Magnetic field effect on topological properties of Dirac semimetals PdTe₂/PtTe₂/PtSe₂

Since the discovery of two-dimensional quantum spin Hall effect in 2006 [1], the development of topological electronic materials (TEMs) has entered a new stage. Various novel TEMs have been discovered one after another, such as topological insulator [2], Weyl semimetal (WSM) [3], triple-point fermion semimetals [4], Dirac semimetal (DSM) [5–8], manifold fermion semimetal [9], topological node line semimetal [10, 11], etc. Recently, TEM has been further extended to high-order topological materials [12, 13] and non-Hermitian topological states of matter [14], which show very rich physical significance and tremendous application potential. The DSM can be further classified into magnetic and non-magnetic DSM according to whether they break time reversal symmetry (TRS). Nonmagnetic or TRS invariant DSM can be obtained through both symmetry-enforced and band inversion routes [11, 15]. Symmetry-enforced DSM only exists in crystal with nonsymmorphic space group symmetry and has four-dimensional irreducible (co)representations [15]. And the Dirac points (DPs) must be located at high-symmetry momenta on the boundary of Brillouin zone (BZ). Recent studies showed that there are 69 nonsymmorphic space groups host symmetry-enforced DPs [11]. Band inversion-induced DSM possesses crossing points of inverted band, which are protected by rotational and mirror symmetries and formed on high symmetry lines (HSLs) [11, 15], such as Na₃Bi [5], Cd₃As₂ [16].

When a system has TRS and inversion symmetry (IS), each band at each k point is double degeneracy, which is a manifestation of Kramers theorem. Due to orbital hybridization, crystal field effect, and spin–orbit coupling (SOC), the degeneracy band generally will be split and then be possibly reversed [17]. The inverted bands must have crossing nodes under symmetry protection. The little groups of non-high symmetry points (HSPs) or lines in the BZ possess lower-dimensional irreducible representation and then non-degenerate bands there. While the group representation of HSPs and lines in the BZ exit possibly higher dimension and results in a higher degree of degeneracy of bands. Discrete crystal rotational symmetry will cause double degenerate inverted bands intersecting on the rotation axis to form band-inverted DPs [11]. The stability of the four-fold degenerate point formed by the above band-inverted mechanism is relatively weak, because the Dirac node can be split by adjusting the parameters (removing band inversion) or breaking the crystal symmetry [7, 11, 16]. Therefore, band inversion-induced DSM as a parent compound can be transformed into other novel states. The three-dimensional Dirac fermions in a DSM are composed of two overlapped Weyl fermions, which can be separated in momentum space if TRS or IS was broken [16, 18, 19].

In our work, we constructed a Wannier-function-based tight-binding (WFTB) model for band inversion-induced DSM PdTe₂/PtTe₂/PtSe₂ with magnetic field B applied along various directions. Atom-centered localized Wannier functions (WFs) are generated based on the calculations of ab initio electronic structure by selecting appropriate initial atomic orbitals. We then constructed a WFTB model by separately adding SOC Hamiltonian and Zeeman energy based on atomic-like Wannier orbitals. Landau levels or Peierls phases are not considered in our WFTB model. Since three systems PdTe₂/PtTe₂/PtSe₂ have the same symmetry and similar band structure near the Fermi level (E_F), we mainly focus on PdTe₂. However, the analysis and main conclusions are also applicable to the other two systems (PtTe₂ and PtSe₂). Some calculations for PtTe₂ and PtSe₂ are provided in the supplementary material (see supplementary material (https://stacks.iop.org/JPCM/34/085802/mmedia)).

Bulk PdTe₂, PtTe₂ and PtSe₂ crystallize in the trigonal CdI₂-type crystal structure with space group $P\bar{3}m1$ (No. 164) and the corresponding point group is D_3d . The crystal structure and the first BZ with HSLs and HSPs are shown in figures 1(a)–(c), respectively. The generators of the D_3d include C_3z , C_2x , and σ_d1, therefore the bulk crystal has twelve symmetry operators. The DPs of PdTe₂/PtTe₂/PtSe₂ are protected by IS, TRS, and C_3v symmetries [8].

