Hidden Weyl points in centrosymmetric paramagnetic metals

In the following, the atomic structure of TaAs₂ [22], TaSb₂ [23], NbSb₂ [24] and NbAs₂ [25] is described.

The reduced unit cell of AB₂ compounds has the general form

$$\begin{eqnarray}{a}_{1} & = & (a,\,b,\,0)\\ {a}_{2} & = & (-a,\,b,\,0)\\ {a}_{3} & = & (-c,\,0,\,d)\end{eqnarray} \tag{ 2 }$$

with the parameters as given in table 1 [22, 24].

Table 1.  Unit cell dimensions (in Å) for AB₂ compounds.

	${\rm{a}}$	${\rm{b}}$	${\rm{c}}$	${\rm{d}}$
TaAs2	4.6655	1.6915	3.8420	6.7330
TaSb2	5.11	1.822	4.1950	7.1502
NbAs2	4.684	1.698	3.8309	6.7933
NbSb2	5.1198	1.8159	4.1705	7.2134

Each unit cell contains two formula units. The atoms are located at the general Wyckoff positions $(x,-x,y)\,\,{\rm{a}}{\rm{n}}{\rm{d}}\,(-x,x,-y)$ for $(x,y)$, as shown in table 2 [22, 24].

Table 2.  Atomic positions $(x,y)$.

	A	B1	B2
TaAs2	$(0.157,0.1959)$	$(0.4054,0.1082)$	$(0.1389,0.5265)$
TaSb2	$(0.152,0.19)$	$(0.405,0.113)$	$(0.147,0.535)$
NbAs2	$(0.1574,0.1965)$	$(0.4059,0.1084)$	$(0.14,0.528)$
NbSb2	$(0.1521,0.1903)$	$(0.4051,0.1127)$	$(0.1475,0.5346)$

Figure 1 shows the reduced unit cell and first BZ of TaAs₂. The k-point path along which the bandstructure calculations are performed is indicated. On the basis reciprocal to that of equation (2), the special k-points are given by

$$\begin{eqnarray}{\rm{\Gamma }} & = & (0,0,0)\\ A & = & (0,0,0.5)\\ L & = & (0.5,0,0.5)\\ M & = & (0.5,0.5,0.5)\\ V & = & (0.5,0,0)\\ Y & = & (0.5,0.5,0).\end{eqnarray} \tag{ 3 }$$

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The electronic structure calculations were performed in VASP [26], with the projector augmented-wave (PAW) [27, 28] pseudopotentials. The PBE approximation [29] was used, and spin–orbit coupling was included in the potentials. The self-consistent field (SCF) calculations were performed on a $11\times 11\times 5$ Γ-centered grid for TaAs₂, and a $10\times 10\times 5$ Γ-centered grid for NbSb₂. The energy cut-off given in the potential files was used, which is $293.2\,\mathrm{eV}$ for NbAs₂ and NbSb₂, and $223.7\,\mathrm{eV}$ for TaAs₂ and TaSb₂.

Additionally, the PBE calculations were tested against the accurate HSE06 hybrid functional [30, 31]. The hybrid SCF calculations for the band structures were performed on a Γ-centered $6\times 6\times 4$ grid for all materials. For the generation of the Wannier tight-binding model of NbSb₂, a Γ-centered $10\times 10\times 5$ grid was used.

The band structure of TaAs₂ and NbSb₂ is shown in figure 2. Both materials exhibit a pair of electron and hole pockets near the M-point, where the minimum band gap is about $318\,\mathrm{meV}$ ($120\,\mathrm{meV}$ without hybrid functionals) in the case of TaAs₂, $151\,\mathrm{meV}$ ($98\,\mathrm{meV}$) for TaSb₂, $261\,\mathrm{meV}$ ($22\,\mathrm{meV}$) for NbAs₂, and $67\,\mathrm{meV}$ ($18\,\mathrm{meV}$) in the case of NbSb₂. A more complete calculation of the band structure can be found, for example, in [32].

