Stability of the Weyl-semimetal phase on the pyrochlore lattice

The mean-field phase diagram in the t_σ–U plane for the undoped case is shown in figure 1. The phase diagram agrees quantitatively with the one in [30] and also qualitatively with [36]. Recently, the pyrochlore material ${\mathrm{Eu}}_{2}{\mathrm{Ir}}_{2}{{\rm{O}}}_{7}$ has been studied by a very similar unrestricted Hartree–Fock approach for a tight-binding model by Wang et al [37]. In this case, the hopping amplitudes were obtained from density-functional calculations without spin–orbit coupling and the spin–orbit coupling and Hubbard-U terms added afterwards. The phase diagram in the plane spanned by spin–orbit coupling strength and U is obtained. The sequence of phases for increasing U for the undistorted pyrochlore structure is similar to figure 1. We return to the comparison below.

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Two main regions can be distinguished: the yellow and orange areas for small U denote phases without magnetic order. In the blue areas for large U, the magnetic moments are nonzero and TRS is broken. Mostly following [30], the phases are denoted as metal (M), semimetal with quadratic band-touching point (SM), topological insulator (TI), metal with all-in-all-out order (mAIAO), TWS, insulator with all-in-all-out magnetic order (AIAO), and insulator with rotated all-in-all-order order (rAIAO). All transitions are continuous except for the first-order transitions towards the rAIAO phase. The continuous transitions can be understood as Lifshitz transitions in the sense that they involve changes in the topology of the intersection of a band with the Fermi energy or of intersections between bands [38, 39]. It is, however, useful to single out second-order transitions at which the mean-field spin polarizations m_i start to deviate from zero. These transitions are highlighted by thin red lines in figure 1. The magnetic order and typical band structures for the various phases are discussed in [30]. We here focus on deviations from and additions to the previous results.

The TWS, mAIAO, and AIAO phases have the same magnetic all-in-all-out order. For a given tetrahedron, the spins at all four corners point either towards or away from the center, as sketched in figure 2. The order breaks TRS and also a ${{\mathbb{Z}}}_{2}$ (Ising) symmetry corresponding to inverting all spins. The occupation and the magnitude of the spin polarization are identical for all sites. The averages of the four spins are

$$\begin{eqnarray}&&{{\bf{m}}}_{1}=\displaystyle \frac{\alpha }{2\sqrt{3}}\,(1,1,1),\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{{\bf{m}}}_{2}=\displaystyle \frac{\alpha }{2\sqrt{3}}\,(1,-1,-1),\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{{\bf{m}}}_{3}=\displaystyle \frac{\alpha }{2\sqrt{3}}\,(-1,1,-1),\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&{{\bf{m}}}_{4}=\displaystyle \frac{\alpha }{2\sqrt{3}}\,(-1,-1,1),\end{eqnarray} \tag{ 25 }$$

with $0\lt | \alpha | \leqslant 1$. The magnetic space group is $\mathrm{Fd}\bar{3}{\rm{m}}^{\prime}$ [40] with the magnetic point group ${\rm{m}}\bar{3}{\rm{m}}^{\prime} ={O}_{h}({T}_{h})$ [41]. The all-in-all-out order is the natural equilibrium state for spins on the frustrated pyrochlore lattice with strong local easy-axis anisotropy and antiferromagnetic interactions [42–44].

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The electronic structure in the three phases is different. For large U, leading to large spin polarizations, the AIAO phase is an insulator. Interestingly, in ${\mathrm{Nd}}_{2}{\mathrm{Ir}}_{2}{{\rm{O}}}_{7}$ and variants with partial substitution of Nd, which lie in this phase, domain walls have been observed to be conducting [45–47]. The appearance of low-energy electronic states in a domain wall is plausible in view of the phase diagram in figure 1: the magnitude $| {{\bf{m}}}_{i}|$ of the spin polarization has to go to zero at a domain wall between the two degenerate AIAO states. A suppression of $| {{\bf{m}}}_{i}|$ tunes the system through the TWS phase to the SM phase, which is conducting even for infinitesimal doping. We note that the conducting domain walls likely do not result from nontrivial topology of the AIAO insulator as this state is theoretically found to be topologically trivial [48].