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The initial positions of atoms and lattice parameters are fully optimized through adopting conjugate gradient method. The obtained lattice constants are given in table 1. We first calculated the electronic structure of PdTe₂/PtTe₂/PtSe₂ without SOC and magnetic field by using the density functional theory (DFT) code VASP [20, 21]. With the DFT calculated bands and initial atomic orbitals, we generated WFs by Wannier90 code [22]. The overlap and projection matrixes are computed at each DFT-sampled k point. In order to maintain the features of atomic orbitals, maximal localization is not applied in the calculations of Wannierization. We checked that the generated WFs are close to atomic-like orbitals. Then we constructed a SOC-free WFTB model from the WFs and added SOC Hamiltonian to the WFTB model such that the model-calculated bands near the E_F agree with the DFT bands. Lastly, we considered Zeeman energy in the WFTB model to study the magnetic field effect on the bands, especially the bands near the DPs.

In the DFT calculations, we adopted the Perdew–Burke–Ernzerhof generalized gradient approximation [23] for the exchange correlation functional with and without SOC. We used projector augmented wave pseudopotentials [24] with an energy cutoff of 500 eV and a 16 × 16 × 8 Monkhorst-Pack k-point mesh. In order to construct the SOC-free WFTB model, we start with an initial set of 12 projected atomic orbitals comprised of s, d_z2, d_x2–y2, d_xy , d_yz , and d_xz orbitals centered at the one Pd/Pt sites and p_x , p_y , and p_z orbitals centered at the two Te/Se sites in the primitive unit cell. We need to include both Pd/Pt-s/d orbitals and Te/Se-p orbitals in the WFTB model due to their large contributions to the bands near the E_F indicated by our DFT calculations (figure 2(c)).

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We constructed the SOC-free WFTB model by using the generated WFs, | R + s _β ⟩, centered at R + s _β , where R is Bravais lattice vector and s _β denote the sites of orbital β (β = 1, ..., 12). The SOC-free Hamiltonian matrix H₀ [25] reads

$$\begin{equation}{H}_{0,\alpha \beta }(\boldsymbol{k})=\left\langle {\psi }_{\boldsymbol{k}\alpha }\right\vert {H}_{0}\left\vert {\psi }_{\boldsymbol{k}\beta }\right\rangle =\sum\limits _{R}{\text{e}}^{\text{i}\boldsymbol{k}\cdot \boldsymbol{R}}{t}_{\alpha \beta }(\boldsymbol{R}-0),\end{equation} \tag{ 1 }$$

$$\begin{equation}{t}_{\alpha \beta }(R-0)=\left\langle 0+{s}_{\alpha }\vert {H}_{0}\vert R+{s}_{\beta }\right\rangle ,\end{equation} \tag{ 2 }$$

where |ψ_k ⟩ are Bloch orbitals and can be expanded by WFs basis. Here t_αβ ( R − 0) is a hopping energy between orbital α at site s _α in the home cell at R = 0 and orbital β at site s _β in the unit cell located at R . We added SOC term to the SOC-free Hamiltonian H₀. The order of the basis in the SOC/Zeeman term is (p_z ↑, p_x ↑, p_y ↑) and (s↑, d_z2↑, d_xz ↑, d_yz ↑, d_x2–y2↑, d_xy ↑) for respective Te/Se-p and Pd/Pt-s/d orbitals. Considering the spin degrees of freedom, the number of WFs basis is now 24. The SOC Hamiltonian is H_SOC = λ L · σ , where λ is the SOC strength parameter, L is the orbital angular momentum, and σ represents Pauli spin matrix. By fitting with the VASP-calculated bands, we found the fitted SOC parameters for PdTe₂/PtTe₂/PtSe₂ listed in table 2. The magnetic field induced Zeeman term is H_Zee = gμ_B SB , where g is Lande factor, μ_B is Bohr magneton and S is the spin angular momentum. Taking Te atom as an example, the SOC matrix and Zeeman matrix of Te atom are respectively

