Aspects of symmetry and topology in the charge density wave phase of 1T–TiSe₂

Transition metal dichalcogenides have layered crystal structures with the weak van der Waals interaction between the adjacent layers. This feature allows them to be exfoliated in few layers or even monolayers, bringing intensive interest for their appealing electronic and optoelectronic properties [1–6]. Owing to the low dimensionality, many materials undergo a charge density wave (CDW) phase transition [2, 7, 8]. 1T–TiSe₂ is a noteworthy examples which shows superconductivity when the CDW is suppressed via doping or pressure [9–14]. The CDW in 1T–TiSe₂ is commensurate, forming a 2 × 2 × 2 superlattice [15, 16]. Simultaneous periodic lattice distortion (PLD) [17] was observed from the phonon softening in the ${L}_{1}^{-}$ mode [${L}_{1}^{-}$ is an irreducible representation (IR) at L] [15, 18, 19], indicating the symmetry of the CDW.

The ${L}_{1}^{-}$ CDW is nontrivial due to its anisotropic character. It is reminiscent of the d-density wave in strongly correlated systems where the local rotation symmetry is broken [20, 21]. Symmetry is important because it bears on the CDW mechanism [22] and determines whether the CDW nodes in the electronic structure influence superconducting T_c. It also drives directional responses [23] and controls the topology of the system. The key role of symmetry, however, has not been hardly explored in the literature [24–27]. With this motivation, we will elucidate the symmetry properties of the CDW state in 1T–TiSe₂ and their connections with the electronic structure and the band topology of the system.

Our analysis starts by considering a system at high temperature, which is described by the bare Hamiltonian ${\hat{H}}_{0}$ with symmetry group ${\mathcal{G}}_{0}$. The symmetry elements R_i of ${\mathcal{G}}_{0}$ will of course commute with the Hamiltonian, i.e. $\left[{\hat{R}}_{i},{\hat{H}}_{0}\right]=0$. When the CDW order develops at low temperatures, the CDW Hamiltonian becomes ${\hat{H}}_{\text{CDW}}={\hat{H}}_{0}+\hat{{\Delta}}$, where $\hat{{\Delta}}$ is to capture the CDW order. As a result, some symmetries ${\hat{R}}_{i}$ will be broken when $\left[{\hat{R}}_{i},{\hat{H}}_{\text{CDW}}\right]=\left[{\hat{R}}_{i},\hat{{\Delta}}\right]\ne 0$. Most obvious is the breaking of the translation symmetry T since $\left[\hat{T},\hat{{\Delta}}\right]\ne 0$. [For a commensurate CDW the translation symmetry can be restored by enlarging the unit cell: ${\hat{T}}_{\text{CDW}}={\hat{T}}^{M}$ for some integer M such that $\left[{\hat{T}}_{\text{CDW}},\hat{{\Delta}}\right]=0$.] So, the symmetry group will be reduced to a subgroup $\mathcal{G}$ of ${\mathcal{G}}_{0}$ which will contain symmetry elements isomorphic to those that are left invariant under the CDW distortion in ${\mathcal{G}}_{0}$.

The symmetry of the CDW order parameter (OP) will be characterized by the symmetry group of the ordering vectors q. We will assume that the OP is a one-dimensional (1D) IR of the little group ${\mathcal{G}}_{0}^{\mathbf{q}}$, a subgroup of ${\mathcal{G}}_{0}$ which contains the point-group symmetries that keep q invariant. By 1D IR we mean that the OP is an eigenstate of point group symmetry ${\hat{R}}_{j}\in {\mathcal{G}}_{0}^{\mathbf{q}}$ with eigenvalue either +1 or −1, i.e. ${\hat{R}}_{j}\hat{{\Delta}}{\hat{R}}_{j}^{{\dagger}}={\pm}\hat{{\Delta}}$. Note that the broken symmetries in the CDW state also include those associated with eigenvalues of −1, which are related to the ordering vectors for the direction of the atomic displacements [28].

Pristine 1T–TiSe₂ crystallizes in a hexagonal lattice structure with point group D_3d and space group ${\mathcal{G}}_{0}=P\overline{3}m1$. Its low-temperature phase is the triple-q CDW state with ordering vectors ${\mathbf{Q}}_{1}=\frac{1}{2}({\mathbf{a}}_{1}^{{\ast}}+{\mathbf{a}}_{3}^{{\ast}})$, ${\mathbf{Q}}_{2}=\frac{1}{2}({\mathbf{a}}_{2}^{{\ast}}+{\mathbf{a}}_{3}^{{\ast}})$, and ${\mathbf{Q}}_{3}=\frac{1}{2}({\mathbf{a}}_{1}^{{\ast}}+{\mathbf{a}}_{2}^{{\ast}}+{\mathbf{a}}_{3}^{{\ast}})$, which connect Γ and the three inequivalent L points (L_1,2,3), see figure 1(a). The little group of L₂ contains point-group elements {$E,{2}_{[100]},{m}_{[100]},\overline{1}$} as the group C_2h , which has four 1D IRs A_g , B_g , A_u , and B_u (table 1). With 3_[001] we obtain conjugate representations for L₂ and L₃. Including translation T of the Bravais lattice, IRs of the OP can be written as

$$\begin{equation}\begin{aligned}\left\{{L}_{1}^{+},{L}_{2}^{+},{L}_{1}^{-},{L}_{2}^{-}\right\}& =\left\{{A}_{g},{B}_{g},{A}_{u},{B}_{u}\right\}\otimes \left({\text{e}}^{-\text{i}{\mathbf{Q}}_{1}\cdot \mathbf{T}},{\text{e}}^{-\text{i}{\mathbf{Q}}_{2}\cdot \mathbf{T}},{\text{e}}^{-\text{i}{\mathbf{Q}}_{3}\cdot \mathbf{T}}\right).\end{aligned}\end{equation} \tag{ 1 }$$

We describe the OP for a given IR by the vector $\overrightarrow {\phi }=({\phi }_{1},{\phi }_{2},{\phi }_{3})\in {\mathbb{R}}^{3}$ where the $\overrightarrow {\phi }$ accompanying the IR determines the symmetry group (isotropy group). The energy is better for the maximal isotropy group [29, 30] so we will take $\overrightarrow {\phi }\propto (1,1,1)$.