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The AB₂ compounds studied here have $C2/m$ symmetry (space group 12). The rotation axis is along the Cartesian y-axis. In reduced coordinates, the symmetry matrices are as follows:

• Identity $E=\,{{\mathbb{I}}}_{3\times 3}$
• Rotation ${C}_{2y}=\left(\begin{array}{ccc}0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & -\,1\end{array}\right)$
• Parity $P=\,-{{\mathbb{I}}}_{3\times 3}$
• Mirror ${M}_{y}={{PC}}_{2y}=\left(\begin{array}{ccc}0 & -\,1 & 0\\ -\,1 & 0 & 0\\ 0 & 0 & 1\end{array}\right).$

From the first-principles wave-functions, the representations corresponding to the two highest valence and two lowest conduction bands at the M-point were determined using the WIEN2k code [34, 35]. They were found to be ${{\rm{\Gamma }}}_{3}^{+},{{\rm{\Gamma }}}_{4}^{+}$ and ${{\rm{\Gamma }}}_{3}^{-},{{\rm{\Gamma }}}_{4}^{-}$, respectively. Their characters are shown in table 3, which comes from table 15 on page 35 in Koster et al [33]. Consequently, the symmetry representations in these four bands are given by

• $\mathrm{Identity}\,E=\,{{\mathbb{I}}}_{4\times 4}$
• Rotation ${C}_{2y}=\left(\begin{array}{cccc}i & 0 & 0 & 0\\ 0 & -\,i & 0 & 0\\ 0 & 0 & i & 0\\ 0 & 0 & 0 & -\,i\end{array}\right)$
• Parity $P=\left(\begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & -\,1 & 0\\ 0 & 0 & 0 & -\,1\end{array}\right)$
• Mirror ${M}_{y}={{PC}}_{2y}=\left(\begin{array}{cccc}i & 0 & 0 & 0\\ 0 & -\,i & 0 & 0\\ 0 & 0 & -\,i & 0\\ 0 & 0 & 0 & i\end{array}\right)$
• Time-reversal ${ \mathcal T }=\left(\begin{array}{cccc}0 & -\,1 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & -\,1\\ 0 & 0 & 1 & 0\end{array}\right)\hat{K}.$

For each of the symmetry operations g, the constraint

$$\begin{eqnarray}&&{ \mathcal H }({\bf{k}})=D(g){ \mathcal H }({g}^{-1}{\bf{k}})D({g}^{-1})\end{eqnarray} \tag{ 4 }$$

is imposed on the 4 × 4 Hamiltonian, where D(g) is the symmetry representation. By applying these constraints on the general form of a four-band Hamiltonian

$$\begin{eqnarray}&&{ \mathcal H }({\bf{k}})=\displaystyle \sum _{i,j\in \{0,x,y,z\}}{C}_{{ij}}({\bf{k}})({\sigma }_{i}\otimes {\sigma }_{j}),\end{eqnarray} \tag{ 5 }$$

we find the Hamiltonian to be of the form

$$\begin{eqnarray}{ \mathcal H }({\bf{k}}) & = & {C}_{00}({\bf{k}})({\sigma }_{0}\otimes {\sigma }_{0})+{C}_{{xx}}({\bf{k}})({\sigma }_{x}\otimes {\sigma }_{x})\\ & & +{C}_{{xy}}({\bf{k}})({\sigma }_{x}\otimes {\sigma }_{y})+{C}_{{xz}}({\bf{k}})({\sigma }_{x}\otimes {\sigma }_{z})\\ & & +{C}_{y0}({\bf{k}})({\sigma }_{y}\otimes {\sigma }_{0})+{C}_{z0}({\bf{k}})({\sigma }_{z}\otimes {\sigma }_{0}),\end{eqnarray} \tag{ 6 }$$

where the ${C}_{{ij}}({\bf{k}})$ are given up to the second order in ${{\bf{k}}}^{* }={\bf{k}}-M$ (in reduced coordinates) by