$$\begin{align}\hfill & \left(\begin{matrix}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1/2\hfill & \hfill \mathrm{i}/2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -\mathrm{i}/2\hfill & \hfill 1/2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \mathrm{i}/2\hfill & \hfill 0\hfill & \hfill -\mathrm{i}/2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1/2\hfill & \hfill \mathrm{i}/2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -1/2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\mathrm{i}/2\hfill \\ \hfill -\mathrm{i}/2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{i}/2\hfill & \hfill 0\hfill \end{matrix}\right)\hfill \\ \hfill & \quad =\left(\begin{matrix}\hfill {H}_{\mathrm{T}\mathrm{e},\mathrm{S}\mathrm{O}\mathrm{C}}^{11}\hfill & \hfill {H}_{\mathrm{T}\mathrm{e},\mathrm{S}\mathrm{O}\mathrm{C}}^{12}\hfill \\ \hfill {H}_{\mathrm{T}\mathrm{e},\mathrm{S}\mathrm{O}\mathrm{C}}^{21}\hfill & \hfill {H}_{\mathrm{T}\mathrm{e},\mathrm{S}\mathrm{O}\mathrm{C}}^{22}\hfill \end{matrix}\right).\hfill \end{align} \tag{ 3 }$$

Here, the 3 × 3 matrices ${H}_{\mathrm{T}\mathrm{e},\mathrm{S}\mathrm{O}\mathrm{C}}^{11}={H}_{\mathrm{T}\mathrm{e},\mathrm{S}\mathrm{O}\mathrm{C}}^{22}$ and ${H}_{\mathrm{T}\mathrm{e},\mathrm{S}\mathrm{O}\mathrm{C}}^{12}={({H}_{\mathrm{T}\mathrm{e},\mathrm{S}\mathrm{O}\mathrm{C}}^{21})}^{+}$ (superscript + denotes Hermitian conjugate) are non-diagonal

$$\begin{align}\hfill & \left(\begin{matrix}\hfill {a}_{z}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {a}_{+}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {a}_{z}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {a}_{+}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {a}_{z}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {a}_{+}\hfill \\ \hfill {a}_{-}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{a}_{z}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {a}_{-}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{a}_{z}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {a}_{-}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{a}_{z}\hfill \end{matrix}\right)\hfill \\ \hfill & \quad =\left(\begin{matrix}\hfill {H}_{\mathrm{T}\mathrm{e},\mathrm{Z}\mathrm{e}\mathrm{e}}^{11}\hfill & \hfill {H}_{\mathrm{T}\mathrm{e},\mathrm{Z}\mathrm{e}\mathrm{e}}^{12}\hfill \\ \hfill {H}_{\mathrm{T}\mathrm{e},\mathrm{Z}\mathrm{e}\mathrm{e}}^{21}\hfill & \hfill {H}_{\mathrm{T}\mathrm{e},\mathrm{Z}\mathrm{e}\mathrm{e}}^{22}\hfill \end{matrix}\right),\hfill \end{align} \tag{ 4 }$$

where a₊ = gμ_B B sin θ e^−iφ , a₋ = gμ_B B sin θ e^iφ , a_z = gμ_B B cos θ, the 3 × 3 matrices ${H}_{\mathrm{T}\mathrm{e},\mathrm{Z}\mathrm{e}\mathrm{e}}^{11}=-{H}_{\mathrm{T}\mathrm{e},\mathrm{Z}\mathrm{e}\mathrm{e}}^{22}$ and ${H}_{\mathrm{T}\mathrm{e},\mathrm{Z}\mathrm{e}\mathrm{e}}^{12}={({H}_{\mathrm{T}\mathrm{e},\mathrm{Z}\mathrm{e}\mathrm{e}}^{21})}^{+}$ are diagonal, the (θ, φ) represents the azimuth angle in the spherical coordinates. When the magnetic field B is applied in different directions, the ${H}_{\mathrm{T}\mathrm{e},\mathrm{Z}\mathrm{e}\mathrm{e}}^{11}/{H}_{\mathrm{T}\mathrm{e},\mathrm{Z}\mathrm{e}\mathrm{e}}^{22}/{H}_{\mathrm{T}\mathrm{e},\mathrm{Z}\mathrm{e}\mathrm{e}}^{12}/{H}_{\mathrm{T}\mathrm{e},\mathrm{Z}\mathrm{e}\mathrm{e}}^{21}$ matrix will change periodically with the azimuth angle. Considered the total number of orbitals in three systems is 24, and the Zeeman matrix is