For smaller spin polarizations, there is a transition from the insulating phase towards a TWS with eight Weyl points lying on the Λ lines, which connect the Γ and L points [30] (in the earlier [17], 24 Weyl points were found). The Weyl points form the corners of a cube, as shown in figure 3. At the transition from the TWS to the AIAO phase for increasing U, the Weyl points annihilate pairwise at the L points. At the transition to the SM phase for decreasing U, all eight Weyl points instead merge at Γ and the spin polarizations m_i vanish continuously. The bands are not split at Γ since the relevant bands transform according to the Γ₈ irrep of O_h. The ensuing nonmagnetic SM phase is thus a semimetal with a quadratic band-touching point [49]. The third, mAIAO region is analyzed further below.

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For large U and small $| {t}_{\sigma }|$, a rAIAO order is found. It can be constructed from the perfect all-in-all-out order by dividing the four spins of one tetrahedron into two pairs. The two spins in the same pair are rotated in the same plane, which contains the locations of the two spins and the center of the opposite tetrahedron edge, in opposite directions. The absolute value of the rotation angle is the same for all spins. If m₁ and m₂ are rotated in the same plane, the resulting average spins read

$$\begin{eqnarray}&&{{\bf{m}}}_{1}=(a,b,b),\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&{{\bf{m}}}_{2}=(a,-b,-b),\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&{{\bf{m}}}_{3}=(-a,b,-b),\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&{{\bf{m}}}_{4}=(-a,-b,b).\end{eqnarray} \tag{ 29 }$$

The net magnetization still vanishes. The all-in-all-out order is the special case $a\,=\,b=\alpha /2\sqrt{3}$. The rotated all-in-all-out order also lowers the symmetry to tetragonal. Since there are three ways to split four spins into two pairs and the free energy is invariant under inverting all spins, there are six degenerate equilibrium states.

The rAIAO phase is the same one that Wang et al [37] find at large U for the ideal pyrochlore structure and denote by AF2. This can be seen by comparing equations (26)–(29) to the spins given in figure 1(b) in [37]. We conclude that the tight-binding model of [37] corresponds to small $| {t}_{\sigma }|$ in our phase diagram.

Our results differ from [18, 30, 37] in that the rotation angle always equals π/2 in these works. Instead, we find the angle to vary, as shown in figure 4. In the limit of large U, the rotation angle approaches π/2. This limit is consistent with Monte Carlo simulations for a pure spin model with Dzyaloshinski-Moriya interaction [50].

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The region denoted as mAIAO in figure 1 and as metallic antiferromagnet in [30] is particularly interesting. Figure 5 reveals that several distinct phases are present. These are best understood by starting from the TWS phase and increasing $| {t}_{\sigma }|$. This pushes the conduction band down away from the Γ point and leads to the appearance of electron-type Fermi pockets at the L points. The chemical potential is then shifted downward from the Weyl points into the valence band. The result is a compensated metal with Weyl points, denoted by TWS/M. For increasing $| {t}_{\sigma }|$, the Weyl cones are also tilted so that the dispersion in one direction eventually becomes flat and then changes sign. The result is a type-II TWS as introduced by Soluyanov et al [16]. This phase is denoted by TWS II.

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Increasing $| {t}_{\sigma }|$ further, the Weyl points move to the L points and annihilate pairwise. The resulting phase does not have any band-touching points but is a compensated metal due to the overlap in energy of bands in different parts of the Brillouin zone. It retains all-in-all-out order and is denoted by mAIAO in figure 5. However, at large $| {t}_{\sigma }|$ and small U, something else happens: at the four L points, four additional pairs of Weyl points are formed. The system then has 16 Weyl points, all of which are of type II. This phase is denoted by TWS II, 16 WP. The newly created and the old Weyl points then move towards each other for increasing $| {t}_{\sigma }|$ or U and finally annihilate pairwise, again leading to the mAIAO phase.

For smaller U, we instead start in the SM phase with quadratic band touching at Γ. For increasing $| {t}_{\sigma }|$, the band structure deforms similarly to what happens in the TWS phase. Fermi pockets appear at the L points and the chemical potential is shifted into the valence band. The resulting compensated metal is denoted by M.

In the following, we consider phases along two representative cuts through the phase diagram, which are indicated in figure 1 by solid lines with dots. As the second control parameter we take the average number ρ of electrons in the J_eff = 1/2 doublet per iridium so that ρ = 1 corresponds to half filling, i.e., the undoped case considered previously. As noted above, doping away from half filling should shift the chemical potential away from the Weyl points in the TWS phase and thus reduce the free energy gain that stabilizes the TWS state. In this section, we only consider states that do not break translational symmetry.