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Table 1. Character table for the point-group symmetry C_2h at the point L. The CDW OP will be one of the IRs ${L}_{1,2}^{{\pm}}$. To avoid confusion, the conduction band at L is labeled with the IR given in parentheses in column 1.

C2h	E	C2	σh	i
${L}_{1}^{+}$ (Ag )	1	1	1	1
${L}_{2}^{+}$ (Bg )	1	−1	−1	1
${L}_{1}^{-}$ (Au )	1	1	−1	−1
${L}_{2}^{-}$ (Bu )	1	−1	1	−1

Referring to the symmetries of OPs in table 1, we can read off the space groups for the four CDW states. The states with mirror eigenvalue −1 do not have a mirror plane but only the glide plane {m_[100]|001}, where the accompanying ${ \overrightarrow {a}}_{3}$ translation produces a complementary −1. Similarly states with the parity eigenvalue −1 respect the combined symmetry ${T}_{{\mathbf{a}}_{3}}i$ ($=\left\{\overline{1}\vert 001\right\}$). As ${T}_{{\mathbf{a}}_{3}}i={T}_{\frac{{\mathbf{a}}_{3}}{2}}i{T}_{\frac{{\mathbf{a}}_{3}}{2}}^{-1}$, there is inversion symmetry with respect the center at $\left(0,0,\frac{1}{2}\right)$. We thus conclude that the space group for ${L}_{1}^{+}$ and ${L}_{2}^{-}$ is $P\overline{3}m1$ (no. 164) and for ${L}_{2}^{+}$ and ${L}_{1}^{-}$ is $P\overline{3}c1$ (no. 165).

The normal state band structure of TiSe₂ using the generalized gradient approximation (GGA) without spin–orbit coupling (SOC) is shown in figure 2(a). There are three bands crossing the Fermi level at Γ and one band crossing at L. The states at Γ consist of the two-fold E_g and one-fold A_2u representations whereas the state at L belongs to the A_g representation. The mean field k ⋅ p Hamiltonian for the CDW can be expressed as

$$\begin{equation}{\hat{H}}_{\text{CDW}}={{\sum }^{\prime }}_{\mathbf{k}}{\psi }_{\mathbf{k}}^{{\dagger}}{\mathcal{H}}_{\text{CDW}}(\mathbf{k}){\psi }_{\mathbf{k}},\end{equation} \tag{ 2 }$$

where the basis ${\psi }_{\mathbf{k}}^{{\dagger}}=({\psi }_{{\Gamma}\mathbf{k}}^{{\dagger}},\enspace {\psi }_{L\mathbf{k}}^{{\dagger}},{\psi }_{A\mathbf{k}}^{{\dagger}},\enspace {\psi }_{M\mathbf{k}}^{{\dagger}})$ is taken and

$$\begin{equation}{\mathcal{H}}_{\text{CDW}}(\mathbf{k})=\left(\begin{matrix}\hfill {\mathcal{H}}_{{\Gamma}L}\hfill & \hfill \mathcal{V}\hfill \\ \hfill {\mathcal{V}}^{{\dagger}}\hfill & \hfill {\mathcal{H}}_{AM}\hfill \end{matrix}\right)(\mathbf{k}).\end{equation} \tag{ 3 }$$

The prime on the summation in equation (2) indicates that only low-energy or k states in the folded Brillouin zone (BZ) are considered (spin index is omitted for brevity). The states at the Γ and L's points as well as at the A and M points are coupled by the CDW gaps. Also, because q vectors connect the Γ and A points, these four points can be written together. The states at Γ and L's can be described by ${\mathcal{H}}_{{\Gamma}L}$:

$$\begin{equation}{\mathcal{H}}_{{\Gamma}L}(\mathbf{k})=\left(\begin{matrix}\hfill {\mathcal{H}}_{{\Gamma}}\hfill & \hfill {{\Delta}}_{{\Gamma}L}\hfill \\ \hfill {{\Delta}}_{{\Gamma}L}^{{\dagger}}\hfill & \hfill {\mathcal{H}}_{L}\hfill \end{matrix}\right)(\mathbf{k}),\end{equation} \tag{ 4 }$$

where ${\mathcal{H}}_{{\Gamma}}$ and ${\mathcal{H}}_{L}$ account for the normal-state band structure around Γ and L's, respectively. In ${\psi }_{{\Gamma}\mathbf{k}}^{{\dagger}}=({\psi }_{{{\Gamma}}_{1}\mathbf{k}}^{{\dagger}},\enspace {\psi }_{{{\Gamma}}_{2}\mathbf{k}}^{{\dagger}},\enspace {\psi }_{{{\Gamma}}_{3}\mathbf{k}}^{{\dagger}})$, the former two operators stand for the E_g states and the last one is for A_2u . In ${\psi }_{L\mathbf{k}}=({\psi }_{{L}_{1}\mathbf{k}}^{{\dagger}},\enspace {\psi }_{{L}_{2}\mathbf{k}}^{{\dagger}},\enspace {\psi }_{{L}_{3}\mathbf{k}}^{{\dagger}})$, the three bands at the three L's are included with k being relative to the corresponding L's. The CDW gap Δ_ΓL is a 3 × 3 matrix, where the matrix element Δ_ij refers to the CDW potential between ${\psi }_{{{\Gamma}}_{i}}^{{\dagger}}$ and ${\psi }_{{L}_{j}}$. We will mainly discuss ${\mathcal{H}}_{{\Gamma}L}$ and ${\mathcal{H}}_{AM}$. The off-diagonal element $\mathcal{V}$, which is induced by the second secondary OP, is expected to be weak.