$$\begin{equation}{H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}=\left(\begin{matrix}\hfill {H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{11}\hfill & \hfill {H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{12}\hfill \\ \hfill {H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{21}\hfill & \hfill {H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{22}\hfill \end{matrix}\right).\end{equation} \tag{ 5 }$$

When the matrix ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{11}/{H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{22}/{H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{12}/{H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{21}$ is direct sum of the Zeeman matrix of three atoms and results in a 12 × 12 matrix. The diagonal matrix ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{11}/{H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{22}$ is the spin up/down part of all atoms and ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{11}=-{H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{22}$, the non-diagonal matrix ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{12}/{H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{21}$ is the up-down spin coupling part of all atoms and ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{12}={({H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{21})}^{+}$. When an external magnetic field is applied, the matrix ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{11}/{H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{22}/{H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{12}/{H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{21}$ also changes periodically, similar to the matrix ${H}_{\mathrm{T}\mathrm{e},\mathrm{Z}\mathrm{e}\mathrm{e}}^{11}/{H}_{\mathrm{T}\mathrm{e},\mathrm{Z}\mathrm{e}\mathrm{e}}^{22}/{H}_{\mathrm{T}\mathrm{e},\mathrm{Z}\mathrm{e}\mathrm{e}}^{12}/{H}_{\mathrm{T}\mathrm{e},\mathrm{Z}\mathrm{e}\mathrm{e}}^{21}$, and hence leads to distinct evolution of positions and energies of magnetic-induced WPs.

The total Hamiltonian for the WFTB model is H = H₀ + H_SOC + H_Zee. We presented the results under magnetic field gμ_B B = 10 meV in the following calculations, unless specified otherwise. And the strength of B that we considered here is experimentally achievable [16]. Furthermore, our findings would not qualitatively change with the field strength if the magnetic field is not extremely high. We choose a larger B field, because the split of each DP is more obvious and easier to analyze.

We check whether our WFTB model is correct by comparing the WFTB band with the first-principles electronic bands. In the absence of magnetic field, the WFTB and VASP-calculated bands for PdTe₂ are in high agreement near the E_F in the absence/presence of SOC as shown in figures 2(a) and (b). The positions and energies of DPs for three considered systems are listed in table 1, which are consistent with the previous calculations [6, 8]. The bands for PtTe₂ and PtSe₂ are presented in the supplementary material. In the vicinity of the DPs, the bands are mainly composed of Te-p orbitals and a small proportion of Pd-s/d orbitals as shown in figure 2(c).

We apply a uniform magnetic field along the C_3z rotation axis, and find each DP was split into four separate WPs along the c axis as illustrated in figure 3(c). The result of magnetic field effects on the rotation axis was also discussed in previous studies [16, 18, 19]. Figures 3(b) and (c) show the development of the four WPs along the c axis in the half BZ under the + B field. The positive/negative sign means the direction of B along with the ±z axis, labeled by + B /− B . The number of field-induced WPs does not change and nor do their chiralities (topological charge χ) within magnetic field strength [0, 10] meV. The chirality of the WPs is identified by calculating Wannier charge center of the Bloch bands which was obtained from our generated total WFTB model H, which was realized by using WannierTools code [26]. The IS partner of the four WPs (W₁₅, W_16a /W_16b , W₁₇) in the k_z > 0 BZ is located in the k_z < 0 BZ as shown in figure 3(d). We only analyzed two WPs W_16a /W_16b near the A–Γ line formed between the 16th (blue conduction band) and 17th band (green valence band) shown in figures 3(b) and (c).