We first consider a cut along the dark green line with dots in figure 1, for which t_σ = −0.9 is fixed and U is varied between 1.0 and 1.4. The phase diagram is shown in figure 6. Several distinct magnetic and also charge orders are found, which are illustrated by the insets. For electron doping, ρ > 1, magnetic order vanishes rapidly. In case of the gapped AIAO phase, this is easy to understand. The AIAO order splits the band touching at the Γ point, regardless of the Fermi energy. For our parameters, the effective mass of the conduction band at the Γ point is larger than that of the valence band. Consequently, the density of states close to the energy of the touching point is asymmetric, being larger in the conduction band. The opening of a gap is then unfavorable for electron doping because the energy increase of occupied conduction-band states overwhelms the energy reduction of valence-band states. This can only be overcome by a stronger interaction U, which is why the transition shifts to larger U for increasing electron doping. The argument is given in more detail in appendix. The situation is not fundamentally different for the TWS phase, except that no full gap is opened but density of states is still shifted away from the Weyl points. We have not found any phases other than the nonmagnetic semimetal for larger ρ. Other phases not considered here, such as incommensurate magnetic order or superconductivity, might occur, though.

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In the following, we discuss the phases for fillings $0\lt \rho \lesssim 1$, starting from small ρ. It should be noted that for a realistic description of a pyrochlore material in this doping range, additional orbitals, in particular the J_eff = 3/2 multiplet need to be taken into account. For small ρ, the ground state is nonmagnetic. Upon increasing ρ, we find a continuous phase transition towards the state marked CDW I in figure 6. This phase has ferrimagnetic order with magnetization along the (111) or an equivalent direction. The average spins take the form

$$\begin{eqnarray}&&{{\bf{m}}}_{1}=(a,a,a),\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&{{\bf{m}}}_{2}=(b,c,c),\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&{{\bf{m}}}_{3}=(c,b,c),\end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}&&{{\bf{m}}}_{4}=(c,c,b),\end{eqnarray} \tag{ 33 }$$

note the arrows in the corresponding inset. Since the spins at sites 2–4 are related to each other by lattice symmetries, but not to the spin at site 1, the occupation number of site 1 is generically different from the others, ${n}_{1}\ne {n}_{2}={n}_{3}={n}_{4}$ (indicated by different colors of the dots in the inset). Hence, this is a CDW state, which, however, does not break translational symmetry. The order is most easily understood by viewing the pyrochlore lattice as an alternating stack of triangular and kagome lattices. In the (111) direction, the triangular layers consist solely of sites with number 1 and the kagome layers of sites 2–4. The spins in the triangular layers point towards one tetrahedron center and away from the other. The spins in the kagome layers are roughly parallel to the ones in the triangular layers but tilted towards and away from the centers of adjacent triangles of the kagome lattice. The projections of these spins onto the plane of the kagome layer are shown in figure 7. The state can also be understood as a ferromagnet that is perturbed by local Ising anisotropies. Since there are four ways to decompose the lattice into triangular and kagome layers and the energy is invariant under inversion of all spins, the equilibrium state is eightfold degenerate.

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Increasing ρ further, we find a transition to a state with average spins

$$\begin{eqnarray}&&{{\bf{m}}}_{1}=(a,b,b),\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&{{\bf{m}}}_{2}=(a,-b,-b),\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&{{\bf{m}}}_{3}=(a,-b,b),\end{eqnarray} \tag{ 36 }$$

$$\begin{eqnarray}&&{{\bf{m}}}_{4}=(a,b,-b)\end{eqnarray} \tag{ 37 }$$

(or other states related by symmetries) and equal occupation numbers. The state is ferrimagnetic with magnetization along (100) or equivalent directions. Note that the spin averages differ from the rAIAO order in equations (26)–(29) only in the signs of m₃ and m₄. Hence, the new configuration is obtained by the same pairwise rotations as the rAIAO order but starting from a state with two spins pointing in and two pointing out of the tetrahedron. We call the new configuration rotated two-in-two-out (r2I2O) order. In the studied parameter range, we always find $| a| \gg | b|$, i.e., the state is close to a collinear ferromagnet. Since there are six possibilities to choose the in-pointing two spins, the degeneracy is sixfold.

