{r}}\psi _{q}^{{\left( \alpha \right)\text{*}}}\psi _{{q'}}^{{\left( \beta \right)}}\sum\limits_G G_{{kq}}^{{\left( \alpha \right)\text{*}}}G_{{kq'}}^{{\left( \beta \right)}},$$
(3)

where g is the dimension of the group. Since the considered representations are equivalent,

$${{G}^{{\left( \beta \right)}}} = A{{G}^{{\left( \alpha \right)}}}{{A}^{{ - 1}}}.$$
(4)

The substitution of Eq. (4) into Eq. (3) gives

$$S_{k}^{{\alpha \beta }} = \frac{{{{A}_{{kk}}}}}{{{{f}_{\alpha }}}}\sum\limits_{qq'} \int {{d}^{3}}{\mathbf{r}}\psi _{q}^{{\left( \alpha \right)\text{*}}}\psi _{{q'}}^{{\left( \beta \right)}}A_{{qq'}}^{{ - 1}},$$
(5)

where we used the orthonormalization condition of the representation:

$$\sum\limits_G G_{{ki}}^{{\left( \alpha \right)\text{*}}}G_{{mn}}^{{\left( \alpha \right)}} = \frac{g}{{{{f}_{\alpha }}}}{{\delta }_{{km}}}{{\delta }_{{in}}},$$
(6)

where \({{f}_{\alpha }}\) is the dimension of the representation. Taking into account that Bloch bands are orthogonal in wave vectors, we obtain

$$S_{k}^{{\alpha \beta }} = \frac{{{{A}_{{kk}}}}}{{{{f}_{\alpha }} - 1}}\sum\limits_{q \ne k} S_{q}^{{\alpha \beta }}A_{{qq}}^{{ - 1}}.$$
(7)

Thus, the overlap integral is not identically zero in the case of different but equivalent irreducible representations. It is easy to show that the overlap integral for nonequivalent representations is identically zero.

To obtain the effective Dirac Hamiltonian and the spectrum in the two-band model in the presence of overlapping bands, we repeat the simplest derivation of the two-band Hamiltonian. The state vector in the two-band model has the form \(\left| {{{\Psi }_{{\mathbf{p}}}}} \right\rangle = {{C}_{u}}\left| {{{u}_{{\mathbf{p}}}}} \right\rangle + {{C}_{v}}\left| {{{v}_{{\mathbf{p}}}}} \right\rangle \), where Cu and \({{C}_{v}}\) are the amplitudes of the Bloch functions |up〉 and \(\left| {{{v}_{{\mathbf{p}}}}} \right\rangle \), respectively. Multiplying the equation \(H\left| {{{\Psi }_{{\mathbf{p}}}}} \right\rangle = {{E}_{{\mathbf{p}}}}\left| {{{\Psi }_{{\mathbf{p}}}}} \right\rangle \) by \(\left\langle {{{u}_{{\mathbf{p}}}}} \right|\) and \(\left\langle {{{v}_{{\mathbf{p}}}}} \right|\) from the left, we obtain the system of equations

$$\left( {\begin{array}{*{20}{c}} {\mathcal{H}_{{uu}}^{{\mathbf{p}}}}&{\mathcal{H}_{{uv}}^{{\mathbf{p}}}} \\ {\mathcal{H}_{{vu}}^{{\mathbf{p}}}}&{\mathcal{H}_{{vv}}^{{\mathbf{p}}}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{{C}_{u}}} \\ {{{C}_{v}}} \end{array}} \right)$$
$$ = {{E}_{{\mathbf{p}}}}\left( {\begin{array}{*{20}{c}} {S_{{uu}}^{{\mathbf{p}}}}&{S_{{uv}}^{{\mathbf{p}}}} \\ {S_{{vu}}^{{\mathbf{p}}}}&{S_{{vv}}^{{\mathbf{p}}}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{{C}_{u}}} \\ {{{C}_{v}}} \end{array}} \right)$$
(8)
$$ = \left[ {{{f}_{0}}({\mathbf{p}}) + {{\sigma }_{i}}{{f}_{i}}({\mathbf{p}})} \right]\left( {\begin{array}{*{20}{c}} {{{C}_{u}}} \\ {{{C}_{v}}} \end{array}} \right),$$