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The k_z positions ${k}_{z}^{16a}/{k}_{z}^{16b}$ and energies E_16a /E_16a of two WPs are listed in table 2 for three considered systems. The ${k}_{z}^{16a}/{k}_{z}^{16b}$ and E_16a /E_16a change linearly with the + B varies ranging from 0 meV to 10 meV as shown in figure 3(e) and so do when B is from −10 meV to 0 meV. Hence, the ${k}_{z}^{16a}/{k}_{z}^{16b}$ and E_16a /E_16a demonstrate a direction symmetrical variation under the +B and −B. The symmetrical change of WPs can be understood by the Zeeman matrix (4) and (5). Firstly, we note the H₀ + H_SOC is unchanged and the H_Zee is the only changing term under magnetic field B . Secondly, when the θ = 0° (+ B ) or θ = 180° (− B ), the non-diagonal part ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{12}$ and ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{21}$ disappear and the Zeeman matrix H_Zee only includes the ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{11}$ and ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{22}$ diagonal part. Thirdly, the + B and − B will change entirely a sign on H_Zee due to the a_z factor in the diagonal part, for example, ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{11}$ and ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{22}$. As a result, the total Hamiltonian will change from H_{+ B} = H₀ + H_SOC + H_Zee to H_{− B} = H₀ + H_SOC − H_Zee. However, it is noteworthy that both H_{+ B} and H_{− B} have the same eigenvalue, resulting in the same two WPs as identified in our calculations. As the strength of magnetic field increases, the k_z coordinate ${k}_{z}^{16a}({k}_{z}^{16b})$ is moving away from (close to) the Γ point, the energy E_16a (E_16b ) is moving away from (close to) the E_F.

Figures 4(a) and (b) show WFTB-calculated energies and the k_z of W_16a /W_16b nodes when the B field along the a axis. The energies of two WPs change in a parabolic-like shape as B field changes from −10 meV to 10 meV as shown in figure 4(a). Interestingly, the energies E_16a /E_16b of both WPs close gradually to the E_F (=0) with the increasing B field, regardless of the B field direction, which is different from the behavior of E_16a /E_16b when B field along z-axis as shown in figure 3(f). Additionally, it is noted that the energy E_16a is slightly lower (higher) than the E_16b when B field along with + a (− a ) axis. This variation reflects roughly a distinct splitting of bands under the reversed magnetic field. In order to understand the different behavior of energies and coordinates of W_16a /W_16b , we here consider a simple atomic extreme 'bands' case under magnetic field. In the absence of SOC, the degenerate spin-up and spin-down bands are split due to the Zeeman effect and form crossing points cp1 and cp2 when applying a + B magnetic field as shown in figure 4(e). However, both cp1 and cp2 exchange their positions and energies when reversing the direction of magnetic field (i.e. − B ) as shown in figure 4(f). When further adding SOC, although spin-up and spin-down bands will mix, mixed bands generally possess non-equal percentage spin bands. As a consequence, the above similar exchange of mixed bands will take place and then exchange positions of two crossing points, leading to finally opposite 'positions' for both W_16a and W_16b under + B and − B field in figures 4(a) and (b). The positions of both WPs demonstrate similar parabolic variation and the dependence of magnetic field direction shown in figure 4(b). When in a -axis magnetic field, the k_z of W_16a /W_16b moves toward the Γ as the magnetic field increases. The energies and k_z coordinates of both W_16a /W_16b under b -axis magnetic field are presented in figures 4(c) and (d). The results are similar qualitatively in addition to the larger differences between ${k}_{z}^{16a}$/E_16a and ${k}_{z}^{16b}$/E_16b compare with those of the corresponding values under a -axis magnetic field. In addition, the positions of both W_16a /W_16b slightly deviate from the k_z -axis under both a - and b -axes magnetic field and other materials have similar results [18].