Next, there is another transition to a state with

$$\begin{eqnarray}&&{{\bf{m}}}_{1}={{\bf{m}}}_{2}=(0,a,-a),\end{eqnarray} \tag{ 38 }$$

$$\begin{eqnarray}&&{{\bf{m}}}_{3}=(c,b,-b),\end{eqnarray} \tag{ 39 }$$

$$\begin{eqnarray}&&{{\bf{m}}}_{4}=(-c,b,-b).\end{eqnarray} \tag{ 40 }$$

As suggested by symmetry, the occupation numbers satisfy ${n}_{1}={n}_{2}\ne {n}_{3}={n}_{4}$. The first two occupations are smaller than the latter two. Hence, this is another CDW state that does not break translational symmetry. Such a charge order, without the magnetic order, has also been found by Kurita et al [51]. We denote this state by CDW II. The first two spins are oriented parallel to the tetrahedron edge connecting sites 3 and 4. The other two spins are nearly parallel to the first two (a and b have the same sign) but canted in opposite directions. The state is ferrimagnetic with magnetization along ($01\bar{1}$) or equivalent directions, i.e., along one of the tetrahedron edges. Since there are six possibilities to choose the two equal spins and the energy is invariant under inversion of all spins, the equilibrium state is twelvefold degenerate.

Close to quarter filling (ρ = 0.5), the order CDW I recurs but the values of a, b, and c in equations (30)–(33) are very similar, i.e., this state is close to a collinear ferromagnet. At quarter filling, Kurita et al [51] have found a CDW with three charge-poor sites, i.e., n₁ > n₂ = n₃ = n₄ and without magnetic order for large nearest-neighbor Coulomb repulsion. For dominating on-site repulsion U, they find a ferromagnet.

For ρ > 0.5, the system returns to the r2I2O state. Interestingly, the CDW I and r2I2O states that meet close to ρ = 0.5 are both Weyl semimetals with two Weyl points. These Weyl points occur at the Fermi energy for quarter filling, i.e., they involve a different pair of bands compared to the TWS close to half filling. As an example, we plot in figure 8 the band structure for t_σ = −0.9, U = 1.3, and ρ = 0.5, for which the r2I2O state is slightly favored over CDW I. The Weyl points are located on the (100) (k_x) axis. These phases are interesting since two is the smallest possible number [52] and since they should show a large anomalous Hall effect [29]. (A multilayer system with two Weyl points has been analyzed by Burkov and Balents [53].) The mechanism favoring a Weyl state around ρ = 0.5 is analogous to the one effective close to half filling, which is discussed below.

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At even larger ρ, the r2I2O order gives way to CDW I again. At large U, this phase surrounds a new phase, denoted by CDW III. It is the most complicated phase we have encountered, characterized by the spins

$$\begin{eqnarray}&&{{\bf{m}}}_{1}=(a,b,c),\end{eqnarray} \tag{ 41 }$$

$$\begin{eqnarray}&&{{\bf{m}}}_{2}=(a,-c,-b),\end{eqnarray} \tag{ 42 }$$

$$\begin{eqnarray}&&{{\bf{m}}}_{3}=(d,e,-e),\end{eqnarray} \tag{ 43 }$$

$$\begin{eqnarray}&&{{\bf{m}}}_{4}=(f,g,-g)\end{eqnarray} \tag{ 44 }$$

and only the occupations n₁ and n₂ being equal. In the specific example, m₁ and m₂ are related by mirror symmetry with respect to the (011) plane and the other two spins are parallel to the same plane. The degeneracy is 24-fold, resulting from a factor of 6 for selecting the pair of equally occupied sites, a factor of 2 from twofold rotation interchanging the two other sites, and a factor of 2 from inverting all spins. Interestingly, all other orders are obtained as special cases: rAIAO for c = b, d = f = −a, and e = −g = b (AIAO is a special case of rAIAO); CDW I with sites interchanged compared to equations (30)–(33) for b = a, d = −c, e = a, and g = f; CDW II for a = 0, c = −b, f = −d, and g = e; and r2I2O for c = b, d = f = a, and e = −g = −b.