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The Hamiltonians can be determined using the symmetry constraints as follows:

$$\begin{equation}{\mathcal{C}}_{3\mathrm{C}\mathrm{D}\mathrm{W}}{\mathcal{H}}_{\text{CDW}}(\mathbf{k}){\mathcal{C}}_{3\mathrm{C}\mathrm{D}\mathrm{W}}^{-1}={\mathcal{H}}_{\text{CDW}}(\mathfrak{R}\mathbf{k}),\end{equation} \tag{ 5 }$$

$$\begin{equation}{\mathcal{I}}_{\text{CDW}}{\mathcal{H}}_{\text{CDW}}(\mathbf{k}){\mathcal{I}}_{\mathrm{C}\mathrm{D}\mathrm{W}}^{-1}={\mathcal{H}}_{\text{CDW}}(-\mathbf{k}),\end{equation} \tag{ 6 }$$

$$\begin{equation}{\mathcal{M}}_{\text{CDW}}{\mathcal{H}}_{\text{CDW}}(\mathbf{k}){\mathcal{M}}_{\mathrm{C}\mathrm{D}\mathrm{W}}^{-1}={\mathcal{H}}_{\text{CDW}}(\mathfrak{M}\mathbf{k}),\end{equation} \tag{ 7 }$$

where $\mathfrak{R}\mathbf{k}$, −k, and $\mathfrak{M}\mathbf{k}$ are momenta under 3_[001], $\overline{1}$, and m_[100], respectively. We define the [100] direction to be x and [001] to be z. The relevant symmetry operators are

$$\begin{equation}{\mathcal{O}}_{\text{CDW}}=\left(\begin{matrix}\hfill {\mathcal{O}}_{\text{CDW,}{\Gamma}L}\hfill & \hfill \hfill \\ \hfill \hfill & \hfill {\mathcal{O}}_{\text{CDW,}AM}\hfill \end{matrix}\right),\end{equation} \tag{ 8 }$$

with

$$\begin{equation}{\mathcal{O}}_{\text{CDW,}{\Gamma}L}=\left(\begin{matrix}\hfill {\mathcal{O}}_{{\Gamma}}\hfill & \hfill \hfill \\ \hfill \hfill & \hfill {\eta }_{\mathcal{O}}{\mathcal{O}}_{L}\hfill \end{matrix}\right)\end{equation} \tag{ 9 }$$

and

$$\begin{equation}{\mathcal{O}}_{\text{CDW,}AM}=\left(\begin{matrix}\hfill {\eta }_{\mathcal{O}}^{3}{\mathcal{O}}_{A}\hfill & \hfill \hfill \\ \hfill \hfill & \hfill {\eta }_{\mathcal{O}}^{2}{\mathcal{O}}_{M}\hfill \end{matrix}\right),\end{equation} \tag{ 10 }$$

where $\mathcal{O}=\left\{{\mathcal{C}}_{3},\mathcal{I},\mathcal{M}\right\}$ and ${\eta }_{\mathcal{O}}$ is the eigenvalue of the OP for operation $\mathcal{O}$. Note that the eigenvalue in front of ${\mathcal{O}}_{A}$ takes ${\eta }_{\mathcal{O}}^{3}$ because Γ is connected to A via the sum of three ordering vectors Q_i . But, the eigenvalue at ${\mathcal{O}}_{M}$ takes ${\eta }_{\mathcal{O}}^{2}$ because a sum of two Q_i connects Γ and M. As ${\mathcal{O}}_{\text{CDW}}$ is block-diagonal, equations (5)–(7) determine the form of ${\mathcal{H}}_{{\Gamma}}$, ${\mathcal{H}}_{L}$, and Δ_ΓL , see appendix A.1 for details. ${\eta }_{\mathcal{O}}$ reflects the symmetries of the OP via the CDW gap matrix as follows

$$\begin{equation}{\mathcal{C}}_{3,{\Gamma}}{{\Delta}}_{{\Gamma}L}(\mathbf{k}){\mathcal{C}}_{3,L}^{-1}={\eta }_{{\mathcal{C}}_{3}}{{\Delta}}_{{\Gamma}L}(\mathfrak{R}\mathbf{k}),\end{equation} \tag{ 11 }$$

$$\begin{equation}{\mathcal{I}}_{{\Gamma}}{{\Delta}}_{{\Gamma}L}(\mathbf{k}){\mathcal{I}}_{L}^{-1}={\eta }_{\mathcal{I}}{{\Delta}}_{{\Gamma}L}(-\mathbf{k}),\end{equation} \tag{ 12 }$$

$$\begin{equation}{\mathcal{M}}_{{\Gamma}}{{\Delta}}_{{\Gamma}L}(\mathbf{k}){\mathcal{M}}_{L}^{-1}={\eta }_{\mathcal{M}}{{\Delta}}_{{\Gamma}L}(\mathfrak{M}\mathbf{k}),\end{equation} \tag{ 13 }$$

where ${\eta }_{\mathcal{I}}$, ${\eta }_{\mathcal{M}}$ (=±1) and ${\eta }_{{\mathcal{C}}_{3}}=1$. The four IRs ${L}_{1}^{+},\enspace {L}_{2}^{+},\enspace {L}_{1}^{-},\enspace {L}_{2}^{-}$ are obtained by taking $({\eta }_{\mathcal{I}},{\eta }_{\mathcal{M}})=(1,1),\enspace (1,-1)\enspace (-1,-1)$, and (−1, 1), respectively (listed also in table 1).