where \(\mathcal{H}_{{ab}}^{{\mathbf{p}}} = \left\langle {{{a}_{{\mathbf{p}}}}} \right|\mathcal{H}\left| {{{b}_{{\mathbf{p}}}}} \right\rangle \) is the matrix element of the Hamiltonian with \(a,b = u,v\); \(S_{{\mathbf{p}}}^{{ab}} = \left\langle {{{a}_{{\mathbf{p}}}}{\text{|}}{{b}_{{\mathbf{p}}}}} \right\rangle = \) \(\int a_{{\mathbf{p}}}^{ * }\left( {\mathbf{r}} \right){{b}_{{\mathbf{p}}}}\left( {\mathbf{r}} \right){{d}^{3}}{\mathbf{r}}\) is the overlap integral of the bands; \({{\sigma }_{i}}\) is the Pauli matrix with \(i = x,y,z\); and \({{f}_{0}}({\mathbf{p}})\) is multiplied by the 2 × 2 identity matrix. Orthogonal Bloch functions are usually used; in this case, the overlap matrix in Eq. (8) is the identity matrix \(S_{{\mathbf{p}}}^{{ab}} = {{\delta }_{{ab}}}\). However, strictly speaking, this is not the case in real materials (e.g., because of multiparticle effects) [14–19]. To demonstrate the possibility of overlapping of Bloch functions, we take them in the simplest form up(r) = \(\sum\nolimits_{\mathbf{R}} {{e}^{{ - i{\mathbf{p}}({\mathbf{r}} - {\mathbf{R}})}}}{{\phi }_{u}}({\mathbf{r}} - {\mathbf{R}})\) and \({{v}_{{\mathbf{p}}}}({\mathbf{r}})\, = \,\sum\nolimits_{\mathbf{R}} {{e}^{{ - i{\mathbf{p}}({\mathbf{r}} - {\mathbf{R}})}}}{{\phi }_{v}}({\mathbf{r}}\, - \,{\mathbf{R}})\), where R is the position vector of an atom at the site of the lattice and \({{\phi }_{u}}\) and \({{\phi }_{v}}\) are the atomic orbitals corresponding to the bands u and \(v\), respectively. Then, the overlap integral can be written in the form

$$\begin{gathered} \int u_{{\mathbf{p}}}^{ * }({\mathbf{r}}){{v}_{{\mathbf{p}}}}({\mathbf{r}}){{d}^{3}}{\mathbf{r}} \\ = \sum\limits_{{\mathbf{RR}}'} {{e}^{{i{\mathbf{p}}({\mathbf{R}} - {\mathbf{R}}')}}}\int \phi _{u}^{ * }({\mathbf{r}} - {\mathbf{R}}{\kern 1pt} '){{\phi }_{v}}({\mathbf{r}} - {\mathbf{R}}){{d}^{3}}{\mathbf{r}}. \\ \end{gathered} $$
(9)

Orbitals corresponding to a certain atom (\({\mathbf{R}} = {\mathbf{R}}{\kern 1pt} '\)) are always orthogonal: \(\int \phi _{a}^{ * }({\mathbf{r}} - {\mathbf{R}}{\kern 1pt} '){{\phi }_{b}}({\mathbf{r}} - {\mathbf{R}}){{d}^{3}}{\mathbf{r}} = {{\delta }_{{ab}}}\). However, orbitals for different atoms \({\mathbf{R}} \ne {\mathbf{R}}{\kern 1pt} '\) are not necessarily orthogonal and the overlap matrix \(S_{{\mathbf{p}}}^{{ab}}\) can have nonzero off-diagonal elements. As a result, the overlapping of Bloch bands is nonzero. It is noteworthy that overlapping at the degeneracy point is identically zero because of the Kramers theorem. Indeed, according to this theorem, Bloch functions in different bands at the crossing point \({{{\mathbf{p}}}_{0}}\) are related as \(\mathcal{T}\left| {{{u}_{{{{{\mathbf{p}}}_{0}}}}}} \right\rangle = \left| {{{v}_{{{{{\mathbf{p}}}_{0}}}}}} \right\rangle \), where \(\mathcal{T}\) is the time reversal operator. Then, it can be shown that \(\left\langle {{{u}_{{{{{\mathbf{p}}}_{0}}}}}|\mathcal{T}{{u}_{{{{{\mathbf{p}}}_{0}}}}}} \right\rangle = 0\) [20, 21]. Equation (8) can be represented in the form