$$\begin{eqnarray}{C}_{00}({{\bf{k}}}^{* }) & = & {C}_{00}^{1}+{C}_{00}^{{x}^{2}+{y}^{2}}({({k}_{x}^{* })}^{2}+{({k}_{y}^{* })}^{2})\\ & & +{C}_{00}^{{xy}}\,{k}_{x}^{* }{k}_{y}^{* }+{C}_{00}^{{xz}-{yz}}({k}_{x}^{* }{k}_{z}^{* }-{k}_{y}^{* }{k}_{z}^{* })\\ & & +{C}_{00}^{{z}^{2}}\,{({k}_{z}^{* })}^{2}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}{C}_{z0}({{\bf{k}}}^{* }) & = & {C}_{z0}^{1}+{C}_{z0}^{{x}^{2}+{y}^{2}}({({k}_{x}^{* })}^{2}+{({k}_{y}^{* })}^{2})\\ & & +{C}_{z0}^{{xy}}\,{k}_{x}^{* }{k}_{y}^{* }+{C}_{z0}^{{xz}-{yz}}({k}_{x}^{* }{k}_{z}^{* }-{k}_{y}^{* }{k}_{z}^{* })\\ & & +{C}_{z0}^{{z}^{2}}\,{({k}_{z}^{* })}^{2}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{C}_{{xx}}({{\bf{k}}}^{* })={C}_{{xx}}^{x-y}({k}_{x}^{* }-{k}_{y}^{* })+{C}_{{xx}}^{z}\,{k}_{z}^{* }\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{C}_{{xy}}({{\bf{k}}}^{* })={C}_{{xy}}^{x-y}({k}_{x}^{* }-{k}_{y}^{* })+{C}_{{xy}}^{z}\,{k}_{z}^{* }\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{C}_{{xz}}({{\bf{k}}}^{* })={C}_{{xz}}^{x+y}({k}_{x}^{* }+{k}_{y}^{* })\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{C}_{y0}({{\bf{k}}}^{* })={C}_{y0}^{x+y}({k}_{x}^{* }+{k}_{y}^{* }).\end{eqnarray} \tag{ 12 }$$

Table 3.  Character table for the relevant double group representations of C_2m [33].

	E	C2y	P	My
${{\boldsymbol{\Gamma }}}_{3}^{+}$	1	$\,i$	$\,1$	i
${{\boldsymbol{\Gamma }}}_{4}^{+}$	1	$-i$	$\,1$	$-i$
${{\boldsymbol{\Gamma }}}_{3}^{-}$	1	$\,i$	−1	$-i$
${{\boldsymbol{\Gamma }}}_{4}^{-}$	1	$-i$	−1	i

These 16 parameters were numerically fitted to the band structure of TaAs₂ using the SciPy [36] package to obtain the values in table 4. The resulting band structure around the M-point is shown in figure 3. Comparing it to the band structure obtained from first-principles reveals that the approximation is accurate in the immediate vicinity of the M-point, but breaks down at around $6 \%$ of the distance along the line M–A. Importantly, the minimum band gap is not preserved in this model. Nevertheless, the model can be used to qualitatively study effects in TaAs₂, owing to the fact that it contains the correct symmetry representations.

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Table 4.  Parameters of the 4 × 4 Hamiltonian of TaAs₂ around M up to the second order.