Although the ${L}_{1}^{-}$ CDW OP has an f-wave symmetry [31], the CDW gap functions may be of different symmetry depending on IRs of the composite bands as we will show below. It can be shown that the gap functions for the ${L}_{1}^{-}$ CDW state to the lowest order of k are:

$$\begin{equation}\begin{aligned}{{\Delta}}_{12}(\mathbf{k})={\lambda }_{11}{k}_{x},\enspace {{\Delta}}_{22}(\mathbf{k})={\lambda }_{21}{k}_{y}+{\lambda }_{21}^{\prime }{k}_{z},\\ {{\Delta}}_{32}(\mathbf{k})=({\lambda }_{31}{k}_{y}+{\lambda }_{31}^{\prime }{k}_{z}){k}_{x},\end{aligned}\end{equation} \tag{ 14 }$$

where λ, λ''s are taken to be real. The other terms can be obtained using the rotation symmetry, see appendix A.2 for details. The gap functions give the couplings among the valence bands at Γ and the folded conduction bands from the L points. Assuming that the valence and conduction bands in the normal state overlap, in the small gap limit, the gap functions in equation (14) will manifest whether bands cross or anticross according to their mirror or twofold rotation eigenvalues. For the large gap case, where multi-band hybridizations will be involved, this two-band picture will fail, and we will resort to topological protections.

Figure 2(b) presents the band structure of the CDW state in the presence of PLD. We can see that the CDW gap size (∼0.1 eV) is not small and the band structure is complex as a result of many band foldings. Now there are two CDW nodes (band crossings between the conduction and valence bands), one on the $\overline{{\Gamma}K}$ line and the other along the $\overline{{\Gamma}M}$ line. Taking into account the rotation and time-reversal symmetries, there are 12 nodes on the k_z = 0 plane. A full BZ exploration of the band structure shows that these nodes persist away from the k_z = 0 plane and yield nodal lines in the BZ, see figure 2(c). Band structures for different k_z values are given in appendix D.

We turn now to discuss the Ginzburg–Landau theory for the ${L}_{1}^{-}$ CDW state. Images of the primary OP (${L}_{1}^{-}$ mode) $\overrightarrow {\phi }=\left({\phi }_{1},{\phi }_{2},{\phi }_{3}\right)$ under various symmetry operations are:

$$\begin{align}{3}_{[001]}:& \left({\phi }_{1},{\phi }_{2},{\phi }_{3}\right)\to \left({\phi }_{2},{\phi }_{3},{\phi }_{1}\right),\\ {m}_{[100]}:& \left({\phi }_{1},{\phi }_{2},{\phi }_{3}\right)\to \left(-{\phi }_{3},-{\phi }_{2},-{\phi }_{1}\right),\\ \overline{I}:& \left({\phi }_{1},{\phi }_{2},{\phi }_{3}\right)\to \left(-{\phi }_{1},-{\phi }_{2},-{\phi }_{3}\right),\\ {T}_{{\mathbf{a}}_{1}}:& \left({\phi }_{1},{\phi }_{2},{\phi }_{3}\right)\to \left(-{\phi }_{1},{\phi }_{2},-{\phi }_{3}\right),\\ {T}_{{\mathbf{a}}_{2}}:& \left({\phi }_{1},{\phi }_{2},{\phi }_{3}\right)\to \left({\phi }_{1},-{\phi }_{2},-{\phi }_{3}\right),\\ {T}_{{\mathbf{a}}_{3}}:& \left({\phi }_{1},{\phi }_{2},{\phi }_{3}\right)\to \left(-{\phi }_{1},-{\phi }_{2},-{\phi }_{3}\right),\end{align} \tag{ 15 }$$

so that the free-energy density which satisfies symmetries of the pristine state to the fourth order becomes:

$$\begin{equation}{F}_{\text{primary}}=\alpha { \overrightarrow {\phi }}^{2}+{\beta }_{1}{ \overrightarrow {\phi }}^{4}+{\beta }_{2}\left({\phi }_{1}^{4}+{\phi }_{2}^{4}+{\phi }_{3}^{4}\right).\end{equation} \tag{ 16 }$$

Here α < 0 when T < T_c for nonzero $\overrightarrow {\phi }$ and β > 0 for stability. Therefore, we seek a secondary OP that is linearly coupled to the primary OP $\overrightarrow {\phi }$, i.e. we look for coincidence of the primary and secondary OPs (appendix B). Symmetry requirements suggest that the secondary OP $\overrightarrow {\zeta }$ will also have three components with the coupling energy

$$\begin{equation}{F}_{\text{coupling}}=\lambda \left({\phi }_{1}{\phi }_{2}{\zeta }_{3}+{\phi }_{2}{\phi }_{3}{\zeta }_{1}+{\phi }_{3}{\phi }_{1}{\zeta }_{2}\right),\end{equation} \tag{ 17 }$$

where $\overrightarrow {\zeta }$ follows the rules in equation (B2) except that m_[100], $\overline{I}$, and ${T}_{{\mathbf{a}}_{3}}$ are without the minus signs. The symmetry conditions dictate that the components of the secondary OP take ordering vectors ${\mathbf{Q}}_{1}^{\prime }=\frac{1}{2}{\mathbf{a}}_{1}^{{\ast}}$, ${\mathbf{Q}}_{2}^{\prime }=\frac{1}{2}{\mathbf{a}}_{2}^{{\ast}}$, and ${\mathbf{Q}}_{3}^{\prime }=\frac{1}{2}({\mathbf{a}}_{1}^{{\ast}}+{\mathbf{a}}_{2}^{{\ast}})$ and that they belong to IR ${M}_{1}^{+}$. In addition to the terms similar to equation (B5), the free energy for the secondary OP can contain the cubic term ζ₁ ζ₂ ζ₃. Besides, coexistence of ${L}_{1}^{-}$ and ${M}_{1}^{+}$ OPs will induce a third (second secondary) OP φ in higher order as