$${{({{S}_{{\mathbf{p}}}})}^{{ - 1}}}\left[ {{{f}_{0}}({\mathbf{p}}) + {{\sigma }_{i}}{{f}_{i}}({\mathbf{p}})} \right]\left( {\begin{array}{*{20}{c}} {{{C}_{u}}} \\ {{{C}_{v}}} \end{array}} \right) = \tilde {\mathcal{H}}\left( {\begin{array}{*{20}{c}} {{{C}_{u}}} \\ {{{C}_{v}}} \end{array}} \right),$$
(10)

where \(S_{{\mathbf{p}}}^{{ab}}\) is the matrix appearing in Eq. (8). Near the Dirac point, fi(p) ≈ \({{v}_{{\text{F}}}}{{p}_{i}}\) and \({{f}_{0}}({\mathbf{p}}) = 0\). The substitution of the elements of the overlap matrix in the form \(S_{{\mathbf{p}}}^{{uu(vv)}} = 1\) (following from the normalization of Bloch functions) and \(S_{{\mathbf{p}}}^{{uv}} = (i{{\vartheta }_{y}} - {{\vartheta }_{x}}){\text{/}}{{v}_{F}} = (S_{{\mathbf{p}}}^{{vu}}){\text{*}}\) into the corresponding effective Hamiltonian \(\tilde {\mathcal{H}} = (S_{{ab}}^{{\mathbf{p}}}{{)}^{{ - 1}}} \times \) \(\left[ {{{f}_{0}}({\mathbf{p}}) + {{\sigma }_{i}}{{f}_{i}}({\mathbf{p}})} \right]\) gives

$$\tilde {\mathcal{H}} = \frac{1}{{1 - {{\beta }^{2}}}}\left( {\boldsymbol{\vartheta} \cdot {\mathbf{p}} + \boldsymbol{\sigma} \cdot \left( {{{v}_{{\text{F}}}}{\mathbf{p}} + i\left[ {{\mathbf{p}} \times \boldsymbol{\vartheta} } \right]} \right)} \right),$$
(11)

where \(\beta = \vartheta {\text{/}}{{v}_{{\text{F}}}}\) and \(\vartheta = \left| \boldsymbol{\vartheta} \right|\). The spectrum of this Hamiltonian has the form

$$\varepsilon \left( p \right) = \frac{{\boldsymbol{\vartheta} \cdot {\mathbf{p}} \pm \sqrt {{{{\left( {\boldsymbol{\vartheta} \cdot {\mathbf{p}}} \right)}}^{2}} + (1 - {{\beta }^{2}})v_{{\text{F}}}^{2}{{p}^{2}}} }}{{1 - {{\beta }^{2}}}}.$$
(12)

As seen, overlapping leads to a new vector parameter \(\boldsymbol{\vartheta} = ({{\vartheta }_{x}},{{\vartheta }_{y}})\)  in the Hamiltonian, which results in the tilt of the spectrum. Figure 1 presents the spectrum at various overlapping parameters. Below, we set \({{v}_{{\text{F}}}}\) = 1.

Fig. 1.
figure 1

(Color online) Energy versus the momentum components px and py given on the horizontal axes according to Eq. (12) at pz = 0 in the presence of overlapping, which is specified for simplicity in the form ϑ = (ϑx, 0, 0), where ϑx = (from top to bottom) 0, 0.3, and 0.6.