All magnetic phases discussed so far in this section are ferrimagnetic. The broad region of ferrimagnetic states is due to a tendency towards ferromagnetism caused by the Stoner mechanism [54], modified by magnetic anisotropies due to strong spin–orbit coupling. We note that Shinaoka et al [43] have obtained various such phases for A₂Mo₂O₇, where A is a rate-earth element, within ab-initio calculations and discussed them based on the interplay of anisotropic exchange, Dzyaloshinsky–Moriya interaction, and local Ising anisotropy. The transitions in the series CDW I–r2I2O–CDW II–CDW I–r2I2O–CDW I–CDW III–CDW I are all of first order with discontinuous changes of the magnetization direction.

For increasing ρ, another first-order transition leads towards antiferromagnetic AIAO order. Only at small U, a region without magnetic or charge order intervenes. The appearance of antiferromagnetic order close to half filling can be understood from a standard perturbative treatment of electron hopping, leading to antiferromagnetic kinetic exchange [55]. The band structure changes from gapped at large U to a TWS for smaller U. Except close to half filling, the AIAO-TWS transition hardly depends on the filling down to $\rho \approx 0.75$ but is instead only controlled by the interaction U. This is surprising since for the heavy doping corresponding to ρ ≈ 0.75, the Weyl points are far above the Fermi energy and their formation should have negligible effect on the free energy. We conclude that it is actually the AIAO magnetic order that is energetically favored in this range, while the TWS is just a secondary consequence of that order. If the interaction U and thus the spin polarizations m_i are small, the ordering splits the quadratic band-touching point of the nonmagnetic SM phase into eight Weyl points. For U ≈ 1.3, these Weyl points are shifted to the L points and annihilate, leading to the insulating AIAO phase.

Finally, the AIAO order gives way to a nonmagnetic phase. A very narrow TWS phase intervenes between the gapped AIAO state and the nonmagnetic state also at larger U, which is not resolved in figure 6. The spin polarization decreases to zero continuously, though very rapidly, as a function of ρ.

The critical interaction U for the transition between the AIAO and TWS phases and the nonmagnetic phase generally increases with ρ, as explained in the appendix. Close to half filling (ρ = 1), the stability of the TWS is enhanced, leading to the nonmonotonic dependence of the critical U on ρ seen in figure 6. The reason is that splitting the quadratic Γ₈ band-touching point into Weyl points reduces the density of states close to the Fermi energy and thus lowers the free energy. As mentioned in the introduction, this mechanism is most efficient if the Fermi energy coincides with the energy of the Γ₈ point, i.e., at half filling.

We briefly consider another cut through the (t_σ, U, ρ) phase diagram, which is indicated by the orange line with dots in figure 1. The phase diagram is shown in figure 9. Overall, the phase diagram is similar to the one in figure 6. The extent of some phases, notably CDW II and CDW III, has expanded but the general sequence of phases is the same. There is now an island of a gapped system with rAIAO order surrounded by the gapped AIAO phase. The rotation angle Φ is smaller than for the rAIAO phase at ρ = 1. The transition between AIAO and rAIAO is of first order. The nonmonotonic dependence of the transition between the TWS and nonmagnetic states on ρ close to half filling is much more pronounced than in figure 6—the stabilization of the TWS phase at ρ ≈ 1 is strongly enhanced. This is mainly due to the cut being nearly parallel to the TWS-to-nonmagnetic transition in figure 1. Moreover, the transition between the AIAO and TWS phases is not flat as in figure 6, but note that now both U and t_σ vary along the vertical axis. The narrow TWS phase between the AIAO and nonmagnetic states is resolved in figure 9.

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In this section, the stability of the Weyl state of the undoped and weakly doped pyrochlore system with respect to breaking the translational symmetry is assessed. We have searched for charge and spin DWs and combinations of the two. As discussed above, breaking of translational symmetry leads to backfolding of bands. Weyl points of opposite chirality that are far apart in the original Brillouin zone can end up close together. A small change of the order parameters could then move them to the same k point, where they can gap out, which could reduce the free energy [3, 25–28]. Since bands close to Weyl points of opposite chirality would hybridize in such a state, it would form an axion insulator [26]. Note that section 3.2 demonstrates that inhomogeneous charge configurations can occur.