$$\begin{equation}{F}_{\text{coupling}}^{\prime }={\lambda }^{\prime }\varphi \left({\phi }_{1}{\zeta }_{1}+{\phi }_{2}{\zeta }_{2}+{\phi }_{3}{\zeta }_{3}\right),\end{equation} \tag{ 18 }$$

where φ, which we call IR ${A}_{2}^{-}$, is parity-odd, mirror-odd and belongs to the ordering vector $\frac{1}{2}{\mathbf{a}}_{3}^{{\ast}}$. Close to T_c, ζ is proportional to ϕ square and thus ψ is proportional to ϕ cube, so that the symmetry characters of ${M}_{1}^{+}$ ($={L}_{1}^{-}\otimes {L}_{1}^{-}$) are the same as ${L}_{1}^{+}$ and those of ${A}_{2}^{-}$ are identical to ${L}_{1}^{-}$. Incidentally, the coincident ${M}_{1}^{+}$ OP will also be present in the other three CDW states, accompanied by the corresponding third OPs.

Although the gap functions in equation (14) all vanish at k = 0, the folded bands still result in threefold degeneracy. But, this threefold degeneracy should not occur as it is not robust in the D_3d group. This can be understood, however, because the concurrent ${M}_{1}^{+}$ mode produces couplings among the bands at L points (also among the M points): the triplet from L's then splits into a singlet A_1u state and a doublet E_u state without breaking symmetry [32]. We emphasize that the parity of the folded bands changes from even to odd due to the CDW OP. In general, the IR of a folded band is the product of IRs of the unfolded band and the OP, written as:

$$\begin{equation}{{\Gamma}}_{\text{folded}}^{\mathbf{k}}={{\Gamma}}_{\text{unfolded}}^{\mathbf{k-Q}}\otimes {{\Gamma}}_{\text{CDW}}^{\mathbf{Q}}.\end{equation} \tag{ 19 }$$

This is our key finding, which is especially useful for investigating topology. By contrast, folded bands from M's to Γ, through the secondary ${M}_{1}^{+}$ OP, split into an A_g and two E_g states. As for the bands from A, their parities change sign because of the ${A}_{2}^{-}$ mode. These arguments are well corroborated by our independent first-principles band structure results, see figure 2(b).

Note that the gap function of two composite bands has the symmetry of the direct product of the IRs of the two bands. For instance, if the closest conduction band is IR A_1u and the valence band is IR A_2u , the associated gap function will be IR A_2g (=A_1u ⊗ A_2u ) that is also known as the i wave outlining a sin(6ϕ) profile, figure 1(b).

We recall at the outset [33] that in the presence of inversion and time-reversal symmetries, SOC-free systems can be classified into semimetals with even or odd number of bulk line nodes, which can be labeled with a set of ${\mathbb{Z}}_{2}$ invariants obtained from the product of parity eigenvalues of filled bands at the four parity-invariant momenta in a BZ plane. The line nodes are gapped by the SOC to yield weak topological insulators [34, 35].

In the CDW state, the parity-product at M, L, and A points in the folded BZ will be trivial, so that only the Γ point remains relevant. The reason is that the original and folded bands hybridize into symmetric and antisymmetric combinations resulting in a parity-product of −1, and even number of such pairs will yield a net product of +1. In the case of no band inversion, the parity-product at Γ of the CDW state is related to the strong topological invariant of the pristine state. Therefore, the strong topological invariant of the CDW state will be determined by the number (mod 2) of band inversions which involve parity switching beyond the topology of the normal state.

Our DFT calculations show that the pristine and the ${L}_{1}^{-}$ CDW states are both topologically trivial. Figure 2(b) shows that the three bands at Γ (E_u and A_2u bands) hybridize with three pristine bands at L points without changing parity yielding a net parity-product of +1. Also, the two bands at A (E_u symmetry) hybridize with two of three bands at the M points and maintain the parity-product of +1. An even number of nodal rings is thus expected to pierce through the k_z = 0 plane as is seen to be the case in figure 2(c). In this connection, we have further analyzed the band structure of the ${L}_{1}^{+}$ CDW state (not shown for brevity) to show that it is a topologically nontrivial semimetal consistent with equation (19).

We have presented an in-depth analysis of the symmetries as well as the topology of the electronic structure of bulk 1T–TiSe₂ CDW. Our first-principles calculations show that the CDW state hosts a nodal band structure in which the nodes are protected by symmetry and topology resembling that of the Dirac nodes in the spin-density-wave phase of iron pnictides [36, 37]. The existing topological theory in this connection only considers spin-density-wave states [38, 39] in terms of ${\mathbb{Z}}_{2}$ classification of three-dimensional insulators, but questions of symmetry properties of the density waves and their connection to the normal states have not been addressed. We resolve these questions by successfully connecting the symmetry and topology of the electronic IRs of the normal and CDW phases.

We emphasize that in our theory when both the time-reversal and inversion symmetries are preserved, inclusion of the SOC gaps band crossings for out-of-plane n-fold (n > 2) rotation axis to protect the Dirac nodes, but it does not change the parity of the bands. Although the CDW OP might change the IR and parity of the folded band, it does not always produce band inversion. We discuss the application of our theory to the band structure based on the GGA density functional, which yields a semimetallic normal state with an energy overlap (indirect negative band gap) between the conduction and valence bands. On the experimental side, there has been a longstanding debate whether the normal state of 1T–TiSe₂ is a semimetal [40–42] or a semiconductor [32, 43]. Our study however is not concerned with such details of the band structure. Our results are intended to be generic in nature and are not sensitive to the density functional used. The question of sensitivity of the band structure of 1T–TiSe₂ to exchange–correlation functional, Hubbard U, and van der Waals interactions has been explored extensively in the literature [44–46]. Correlations are generally found to reduce the band overlap in better accord with experimental results. This is also the case in our first-principles band structure of 1T–TiSe₂ obtained with the modified Becke–Johnson (mBJ) meta-GGA density functional, which correctly captures the bandgap correction and reproduces an insulating electronic state, see appendix E for details. Our theory is thus a 'weak-pairing theory' in which the Fermi energy is larger than the CDW gap. For these reasons, it is natural for us to base our analysis on the GGA band structure.