Full size image

The Hamiltonian (11) is non-Hermitian, but its spectrum is real because this Hamiltonian is pseudo-Hermitian, i.e.,

$${{\tilde {\mathcal{H}}}^{\dag }} = \eta \tilde {\mathcal{H}}{{\eta }^{{ - 1}}},$$
(13)

where the operator \(\eta \) can be represented in the form \(\eta = {{D}^{\dag }}D\), where D is a certain nondegenerate operator. The pseudo-Hermiticity condition (13) is a more general condition of the reality of the spectrum of the Hamiltonian [22–25]. The operator D for a Hermitian Hamiltonian is unitary and condition (13) is reduced to the conventional Hermiticity condition. To demonstrate that condition (13) is always valid for the Hamiltonian (11) because \(\beta < 1\), it is necessary to prove that the condition

$$\eta \boldsymbol{\sigma} \cdot \left( {{\mathbf{p}} + i\left[ {{\mathbf{p}} \times \boldsymbol{\vartheta} } \right]} \right){{\eta }^{{ - 1}}} = \boldsymbol{\sigma} \cdot \left( {{\mathbf{p}} - i\left[ {{\mathbf{p}} \times \boldsymbol{\vartheta} } \right]} \right),$$
(14)

where \(\eta = {{D}^{\dag }}D\), is satisfied at \(\vartheta < 1\). The Hermitian matrix \(\eta = {{D}^{\dag }}D\) can be represented in the general form

$$\eta {\text{ }} = {{a}_{0}} + {\boldsymbol{\sigma}}\;\cdot\;{\mathbf{a}},$$
(15)

where \({{a}_{0}},{\mathbf{a}} \in \mathbb{R}\). Condition (14) is now written in the form

$$\begin{gathered} \left( {{{a}_{0}} + \boldsymbol{\sigma} \cdot {\mathbf{a}}} \right)\left( {1 + \boldsymbol{\sigma} \cdot \boldsymbol{\vartheta} } \right)\boldsymbol{\sigma} \cdot {\mathbf{p}} \\ = \left( {1 - \boldsymbol{\sigma} \cdot \boldsymbol{\vartheta} } \right)\boldsymbol{\sigma} \cdot {\mathbf{p}}\left( {{{a}_{0}} + \boldsymbol{\sigma} \cdot {\mathbf{a}}} \right). \\ \end{gathered} $$
(16)

This condition is valid at \({{a}_{0}} = 1\) and \({\mathbf{a}} = - \boldsymbol{\vartheta} \). The matrix \(\eta = {{D}^{\dag }}D\) has the property \(\det \eta = \det ({{D}^{\dag }}D) = \) \({{\left| {\det (D)} \right|}^{2}} > 0\). Therefore, \(a_{0}^{2} - a_{i}^{2} > 0\). In other words, if \(a_{0}^{2} - a_{i}^{2} < 0\), the representation \(\eta = {{D}^{\dag }}D\) is impossible and the condition of the reality of the spect-rum (13) is not satisfied. Thus, the spectrum of the Hamiltonian (11) is real under the condition \(\vartheta < 1\). In particular, this means that the topological characteristics of this Hamiltonian are the same as those for the simple Dirac Hamiltonian σ · p.

It could be thought that the conditions of the existence of contact points of these bands first obtained in [13] change for overlapping bands. In particular, this overlapping could lead to the disappearance of contact points and to opening of a gap. We repeat the determination of the energy spectrum near contact points in the \({\mathbf{kp}}\) approximation in the presence of overlapping. In the \({\mathbf{kp}}\) approximation (the electron–electron interaction V is taken into account in the Hartree approximation),

$$\left[ { - \frac{{{{\hbar }^{2}}{{\nabla }^{2}}}}{{2m}} - \frac{{i{{\hbar }^{2}}}}{m}{\mathbf{k}} \cdot \boldsymbol{\nabla} + \frac{{{{\hbar }^{2}}}}{{2m}} + V} \right]\left| {{{u}_{{\mathbf{k}}}}} \right\rangle = {{E}_{k}}\left| {{{u}_{{\mathbf{k}}}}} \right\rangle .$$
(17)