In the absence of a DW, the Weyl points form the corners of a cube with neighboring points having opposite chirality, see figure 3. The optimal situation for the described mechanism is a DW in the (001) or equivalent directions. The backfolding can then place all Weyl points close together in pairs of opposite chirality. For a DW in the (111) or equivalent directions, only two Weyl points of opposite chirality can end up close together, namely the two with their separation vector parallel to the ordering vector Q of the DW. To allow for DW solutions, we consider supercells consisting of n primitive unit cells such that the translational symmetry is reduced in the (001) or (111) direction. The average occupations n_i and spin polarizations m_i are allowed to be distinct for all 4n sites in the supercell.

We have performed iterative mean-field calculations for DWs along (001) with n = 3–5 and along (111) with n = 2–5, for various values of t_σ and U. We have started from up to 30 sets of random initial values for all mean-field parameters n_i and m_i for each case. Furthermore, we have also initialized the iteration with DW states imposed by hand that are discussed below. No DW states breaking translational symmetry have been found. Under iteration, all spin polarizations flow to the AIAO order and all charges to uniform occupation. Even if the locations of the Weyl points in the uniform state are fine tuned, by changing U and t_σ, to fold back on top of each other, we do not obtain a stable DW. This has been tested for DWs along (001) with n = 3, 4 and along (111) with n = 2–4. To check whether small shifts of the Fermi energy have any effect, we have varied ρ between 0.98 and 1.02 in all cases. In the remainder of this section, we discuss why the TWS state is robust.

We first demonstrate that the Weyl points can indeed be moved and induced to annihilate by imposing a DW. We take parameters t_σ = −0.775 and U = 1.515, which lie within the TWS phase in figure 1. The band structure along the k_z direction through a pair of Weyl points is shown in figure 10(a). It is of course possible to describe this situation using a larger unit cell. For the example of n = 3 primitive unit cells stacked in the (001) direction, the bands are shifted as indicated by the arrows in figure 10(a), which leads to the band structure in figure 10(b). Now, a CDW is switched on by adding

$$\begin{eqnarray}&&{H}_{\mathrm{CDW}}=-U\delta \displaystyle \sum _{{\bf{r}}}\cos \left(\displaystyle \frac{\pi }{n}\,{\bf{r}}\cdot \hat{{\bf{z}}}\right)\displaystyle \sum _{\sigma }{c}_{{\bf{r}}\sigma }^{\dagger }{c}_{{\bf{r}}\sigma }\end{eqnarray} \tag{ 45 }$$

to the Hamiltonian. Here, r traverses all sites of the pyrochlore structure and $\hat{{\bf{z}}}$ is the unit vector in the (001) direction. In a selfconsistent calculation, H_CDW would represent the Hartree potential resulting from the CDW. Translational symmetry is broken for n = 2, 3, ...

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The band structures in figure 10 correspond to a vanishing CDW amplitude, δ = 0. For $\delta \ne 0$, the band energies change and energy gaps open at the new zone boundaries. The band touching at the Weyl points cannot be gapped out since these points are topologically protected by a nonzero Chern number. Hence, it can only change if Weyl points are annihilated or created in pairs. Figure 11 illustrates how the Weyl points move for increasing amplitude δ. We find that for sufficiently large δ, the Weyl points always meet and gap out. However, it is not always the case that the Weyl points that start out close together annihilate. This simple scenario is only realized in figure 11(a). For smaller U, figures 11(b) and (c), the Weyl points are further apart for δ = 0 and the annihilation is more complex.

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A closer look at figure 11 reveals that the Weyl points do not all annihilate at the same value of δ. Instead, two pairs annihilate for much smaller δ than the other two. This is consistent with symmetry since the CDW reduces the magnetic point group to D_2h(C_2h), which contains a twofold rotation about the z-axis but no fourfold rotations. D_2h(C_2h) also does not contain threefold rotation axes so that the Weyl points are not pinned to the (111) direction.

Since a supercell consisting of n primitive unit cells stacked in the (001) direction comprises 2n rectangular layers, CDWs with half integer $n=3/2,5/2,\,\ldots$ in equation (45) are possible. Consecutive chains of maximally occupied sites are then rotated by 90° with respect to each other and the supercell contains 2n primitive unit cells. In this case, the magnetic point group is D_4h(D_2h), which retains fourfold rotation combined with time reversal. Consequently, all eight Weyl points are linked by symmetries. However, the threefold rotation symmetries are still broken. This case is included in our unrestricted Hartree–Fock calculations. If we choose a supercell containing a number m of primitive unit cells and m is odd, the mean-field equations are free to select a DW with integer n = m or with half-integer n = m/2.