Figure 3 shows the pressure effect on the electronic structure of bulk 1T–TiSe₂. The energy overlap is seen to be retained under moderate pressures as is the CDW order. This suggests that a topological phase transition in the CDW state can be achieved by applying hydrostatic pressure. By showing how symmetry-related topology can be obtained directly from the electronic structure, our study provides a guide in search of topological CDW phases. Our analysis can be generalized straightforwardly to consider other spontaneous symmetry-breaking phases.

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SMH is supported by the Ministry of Science and Technology (MoST) in Taiwan under Grant No. 105-2112-M-110-014-MY3 and also by the NCTS of Taiwan. He also acknowledges support from Academia Sinica's Short-term Visiting Program for Domestic Scholars. The work at TIFR Mumbai is supported by the Department of Atomic Energy of the Government of India under Project No. 12-R&D-TFR-5.10-0100. The work at Northeastern University was supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences Grant No. DE-SC0019275 (materials discovery for QIS applications) and benefited from Northeastern University's Advanced Scientific Computation Center and the National Energy Research Scientific Computing Center through DOE Grant No. DE-AC02-05CH11231.

The data that support the findings of this study are available upon reasonable request from the authors.

Here we discuss the construction of our mean-field 2 × 2 × 2 CDW Hamiltonian without SOC for bulk 1T–TiSe₂. It is written as

$$\begin{equation}{\hat{H}}_{\text{CDW}}={{\sum }^{\prime }}_{\mathbf{k}}{\psi }_{\mathbf{k}}^{{\dagger}}{\mathcal{H}}_{\text{CDW}}(\mathbf{k}){\psi }_{\mathbf{k}},\end{equation} \tag{ A1 }$$

where the prime indicates that k runs over states near the Fermi surfaces (FS) and

$$\begin{equation}{\mathcal{H}}_{\text{CDW}}(\mathbf{k})=\left(\begin{matrix}\hfill {\mathcal{H}}_{{\Gamma}L}\hfill & \hfill \mathcal{V}\hfill \\ \hfill {\mathcal{V}}^{{\dagger}}\hfill & \hfill {\mathcal{H}}_{AM}\hfill \end{matrix}\right)(\mathbf{k})\end{equation} \tag{ A2 }$$

in the basis ${\psi }_{\mathbf{k}}^{{\dagger}}=({\psi }_{{\Gamma}\mathbf{k}}^{{\dagger}},\enspace {\psi }_{L\mathbf{k}}^{{\dagger}},{\psi }_{A\mathbf{k}}^{{\dagger}},\enspace {\psi }_{M\mathbf{k}}^{{\dagger}})$. Based on our first-principles calculations, the normal-state band structure near the Fermi level consists of three valence states at the Γ point, two valence states at the A point, and a conduction band whose minimum locates at each L point. The L-point conduction band adiabatically evolves into bands at the M point, see figure 2(a) of the main text. The two valence bands at Γ are comprised of E_g IR whereas the third band belongs to A_2u (table 2). The bands at A have the IR E_u whereas at the L and M, they have A_g IR (table 1). For ${\psi }_{{\Gamma}\mathbf{k}}^{{\dagger}}$, which contains three creation operators for the hole pockets around Γ, ${\psi }_{{\Gamma}\mathbf{k}}^{{\dagger}}=({\psi }_{{{\Gamma}}_{1}\mathbf{k}}^{{\dagger}},\enspace {\psi }_{{{\Gamma}}_{2}\mathbf{k}}^{{\dagger}},\enspace {\psi }_{{{\Gamma}}_{3}\mathbf{k}}^{{\dagger}})$, we assign E_g to the first two operators and A_2u to the last operator. For ${\psi }_{A\mathbf{k}}^{{\dagger}}$, we denote ${\psi }_{A\mathbf{k}}^{{\dagger}}=({\psi }_{{A}_{1}\mathbf{k}}^{{\dagger}},\enspace {\psi }_{{A}_{2}\mathbf{k}}^{{\dagger}})$. As for ψ_{L k} , it contains three creation operators for the electron pockets around the three L points, ${\psi }_{L\mathbf{k}}^{{\dagger}}=({\psi }_{{L}_{1}\mathbf{k}}^{{\dagger}},\enspace {\psi }_{{L}_{2}\mathbf{k}}^{{\dagger}},\enspace {\psi }_{{L}_{3}\mathbf{k}}^{{\dagger}})$. Replacing L by M, we obtain ${\psi }_{M\mathbf{k}}^{{\dagger}}$.