According to the standard procedure, \( - \frac{{i{{\hbar }^{2}}}}{m}\boldsymbol{\kappa} \cdot \boldsymbol{\nabla} \) near the contact k + κ, where κ is a small deviation, is considered as a perturbation. In the presence of overlapping, the energy change δE(k + κ) under small deviation from the contact is given by the expression

$$\begin{gathered} \delta E\left( {{\mathbf{k}} + \boldsymbol{\kappa} } \right) \\ = \pm \frac{{\sqrt {{{{\left( {\boldsymbol{\kappa} \cdot \overline {\mathbf{f}} } \right)}}^{2}} + {{{\left( {\boldsymbol{\kappa} \cdot {\text{Re}}{\mathbf{g}}} \right)}}^{2}} + {{{\left( {\boldsymbol{\kappa} \cdot \overline {{\text{Im}}{\mathbf{g}}} } \right)}}^{2}}} }}{{1 - \vartheta _{x}^{2}}} + o\left( {{{\kappa }^{2}}} \right), \\ \end{gathered} $$
(18)

where \(\overline {\mathbf{f}} = {\mathbf{f}} + i{{\vartheta }_{x}}{\text{Im}}{\mathbf{g}}\), \(\overline {{\text{Im}}{\mathbf{g}}} = {\text{Im}}{\mathbf{g}} - i{{\vartheta }_{x}}{\mathbf{f}}\), ϑ = (ϑx, 0, 0) is set for simplicity, and

$${\mathbf{f}} = - i\frac{{{{\hbar }^{2}}}}{m}\left[ {\left\langle {{{u}_{{\mathbf{k}}}}{\text{|}}\nabla {{u}_{{\mathbf{k}}}}} \right\rangle - \left\langle {{{v}_{{\mathbf{k}}}}{\text{|}}\nabla {{v}_{{\mathbf{k}}}}} \right\rangle } \right],$$
(19)
$${\mathbf{g}} = i\frac{{{{\hbar }^{2}}}}{m}\left\langle {{{u}_{{\mathbf{k}}}}{\text{|}}\nabla {{v}_{{\mathbf{k}}}}} \right\rangle .$$
(20)

For the existence of a contact, it is sufficient that none of three vectors \({\mathbf{f}} + i{{\vartheta }_{x}}{\text{Im}}{\mathbf{g}},{\text{Re}}{\mathbf{g}},{\text{Im}}{\mathbf{g}} - i{{\vartheta }_{x}}{\mathbf{f}}\) be identically zero. If any of these vectors is identically zero, the direction κ perpendicular to the other two vectors will always exist. The spectrum in this direction will be quadratic in momentum (because the linear term of the Taylor series is identically zero); i.e., any arbitrarily small perturbation can induce a band gap. The vector \({\text{Re}}{\mathbf{g}}\) in the presence of a center of inversion is always zero. Consequently, an isolated contact point of the bands will be absent in this case. None of the vectors is identically zero in the absence  of the center of inversion. The equality \({\text{Im}}{\mathbf{g}} - i{{\vartheta }_{x}}{\mathbf{f}} = 0\) can be expected in certain directions κ in the presence of overlapping. However, this equality is possible only under the condition \({\text{Im}}{\mathbf{g}} = {\mathbf{f}} \equiv 0\) because the vector f is real. However, even if the vec-tor f becomes complex for some reasons, the condition \({\text{Im}}{\mathbf{g}} - i{{\vartheta }_{x}}{\mathbf{f}} = 0\) is insufficient for the disappearance of the contact. An additional condition \({\mathbf{f}},{\text{Re}}{\mathbf{g}} \bot \kappa \) for the direction κ is necessary. Thus, the radicand in Eq. (18) can vanish at a single point only if three vectors f, Re g, and Im g are coplanar or at least one of them is identically zero. This conclusion completely coincides with conclusions in [13]; i.e., overlapping affects only the shape of the spectrum and does not affect the fundamental conditions of the existence of the Dirac point.