$[\mathrm{eV}]$	${C}_{00}^{1}$	$=\,7.066$	${C}_{z0}^{1}$	$=\,-0.224$
$[\mathrm{eV}\mathring{\rm A} ]$	${C}_{{xz}}^{x+y}$	$=\,1.272$	${C}_{y0}^{x+y}$	$=\,1.270$
	${C}_{{xx}}^{x-y}$	$=\,-0.061$	${C}_{{xy}}^{x-y}$	$=\,-1.999$
	${C}_{{xx}}^{z}$	$=\,-0.554$	${C}_{{xy}}^{z}$	$=\,-0.253$
$[\mathrm{eV}{\mathring{\rm A} }^{2}]$	${C}_{00}^{{x}^{2}+{y}^{2}}$	$=\,-71.21$	${C}_{z0}^{{x}^{2}+{y}^{2}}$	$=\,56.30$
	${C}_{00}^{{xy}}$	$=\,-137.1$	${C}_{z0}^{{xy}}$	$=\,123.1$
	${C}_{00}^{{xz}-{yz}}$	$=\,1.52$	${C}_{z0}^{{xz}-{yz}}$	$=\,-1.49$
	${C}_{00}^{{z}^{2}}$	$=\,-0.84$	${C}_{z0}^{{z}^{2}}$	$=\,-1.88$

In the absence of a magnetic field, there is no direct band gap closure in AB₂ compounds. Since the valence bands thus form a well-defined manifold, they can be classified, just like insulators, according to the topology of these valence bands. Because time-reversal symmetry is fulfilled, a ${{\mathbb{Z}}}_{2}$ classification is possible.

All compounds were found to be weak topological insulators, with ${{\mathbb{Z}}}_{2}$ indices $0;(111)$. That is, all time-reversal invariant planes ${k}_{i}=0,\,0.5$ have a non-trivial ${{\mathbb{Z}}}_{2}$ index ${\rm{\Delta }}=1$. This result was derived from first-principles using the Z2Pack code [37], and agrees with previous studies [15, 17, 32]. The corresponding evolution of the Wannier charge centers is shown, for the case of TaAs₂, in figure 4.

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Figure 5 shows the surface density of states for a slab of TaAs₂, with surfaces parallel to the mirror plane perpendicular to the Cartesian y-axis (the light blue plane shown in figure 1). The presence of topological surface states confirms the conclusion that the material is a weak topological insulator. The surface spectrum was calculated by the iterative Green's function [38], which was implemented in WannierTools [39].

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Here we study the effects of the magnetic field on TaAs₂ by adding a Zeeman splitting term to the ${\bf{k}}\cdot {\bf{p}}$ model derived in section 2.3 (equation (6)). The splitting term is given by

$$\begin{eqnarray}&&{\rm{\Delta }}{ \mathcal H }={c}_{x}{\sigma }_{0}\otimes {\sigma }_{y}+{c}_{y}{\sigma }_{0}\otimes {\sigma }_{z}+{c}_{z}{\sigma }_{0}\otimes {\sigma }_{x},\end{eqnarray} \tag{ 13 }$$

where c_i is the strength of the Zeeman splitting induced by the magnetic field in that direction, that is

$$\begin{eqnarray}&&{c}_{i}=\displaystyle \sum _{j}\,{g}_{{ij}}{\mu }_{{\rm{B}}}{H}_{j}.\end{eqnarray} \tag{ 14 }$$

This assumes that the g-factor is equal for all bands. The limitations of this approximation are discussed in section 3.4.

3.2.1. Magnetic field along the rotation axis $\hat{y}$

When a magnetic field is applied along the rotation axis $\hat{y}$, the Zeeman term equation (13) takes the form

$$\begin{eqnarray}&&{\rm{\Delta }}{ \mathcal H }={c}_{y}{\sigma }_{0}\otimes {\sigma }_{z}.\end{eqnarray} \tag{ 15 }$$

This term preserves all spatial symmetries of the system, breaking only time-reversal.

Along the M–A line, the C_xx and C_yy contributions to the Hamiltonian vanish since ${k}_{x}^{* }={k}_{y}^{* }$ and ${k}_{z}^{* }=0$. Consequently, the energy eigenvalues are given by

$$\begin{eqnarray}&&E({\bf{k}})={C}_{00}({\bf{k}})\pm {c}_{y}\mp \sqrt{{C}_{{xz}}{({\bf{k}})}^{2}+{C}_{y0}{({\bf{k}})}^{2}+{C}_{z0}{({\bf{k}})}^{2}}.\end{eqnarray} \tag{ 16 }$$

The Zeeman term counteracts the original splitting (square root term), such that for a sufficient magnetic field there will be a direct band gap closure. Away from the M−A line, the band gap remains open, giving rise to a Weyl point.