We will mainly consider the CDW potentials among the bands at Γ and L points. Other potentials can be constructed along similar lines. We first examine

$$\begin{equation}{\mathcal{H}}_{{\Gamma}L}(\mathbf{k})=\left(\begin{matrix}\hfill {\mathcal{H}}_{{\Gamma}}\hfill & \hfill {{\Delta}}_{{\Gamma}L}\hfill \\ \hfill {{\Delta}}_{{\Gamma}L}^{{\dagger}}\hfill & \hfill {\mathcal{H}}_{L}\hfill \end{matrix}\right)(\mathbf{k}),\end{equation} \tag{ A3 }$$

where ${\mathcal{H}}_{{\Gamma}}$ and ${\mathcal{H}}_{L}$ account for the normal-state band structure around Γ and L's, respectively. In these expression, ${\mathcal{H}}_{{\Gamma}}$, ${\mathcal{H}}_{L}$ and Δ_ΓL are all 3 × 3 matrices written as

$$\begin{equation}{\mathcal{H}}_{{\Gamma}}(\mathbf{k})=\left(\begin{matrix}\hfill {h}_{11}\hfill & \hfill {h}_{12}\hfill & \hfill {h}_{13}\hfill \\ \hfill {h}_{12}^{{\ast}}\hfill & \hfill {h}_{22}\hfill & \hfill {h}_{23}\hfill \\ \hfill {h}_{13}^{{\ast}}\hfill & \hfill {h}_{23}^{{\ast}}\hfill & \hfill {h}_{33}\hfill \end{matrix}\right)(\mathbf{k}),\end{equation} \tag{ A4 }$$

$$\begin{equation}{\mathcal{H}}_{L}(\mathbf{k})=\left(\begin{matrix}\hfill {\xi }_{1}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\xi }_{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_{3}\hfill \end{matrix}\right)(\mathbf{k}),\end{equation} \tag{ A5 }$$

$$\begin{equation}{{\Delta}}_{{\Gamma}L}(\mathbf{k})=\left(\begin{matrix}\hfill {{\Delta}}_{11}\hfill & \hfill {{\Delta}}_{12}\hfill & \hfill {{\Delta}}_{13}\hfill \\ \hfill {{\Delta}}_{21}\hfill & \hfill {{\Delta}}_{22}\hfill & \hfill {{\Delta}}_{23}\hfill \\ \hfill {{\Delta}}_{31}\hfill & \hfill {{\Delta}}_{32}\hfill & \hfill {{\Delta}}_{33}\hfill \end{matrix}\right)(\mathbf{k}).\end{equation} \tag{ A6 }$$

Δ_ij (k) will be regarded as the CDW potential between Γ_i and L_j . The Hamiltonians can then be written down according to the symmetry constraints as:

$$\begin{equation}{\mathcal{C}}_{3}{\mathcal{H}}_{\text{CDW}}(\mathbf{k}){\mathcal{C}}_{3}^{-1}={\mathcal{H}}_{\text{CDW}}(\mathfrak{R}\mathbf{k}),\end{equation} \tag{ A7 }$$

$$\begin{equation}{\mathcal{I}\mathcal{H}}_{\text{CDW}}(\mathbf{k}){\mathcal{I}}^{-1}={\mathcal{H}}_{\mathrm{C}\mathrm{D}\mathrm{W}}(-\mathbf{k}),\end{equation} \tag{ A8 }$$

$$\begin{equation}\mathcal{M}{\mathcal{H}}_{\text{CDW}}(\mathbf{k}){\mathcal{M}}^{-1}={\mathcal{H}}_{\text{CDW}}(\mathfrak{M}\mathbf{k}),\end{equation} \tag{ A9 }$$

where $\mathfrak{R}\mathbf{k}$, −k, and $\mathfrak{M}\mathbf{k}$ are momenta under (counterclockwise) 3_[001], $\overline{1}$, and m_[100], respectively. We define the [100] direction to be x and [001] to be z. The relevant symmetry operators would be k-dependent and are block-diagonal as

$$\begin{equation}{\mathcal{C}}_{3}=\left(\begin{matrix}\hfill {\mathcal{C}}_{3,{\Gamma}}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill {\mathcal{C}}_{3,L}\hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill {\mathcal{C}}_{3,A}\hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {\mathcal{C}}_{3,M}\hfill \end{matrix}\right),\qquad \mathcal{I}=\left(\begin{matrix}\hfill {\mathcal{I}}_{{\Gamma}}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill {\mathcal{I}}_{L}\hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill {\mathcal{I}}_{A}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {\mathcal{I}}_{M}\hfill \end{matrix}\right),\qquad \mathcal{M}=\left(\begin{matrix}\hfill {\mathcal{M}}_{{\Gamma}}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill {\mathcal{M}}_{L}\hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill {\mathcal{M}}_{A}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {\mathcal{M}}_{M}\hfill \end{matrix}\right).\end{equation} \tag{ A10 }$$

We choose

$$\begin{equation}{\mathcal{C}}_{3,{\Gamma}}=\left(\begin{matrix}\hfill -\frac{1}{2}\hfill & \hfill \frac{\sqrt{3}}{2}\hfill & \hfill 0\hfill \\ \hfill -\frac{\sqrt{3}}{2}\hfill & \hfill -\frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{matrix}\right),\qquad {\mathcal{C}}_{3,A}=\left(\begin{matrix}\hfill -\frac{1}{2}\hfill & \hfill \frac{\sqrt{3}}{2}\hfill \\ \hfill -\frac{\sqrt{3}}{2}\hfill & \hfill -\frac{1}{2}\hfill \end{matrix}\right),\qquad {\mathcal{C}}_{3,L}={\mathcal{C}}_{3,M}=\left(\begin{matrix}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{matrix}\right),\end{equation} \tag{ A11 }$$

$$\begin{equation}{\mathcal{I}}_{{\Gamma}}=\left(\begin{matrix}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill \end{matrix}\right),\qquad {\mathcal{I}}_{A}=\left(\begin{matrix}\hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{matrix}\right),\qquad {\mathcal{I}}_{L}={\mathcal{I}}_{M}=\left(\begin{matrix}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{matrix}\right),\end{equation} \tag{ A12 }$$

$$\begin{equation}{\mathcal{M}}_{{\Gamma}}=\left(\begin{matrix}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{matrix}\right),\qquad {\mathcal{M}}_{A}=\left(\begin{matrix}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{matrix}\right),\qquad {\mathcal{M}}_{L}={\mathcal{M}}_{M}=\left(\begin{matrix}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{matrix}\right).\end{equation} \tag{ A13 }$$