As shown above, overlapping of the bands results in the tilt of the spectrum. However, if electron and hole Fermi pockets cross at the Dirac point, the spectrum is tilted without overlapping. Overlapping leads only to an additional tilt. The total Hamiltonian including the tilt and overlapping has the form (the derivation is the same as for the direct spectrum only with f0(p) = ω · p)

$$\begin{gathered} \tilde {\mathcal{H}} = \frac{1}{{1 - {{\beta }^{2}}}}\left[ {\left( {\boldsymbol{\omega} + \boldsymbol{\vartheta} } \right) \cdot {\mathbf{p}}} \right. \\ \left. { + \;\boldsymbol{\sigma} \cdot \left( {{\mathbf{p}} + \left( {\boldsymbol{\omega} \cdot {\mathbf{p}}} \right)\boldsymbol{\vartheta} + i\left[ {{\mathbf{p}} \times \boldsymbol{\vartheta} } \right]} \right)} \right]. \\ \end{gathered} $$
(21)

This Hamiltonian contains two parameters ω and \(\boldsymbol{\vartheta} \). It is seen that the tilt disappears completely at \(\boldsymbol{\vartheta} = - \boldsymbol{\omega} \). Thus, overlapping of the bands can lead to the direct spectrum. The tilt vector in the system with two Dirac points has opposite signs at different points. This means that overlapping results in the disappearance of the tilt at \(\boldsymbol{\vartheta} = - \boldsymbol{\omega} \) at only one of the points. Thus, a phase where the spectrum is tilted at one Dirac point and is direct at the other point is possible. An equilibrium current could appear in this case because of a nonzero total tilt from two points. However, the anisotropy of the spectrum is such that the equilibrium current remains zero. Figure 2 shows spectra (isoenergy surfaces) at various overlapping parameters in the presence of the tilt.

Fig. 2.
figure 2

(Color online) Spectrum (isoenergy surfaces) of the Hamiltonian (21) at pz = 0 and at the tilt ω = (–0.7, 0, 0) at overlapping ϑ = (ϑx, 0, 0), where ϑx = (a) 0, (b) 0.4, (c) 0.7, and (d) 0.8.

Full size image

Finally, we can show that the Hamiltonian (21) describes the spinor field in the curved spacetime by setting \(e_{\nu }^{\mu } = {{v}_{{\text{F}}}}\delta _{\nu }^{\mu } + \delta _{\nu }^{0}{{\omega }^{i}}\delta _{i}^{\mu } + {{\vartheta }_{i}}\delta _{\nu }^{i}\delta _{0}^{\mu }\) in the general Lagrangian

$$\mathcal{L} = i\overline \psi {{\gamma }^{\nu }}e_{\nu }^{\mu }{{\partial }_{\mu }}\psi $$
(22)

in the tetrad formalism. Such Weyl fermions were proposed in [26, 27]. Nonzero components \(e_{i}^{0},e_{0}^{i}\) of the tetrad (they are \(e_{i}^{0} = {{\vartheta }_{i}},e_{0}^{i} = {{\omega }^{i}}\)) indicate that the spacetime is curved. One of the main conclusions of this work is that the Hamiltonian (21) corresponds to this Lagrangian. This can be seen from the Euler–Lagrange equation. Thus, overlapping of the bands is effectively manifested as the curvature of the spacetime for the Dirac spinor field. This provides a unique possibility of simulating various exotic phenomena in such Dirac systems caused by the curvature of the spacetime. Inhomogeneous tilted Weyl spectra were used in [28] (see also [29]) to effectively simulate the event horizon near black and white holes. However, such spectra do not include a relation between the spatial components of the Dirac matrix and the time component of the momentum (tetrad \(e_{i}^{0}\)) and thereby cannot be used to simulate all features of the curved spacetime. This demerit is compensated by overlapping, which results in the contribution of the component \(e_{i}^{0}\), which is missed for a symmetry reason.

To summarize, we have studied effects of band overlapping on the Dirac Hamiltonian. They should be taken into account to analyze various phenomena in Dirac systems with a tilted spectrum (see, e.g., [30–33]).