When the Zeeman splitting is gradually switched on, two pairs of Weyl points form at about ${c}_{y}=0.11\,\mathrm{eV}$. Increasing the Zeeman splitting leads to a separation between the two nodes in a pair, with one node each moving towards the M-point. Finally, at ${c}_{y}\approx 0.25\,\mathrm{eV}$, these two nodes meet at M and annihilate. This process is shown in figure 6.

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The existence of these Weyl points was confirmed by verifying that the nodes are a source or a sink of Berry curvature. For this purpose, the Chern number of spheres surrounding the points was calculated by tracking the hybrid Wannier charge centers (HWCCs) on loops around the sphere [2, 37, 40, 41], using the Z2Pack software [37]. Figure 7 shows the evolution of the sum of the HWCC for two of the four nodes found at ${c}_{y}=0.12\,\mathrm{eV}$, demonstrating that the two points are Weyl nodes of opposite chirality.

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3.2.2. General magnetic field direction

Finally, the effects of a magnetic field in a general direction were studied. It turns out that even though such a field breaks the spatial symmetries of the system, Weyl nodes still appear under a strong enough magnetic field. When a magnetic field is applied in an $\hat{x}$- or $\hat{z}$-direction, a single pair of Weyl points emerges from the M-point. These Weyl nodes are located on the ${k}_{x}=-{k}_{y}$ plane, as shown in table 5.

Table 5.  The position ${{\bf{k}}}^{* }={\bf{k}}-M$ (in reduced coordinates) and chirality of the Weyl points for Zeeman splitting in the $\hat{x}$- and $\hat{z}$-direction.

Splitting $[\mathrm{eV}]$	Weyl position ${{\bf{k}}}^{* }$	Chirality
cx = 0.225	$(-0.0042,0.0042,0.00093)$	−1
	$(0.0042,-0.0042,-0.00093)$	1
cx = 0.25	$(-0.025,0.025,0.0054)$	−1
	$(0.025,-0.025,-0.0054)$	1
cx = 0.3	$(-0.044,0.044,0.0098)$	−1
	$(0.044,-0.044,-0.0098)$	1
cz = 0.225	$(0.0011,-0.0011,-0.018)$	−1
	$(-0.0011,0.0011,0.018)$	1
cz = 0.25	$(0.0066,-0.0066,-0.11)$	−1
	$(-0.0066,0.0066,0.11)$	1
cz = 0.3	$(0.012,-0.012,-0.18)$	−1
	$(-0.012,0.012,0.18)$	1

Figure 8 shows the number of Weyl points as a function of the Zeeman splitting. To obtain this phase diagram, the candidate Weyl points were identified using a quasi-Newton algorithm to find the minima in the band gap (using SciPy.optimize.minimize [36]), for different initial guesses. In the second step, the Chern number on the small sphere (radius ${10}^{-4}\,{\mathring{\rm A} }^{-1}$) surrounding the candidate points was evaluated (using Z2Pack [37]), keeping only the points with a non-zero Chern number. Finally, duplicate points were eliminated by checking whether two points lie within the diameter of the sphere of one another.

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Having studied the effects of Zeeman splitting on the ${\bf{k}}\cdot {\bf{p}}$ model for TaAs₂, we now study a more realistic tight-binding model for NbSb₂, derived from a first-principles calculation with hybrid functionals using the Wannier90 code [42, 43]. NbSb₂ was chosen because it has the smallest direct band gap of the four materials, making it the most promising candidate for hosting Weyl points at realistic magnetic field strength.