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We also investigated the band's evolution when a B field is applied in ac/bc/ab plane for three considered systems. Here, we only focus on the PdTe₂ and the results for PtTe₂ and PtSe₂ are given in supplementary material. When the B field is rotated from (θ, φ) = (0°, 0°) to (θ, φ) = (180°, 0°) with angle step 5°, we also identified two pairs of WPs that are close to the k_z -axis between the 16th and 17th band and the distribution of magnetic-induced WPs is discussed in reference [18]. The evolution of k_z and energy of W_16a /W_16b with k_z > 0 as a function of angle φ are shown in figures 5(a) and (b). The W_16a /W_16b with larger/smaller k_z is further/closer to the point Γ at each angle. The k_z of WPs shows both mirror symmetries over [0°, 180°] and center IS over [0°, 360°]. The chirality of W_16a /W_16b WPs changes from +1/−1 (θ = [0°, 90°]) to −1/+1 (θ = [90°, 180°]). The two WPs merge when the θ is 90°. These symmetries can be rational by the Zeeman term. The angle θ is only variable for ac plane B field and have cos θ/sin θ factor in the diagonal part ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{11}$, ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{22}$ and non-diagonal part ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{12}$, ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{21}$ of H_Zee, which leads to a periodical change of H as angle θ. Consequently, the k_z of WPs within θ = [0°, 90°] is mirror-symmetric to that of WPs within θ = [90°, 180°] due to the symmetry of sine/cosine function. The k_z of WPs within θ = [0°, 90°] decrease as the increasing θ. The k_z of W_16a /W_16b in unit 1 Å⁻¹ are 0.487/0.480 for PdTe₂, 0.459/0.452 for PtTe₂, 0.352/0.343 for PtSe₂ at θ = 0° and 0.484/0.484 for PdTe₂, 0.456/0.456 for PtTe₂ and 0.347/0.347 for PtSe₂ at θ = 90°. The energies E_16a /E_16b show a similar variation tendency as angle θ as depicted in figure 5(b).

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When the B field is rotated from (θ, φ) = (0°, 90°) to (θ, φ) = (180°, 90°) in the bc plane, the k_z and energy evolution of W_16a /W_16b as a function of angle θ is shown in figures 5(c) and (d). The symmetrical evolution of the positions and energies of W_16a /W_16b as well as the change in chirality is similar to a -axis magnetic field. These symmetries are reasonable according to the analysis of Zeeman terms. The angle θ is only variable for bc plane B field and has a cos θ/i sin θ factor in the diagonal/non-diagonal part ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{11}$ and ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{22}$/non-diagonal part ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{12}$ and ${H}_{\mathrm{Z}\mathrm{e}\mathrm{e}}^{21}$ of H_Zee, which leads to a periodical change of H as angle θ. The k_z of W_16a /W_16b in unit 1 Å⁻¹ are 0.488/0.480 for PdTe₂, 0.459/0.452 for PtTe₂, 0.352/0.343 for PtSe₂ at θ = 0° and 0.484/0.483 for PdTe₂, 0.456/0.456 for PtTe₂ and 0.348/0.347 for PtSe₂ at θ = 90°.