If the Weyl points are lying close together in the backfolded Brillouin zone, figure 11(a), we find that typically a small amplitude δ is sufficient to annihilate them. However, this does not lead to the opening of a global gap. Rather, we find a semimetal with Weyl rings, i.e., one-dimensional loops of crossings of otherwise nondegenerate bands. Although this state ultimately does not help to understand the stability of the TWS, it is worth to briefly discuss it as an alternative topological state. The Weyl loops are lying in the k_z = π/2n plane, which forms a face of the backfolded Brillouin zone. They are shown in figure 12. The k_z = π/2n plane is invariant under the mirror reflection m_z, which is the only nontrivial symmetry in the little magnetic group C_1h of that plane. The two touching bands transform according to different irreps of C_1h so that crossings are allowed by symmetry. The crossing of two bands in a two-dimensional manifold is generically one dimensional. Away from the high-symmetry plane, the little group is trivial (C₁) and all band crossings are avoided. The Weyl rings can also be understood based on the theory of symmetry-protected topological states [3, 56]. Weyl rings protected by a mirror symmetry have been found by band-structure calculations for GdPtBi in a magnetic field [24]. The origin of the loops is distinct from the case discussed in [57] for pyrochlore iridates, where they are due to an approximate symmetry. The appearance of Weyl rings is not favorable for the formation of DWs since they increase the density of states close to the Fermi energy, even if the band structure is fine tuned so that the loop appears at the Fermi energy. Hence, DWs that break the mirror symmetry generally have a lower free energy.

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For CDWs in the (111) direction, a supercell is formed by stacking n primitive unit cells along (111). A CDW with n = 1 corresponds to a case where the alternating triangular and kagome layers have different occupation numbers. This is the CDW I phase introduced in section 3.2. For n ≥ 2, the Brillouin zone is backfolded. As noted above, only the two Weyl points with connecting vector parallel to (111) can end up close together. We find that they annihilate and gap out for sufficiently large δ. The CDW breaks lattice symmetries in such a way that these two Weyl points are no longer related to the other six by symmetry. CDWs along (111) do not remove the threefold rotation symmetry about this direction. Thus the Weyl points that can annihilate are pinned to the (111) direction.

The most favorable situation for DW formation is realized when Weyl points of opposite chirality are mapped to the same point in the backfolded Brillouin zone. Then an infinitesimal DW amplitude should be able to gap them out. As noted above, we have not obtained stable DW solutions even in that case. To understand why not, we consider the deviations Δn_i and Δm_i of n_i and m_i, respectively, from the TWS state as small perturbations. The free energy does not contain a term of first order in Δn_i and Δm_i. For the occupations, the first-order contribution to the mean-field energy is proportional to ${\sum }_{i}{n}_{i}^{(0)}\,{\rm{\Delta }}{n}_{i}$, where ${n}_{i}^{(0)}$ is the unperturbed occupation number at site i. However, these are uniform in the TWS state so that the contribution is proportional to ${\sum }_{i}{\rm{\Delta }}{n}_{i}$. This sum vanishes since the number of electrons is fixed. For the spin polarizations, the first-order correction to the energy is proportional to ${\sum }_{i}{{\bf{m}}}_{i}^{(0)}\cdot {\rm{\Delta }}{{\bf{m}}}_{i}$, where ${{\bf{m}}}_{i}^{(0)}$ is the unperturbed spin polarization⁴ . Hence, only the component ${\rm{\Delta }}{m}_{i}\equiv ({{\bf{m}}}_{i}^{(0)}/| {{\bf{m}}}_{i}^{(0)}| )\cdot {\rm{\Delta }}{{\bf{m}}}_{i}$ parallel to ${{\bf{m}}}_{i}^{(0)}$ contributes. Since the unperturbed ${{\bf{m}}}_{i}^{(0)}$ have uniform magnitude, the first-order correction to the energy is proportional to ${\sum }_{i}{\rm{\Delta }}{m}_{i}$. The Δm_i can now be decomposed into a uniform contribution $\overline{{\rm{\Delta }}{m}_{i}}$, where the average is over all sites i, and a site-dependent one, ${\rm{\Delta }}{m}_{i}-\overline{{\rm{\Delta }}{m}_{i}}$. We already know that the TWS state is stationary for uniform perturbations so that the free energy cannot depend linearly on $\overline{{\rm{\Delta }}{m}_{i}}$. We can thus subtract the uniform part from Δm_i without changing the free energy to linear order. But then we obtain ${\sum }_{i}{\rm{\Delta }}{m}_{i}=0$ and the magnetic contribution is found to vanish to linear order.