The Zeeman splitting for this model can again be expressed by adding the corresponding terms to the Hamiltonian

$$\begin{eqnarray}&&{\rm{\Delta }}{ \mathcal H }={c}_{x}{\sigma }_{x}\otimes {{\mathbb{I}}}_{22\times 22}+{c}_{y}{\sigma }_{y}\otimes {{\mathbb{I}}}_{22\times 22}+{c}_{z}{\sigma }_{z}\otimes {{\mathbb{I}}}_{22\times 22},\end{eqnarray} \tag{ 17 }$$

where the change in the splitting terms (compared to equation (13)) is due to the different orbital basis used for the tight-binding model. We search for Weyl points between the last valence band and the first conduction band.

First, we study the effect of applying a magnetic field in the y-direction. Figure 9 shows the effect of this splitting along the M−A line. For ${c}_{y}\approx 0.06\,\mathrm{eV}$, two pairs of Weyl points appear close to the M−A line. The reason these points are not exactly on the line is because the crystal symmetry is broken when constructing the Wannier-based tight-binding model [42, 43]. Apart from the numerical difference, this effect is analogous to the case of the ${\bf{k}}\cdot {\bf{p}}$ model for TaAs₂, where the two pairs of Weyl points appeared at ${c}_{y}=0.11\,\mathrm{eV}$.

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Table 6 shows the Weyl point positions, chirality and type for selected values of Zeeman splitting. It shows that Weyl points appear even at smaller values of c_y away from the M−A line. This is a crucial difference from the ${\bf{k}}\cdot {\bf{p}}$ model, which is only valid near the M-point. Furthermore, all Weyl points found for these splitting values are of type-II [2]. Type-II Weyl points have a tilted energy spectrum, making their Fermi surface open instead of point-like. As a consequence, their chiral anomaly—and their effect on magnetoresistance—is expected to be anisotropic.

Table 6.  The Weyl point positions (in reduced coordinates), chirality and type for different values of Zeeman splitting in the tight-binding model for NbSb₂.

Splitting $[\mathrm{eV}]$	Position ${\bf{k}}$	Chirality	Type
cx = 0.045	(0.4393, 0.4460, 0.5004)	+1	II
	$(0.4359,0.4444,0.5026)$	−1	II
	$(0.5641,0.5556,0.4974)$	+1	II
	$(0.5607,0.5540,0.4996)$	−1	II
cy = 0.03	$(0.3670,0.5141,0.0977)$	+1	II
	$(0.3655,0.5142,0.1004)$	−1	II
	$(0.6345,0.4858,0.8997)$	+1	II
	$(0.6330,0.4858,0.9023)$	−1	II
cy = 0.04	$(0.3724,0.5116,0.0890)$	+1	II
	$(0.3627,0.5135,0.1055)$	−1	II
	$(0.6373,0.4865,0.8945)$	+1	II
	$(0.6276,0.4884,0.9110)$	−1	II
	$(0.9028,0.0340,0.5451)$	+1	II
	$(0.9018,0.0354,0.5390)$	−1	II
	$(0.0982,0.9646,0.4610)$	+1	II
	$(0.0974,0.9658,0.4545)$	−1	II
cy = 0.06	$(0.3791,0.5068,0.0775)$	+1	II
	$(0.3592,0.5131,0.1108)$	−1	II
	$(0.6407,0.4869,0.8892)$	+1	II
	$(0.6211,0.4929,0.9222)$	−1	II
	$(0.9033,0.0328,0.5532)$	+1	II
	$(0.9006,0.0364,0.5314)$	−1	II
	$(0.0994,0.9636,0.4686)$	+1	II
	$(0.0968,0.9671,0.4467)$	−1	II
	$(0.4493,0.4555,0.5031)$	+1	II
	$(0.4309,0.4320,0.4825)$	−1	II
	$(0.5691,0.5680,0.5175)$	+1	II
	$(0.5507,0.5445,0.4969)$	−1	II
cz = 0.0475	$(0.4494,0.4384,0.4853)$	+1	II
	$(0.4420,0.4366,0.4816)$	−1	II
	$(0.5580,0.5634,0.5184)$	+1	II
	$(0.5506,0.5616,0.5147)$	−1	II

Finally, a phase diagram was calculated showing the number of Weyl points as a function of the magnetic field (see figure 10). Unlike for the ${\bf{k}}\cdot {\bf{p}}$ model, the number of Weyl points keeps increasing when the applied Zeeman term grows stronger. Again, the reason for this difference is that Weyl points also form far away from the M-point, where the ${\bf{k}}\cdot {\bf{p}}$ approximation is no longer applicable.