The k_z and E_16a /E₁₆ of WPs are presented in figures 5(e) and (f) when the B field varies in the ab plane, i.e. θ = 90°, 0° ⩽ φ < 360°. The distribution of k_z and energies demonstrates a broken rotational symmetry which can be understood by the analysis of Hamiltonian. Specifically, the distribution of both k_z and E_16a /E₁₆ is approximate left-right mirror symmetrical between [0°, 90°] and [180°, 90°] and strict up-down mirror symmetrical between [0°, 180°] and [360°, 180°]. Firstly, the variable φ dominates the change of non-diagonal term of H_Zee including the sine/cosine factor. For example, the non-diagonal part contains a₊ and a₋ in the equation (4). Secondly, the characteristic variation of sine function causes approximate rational symmetries of the k_z and E_16a /E_16b among four quadrants, though sine function shows a strict symmetry for φ and 180° − φ. Specifically, the φ and 180° − φ show an approximate symmetry because the H_SOC + H_Zee is different at φ and 180° − φ, which in turn leads to slight different eigenvalues. However, the φ and 360° − φ show a strict symmetry because the H_SOC + H_Zee is same at φ and 360° − φ, which then leads to the same eigenvalues, and strict 'mirror' symmetry of WPs when φ lies between [0°, 180°] and [180°, 360°].

Additionally, when the B field is along the direction other than the c axis, the 16th and 17th bands have an on-axis bandgap on the A–Γ line and similar studies arediscussed in reference [18], and the bandgap is the largest under the bc plane B field. Notably, the crossing points are possible near the k_z axis. Figure 6(a) shows typical WFTB-calculated bands for the B field at (θ, φ) = (45°, 90°), and the bands indicate a gap between the 16th band (blue) and 17th band (green) on the A–Γ line. The gap as a function of θ is given in figure 6(b) and the θ and 180° − θ shows an approximate mirror symmetry. The gaps are zero for both θ = 0° and 180°, which implies the existence of nodes and agrees with the result of z-axis B field. The maximal gaps appear at around θ ≈ 50° and 130°.

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In summary, we have developed a WFTB model for the topological DSM PdTe₂/PtTe₂/PtSe₂, which reproduces the DFT-calculated bands well. Based on the WFTB model, we investigate the magnetic effect on topological properties of the considered systems under magnetic field B along various directions in three-dimensional space. The magnetic field-induced the phase transition of DSM-WSM and the topological properties of the resulting WSM are discussed systematically. The DSM PdTe₂/PtTe₂/PtSe₂ would transit into WSM with four pairs of Weyl points in the entire BZ under the axial-oriented magnetic field. Under the c -axis B field, each DP on the c axis will be split into four WPs, and their positions are still located on the c axis. The position and energy evolution of four splitting WPs vary linearly with the strength of magnetic field. Under the a - and b -axes magnetic field, however, the positions of the splitting WP deviate slightly from c axis, and their coordinates and energies change in parabolic-like curves with the increasing magnetic field. The symmetrical characterization of WPs' distribution under an inverted magnetic field can be rationalized by analyzing the Zeeman term in the WFTB model. When we further apply a magnetic field in the three typical planes (ac, bc, and ab), the results are more diverse compared to the axial magnetic field. When the B field is applied in the ac and bc plane, the systems also turn into WSM with four pairs of WPs. The k_z and energies of WPs within θ = [0°, 90°] and θ = [90°, 180°] is symmetrical. Under the ab plane magnetic field, the distribution of k_z and energies show broken rotational symmetry over angle φ = [0, 360°] due to the variation of non-diagonal part of Hamiltonian. Additionally, when the B field is along the direction other than the c axis, the system opens a gap on the A–Γ path, and the gap changes with the direction of a magnetic field. When the magnetic field intensity is increased to 0.1 eV, these results are still reliable and the corresponding calculations are provided in the supplementary material (see supplementary material). These results are useful for improving the understanding of the magnetic field-induced effect of other DSMs. Furthermore, our findings are meaningful for the possible applications of DSMs in the topological electronic devices under external magnetic field.

This work is supported by the National Natural Science Foundation of China (Grant No. 11204261), Natural Science Foundation of Hunan Province (Grant No. 2018JJ2381), Postgraduate Scientific Research Innovation Project of Hunan Province (Grant No. XDCX2021B127), a Key Project of Education Department of Hunan Province (Grant No. 19A471), and Postgraduate Scientific Research Innovation Project of Hunan Province (Grant No. CX20210615).

All data that support the findings of this study are included within the article (and any supplementary files).

There are no conflicts to declare.