Due to the vanishing of the first-order contributions, the uniform TWS state remains a stationary point of the free energy when DWs are allowed. We first consider the stability of this stationary point understood as a fixed point of the iterative mapping described by the mean-field equations. The space of perturbations Δn_i and Δm_i has 16n − 1 independent dimensions, see section 2. If there were an unstable direction, a random set of Δn_i and Δm_i would have nonzero overlap with it with probability one. In this sense, it is sufficient to check the stability for a single random set of starting values. However, if the initial overlap is very small, the iteration might not pick up the unstable direction during a feasible number of iterations. For this reason, we have always taken several sets of random starting values. Since they all converge to the uniform stationary point, we conclude that all directions are stable. It remains to understand the reason for this.

A DW should be most strongly stabilized when it opens a global gap opens at the Fermi energy. This can indeed happen: for t_σ = −0.775, U = 1.55, and n = 3, we have considered 10⁵ sets of small random deviations and find a global gap in 8.8% of cases. Complete splitting between the bands without a global gap is found with a frequency of 13.9% and persisting Weyl points with 77.3%. We will now give an argument as to why a DW is not stabilized even for a global gap.

The stability analysis of the uniform state is not straightforward since the stationary point is a minimum as a function of the spin polarizations m_i but is a maximum as a function of the occupations n_i [58]. This can be understood based on minimization of the grand-canonical potential or within the functional-integral approach [58]⁵ . The correct extremization procedure is to first maximize the free energy for fixed Δm_i as a function of the Δn_i satisfying ${\sum }_{i}{\rm{\Delta }}{n}_{i}=0$. Let us denote the result by ${ \mathcal F }({\rm{\Delta }}{{\bf{m}}}_{1},{\rm{\Delta }}{{\bf{m}}}_{2},\,\ldots )$. Then, ${ \mathcal F }$ is minimized with respect to the Δm_i. This means that standard stability analysis applies to ${ \mathcal F }({\rm{\Delta }}{{\bf{m}}}_{1},{\rm{\Delta }}{{\bf{m}}}_{2},\,\ldots )$.

Let us denote the gap by 2Δ, i.e., occupied states close to the Weyl points are shifted downward in energy by Δ. The momentum-space volume that is affected should scale with Δ³ since the unperturbed dispersion is approximately linear. The typical energy gain of occupied states within this volume is proportional to −Δ. The total gain in the free energy ${ \mathcal F }$ should thus scale as −Δ⁴. As noted above, Δ is of first order in the perturbation, i.e., in an overall scaling factor λ for all Δm_i. This gives a negative contribution of order −λ⁴ to the free energy. However, there is another contribution from the decoupling term in the mean-field ansatz, ${NU}{\sum }_{i}{{\bf{m}}}_{i}\cdot {{\bf{m}}}_{i}$. This term is positive and scales as λ². Hence, the uniform TWS stationary point is stable against small perturbation. It is instructive to contrast the case that a normal Fermi surface with Fermi wave number k_F is gapped out. In that case, the affected k-space volume scales as ${k}_{F}^{2}\,{\rm{\Delta }}$ and the total energy gain as $-{k}_{F}^{2}\,{{\rm{\Delta }}}^{2}$ and thus as −λ². This energy can compete with the quadratic decoupling term. This is the situation for the instability of a compensated metal towards an excitonic insulator [59, 60]⁶ .

A few comments are in order. Our argument does not exclude a first-order transition to a state with large Δm_i. We have not found any such state, though. Furthermore, depending on the details of the model, there likely exists a second-order contribution to the total energy gain due to small shifts of the (large) energies of states far from the Weyl points. However, the coefficient of the corresponding quadratic term is expected to be small since it describes the effect of states deep in the Fermi sea. If so, it cannot overcome the large, positive decoupling term. In other words, the TWS is rather stable against DW formation since the occupied density of states close to the Fermi energy is small to start with. Thus changes in the electronic dispersion at the mean-field level cannot lead to a large reduction that could overcome the positive decoupling term.