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For some values of splitting, the phase diagram shows an odd number of Weyl points, which is physically impossible. The reason for this is that the numerical procedure used to identify the number of Weyl points may not find a Weyl point if it is too close to another one. Since this problem only occurs rarely (see figure 11), the phase diagram is still valid overall. Also, the procedure ensures that no Weyl point can be counted twice, so the phase diagram represents a lower limit for the real number of Weyl points. Thus, the general result, in which the number of Weyl points increases with stronger Zeeman splitting, remains valid.

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In the previous sections, the effect of the magnetic field was modeled by applying Zeeman splitting to the model Hamiltonian. The discussion was simplified by assuming that the g-factor was equal for all energy bands, and independent of ${\bf{k}}$. Here, we discuss how the results might change if this assumption is not made.

If the g-factor is k-dependent, but still the same for all energy bands, the results above will change quantitatively, but not qualitatively. The reason for this is that the Weyl node that appears at a specific k-point will still be there, but for a different magnetic field. That is, the order in which the Weyl nodes at different k-points appear might change, but not the overall picture of an increasing number of Weyl points with a stronger magnetic field.

The same is true if the g-factor varies for different energy bands, as long as the sign of the g-factor remains the same. Because the appearance of Weyl points is due to the relative Zeeman splitting between the last valence and first electron bands, it does not matter how much the splitting of each band contributes.

If the g-factors in the relevant bands have opposite signs, however, there is a qualitative change in behavior. This is illustrated in the following with the example of the ${\bf{k}}\cdot {\bf{p}}$ model of TaAs₂, discussed in sections 2.3, 3.2. To account for the opposite sign of the g-factor for the valence and conduction bands, the Zeeman splitting term (equation (13)) is changed to

$$\begin{eqnarray}&&{\rm{\Delta }}{ \mathcal H }={c}_{x}{\sigma }_{z}\otimes {\sigma }_{y}+{c}_{y}{\sigma }_{z}\otimes {\sigma }_{z}+{c}_{z}{\sigma }_{z}\otimes {\sigma }_{x}.\end{eqnarray} \tag{ 18 }$$

With c_y splitting, the energy bands on the mirror plane are then given by

$$\begin{eqnarray}&&E({\bf{k}})={C}_{00}({\bf{k}})\pm {c}_{y}\mp \sqrt{{C}_{{xx}}{({\bf{k}})}^{2}+{C}_{{xy}}{({\bf{k}})}^{2}+{C}_{z0}{({\bf{k}})}^{2}}.\end{eqnarray} \tag{ 19 }$$

As in equation (16), the Zeeman term counteracts the original splitting. The difference from the previous case is that this equation holds on an entire plane in reciprocal space instead of just a line. As a consequence, we can expect the appearance of a nodal line with sufficient Zeeman splitting. Indeed, a nodal line appears for ${c}_{y}\gtrsim 0.2242\,\mathrm{eV}$, as shown in figure 12. The Berry phase on a closed path around this nodal line was calculated to be π, using the Z2Pack [37] software. This verifies the topological nature of the nodal line.

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In conclusion, the qualitative result obtained above remains intact when the g-factors are assumed to be k-dependent, and different for the valence and conduction bands, as long as they keep the same sign. A more adequate model of the magnetic field is needed to establish the exact qualitative and quantitative nature of the topological phases with the applied magnetic field. The current results indicate that Weyl nodes will appear, at least for some directions of magnetic field.