Note that here we adopt a representation in which ${\mathcal{I}}_{L}$, ${\mathcal{I}}_{L}$, ${\mathcal{M}}_{{\Gamma}}$ and ${\mathcal{M}}_{A}$ matrices are diagonal, where the diagonal elements are the eigenvalues of the corresponding symmetries at various symmetry points. In this basis, the ${\mathcal{C}}_{3,{\Gamma}}$ and ${\mathcal{C}}_{3,A}$ matrices are non-diagonal for the E_g and E_u states. As for the L and M points, they involve large momentum transfers. For instance, we set $\left\vert {\psi }_{{L}_{1}\mathbf{k}}\right\rangle {\mapsto}\left\vert {\psi }_{{L}_{2}\mathfrak{R}\mathbf{k}}\right\rangle$ under 3_[001]; while under m_[100], which leaves L₂ invariant and interchanges L₁ and L₃, $\left\vert {\psi }_{{L}_{2}\mathbf{k}}\right\rangle {\mapsto}\left\vert {\psi }_{{L}_{2}\mathfrak{M}\mathbf{k}}\right\rangle$ and $\left\vert {\psi }_{{L}_{1}\mathbf{k}}\right\rangle {\leftrightarrow}\left\vert {\psi }_{{L}_{3}\mathfrak{M}\mathbf{k}}\right\rangle$.

A.1. Bare Hamiltonians

A.1.1. Hamiltonian at Γ

We will write down the Hamiltonian at ${\Gamma}\enspace {\mathcal{H}}_{{\Gamma}}$. Here it is easier to use a basis that makes the three-fold rotation operator diagonal. The basis is unitarily transformed as ${~{\psi }}_{{\Gamma}\mathbf{k}}^{{\dagger}}={\psi }_{{\Gamma}\mathbf{k}}^{{\dagger}}\mathcal{U}$ and the symmetry operators transform, for instance, as ${~{\mathcal{C}}}_{3,{\Gamma}}={\mathcal{U}}^{-1}{\mathcal{C}}_{3,{\Gamma}}\mathcal{U}$, where the unitary matrix

$$\begin{equation}\mathcal{U}=\left(\begin{matrix}\hfill \frac{1}{\sqrt{2}}\hfill & \hfill \frac{1}{\sqrt{2}}\hfill & \hfill 0\hfill \\ \hfill \frac{i}{\sqrt{2}}\hfill & \hfill -\frac{i}{\sqrt{2}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{matrix}\right).\end{equation} \tag{ A14 }$$

In this 'tilde' basis, the symmetry operators are

$$\begin{equation}\begin{aligned}{~{\mathcal{C}}}_{3,{\Gamma}}=\left(\begin{matrix}\hfill {\text{e}}^{\text{i}\frac{2\pi }{3}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\text{e}}^{-\text{i}\frac{2\pi }{3}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{matrix}\right)\text{,}\;\qquad {~{\mathcal{M}}}_{{\Gamma}}=\left(\begin{matrix}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{matrix}\right)\text{,}\;\\ {~{\mathcal{I}}}_{{\Gamma}}=\left(\begin{matrix}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill \end{matrix}\right)\text{,}\;\qquad {~{\mathcal{T}}}_{{\Gamma}}=\left(\begin{matrix}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{matrix}\right)K,\end{aligned}\end{equation} \tag{ A15 }$$

where the last operator is the time-reversal operator with complex conjugation K. This tilde Hamiltonian follows the symmetry constraints

$$\begin{equation}{~{\mathcal{C}}}_{3,{\Gamma}}{~{\mathcal{H}}}_{{\Gamma}}(\mathbf{k}){~{\mathcal{C}}}_{3,{\Gamma}}^{-1}={~{\mathcal{H}}}_{{\Gamma}}(\mathfrak{R}\mathbf{k}),\end{equation} \tag{ A16 }$$

$$\begin{equation}{~{\mathcal{I}}}_{{\Gamma}}{~{\mathcal{H}}}_{{\Gamma}}(\mathbf{k}){~{\mathcal{I}}}_{{\Gamma}}^{-1}={~{\mathcal{H}}}_{{\Gamma}}(-\mathbf{k}),\end{equation} \tag{ A17 }$$

$$\begin{equation}{~{\mathcal{T}}}_{{\Gamma}}{~{\mathcal{I}}}_{{\Gamma}}{~{\mathcal{H}}}_{{\Gamma}}(\mathbf{k}){\left({~{\mathcal{T}}}_{{\Gamma}}{~{\mathcal{I}}}_{{\Gamma}}\right)}^{-1}={~{\mathcal{H}}}_{{\Gamma}}(\mathbf{k}),\end{equation} \tag{ A18 }$$

$$\begin{equation}{~{\mathcal{M}}}_{{\Gamma}}{~{\mathcal{H}}}_{{\Gamma}}(\mathbf{k}){~{\mathcal{M}}}_{{\Gamma}}^{-1}={~{\mathcal{H}}}_{{\Gamma}}(\mathfrak{M}\mathbf{k}).\end{equation} \tag{ A19 }$$

By taking ${~{\mathcal{H}}}_{{\Gamma}}(\mathbf{k})=\left(\begin{matrix}\hfill {~{h}}_{11}\hfill & \hfill {~{h}}_{12}\hfill & \hfill {~{h}}_{13}\hfill \\ \hfill {~{h}}_{12}^{{\ast}}\hfill & \hfill {~{h}}_{22}\hfill & \hfill {~{h}}_{23}\hfill \\ \hfill {~{h}}_{13}^{{\ast}}\hfill & \hfill {~{h}}_{23}^{{\ast}}\hfill & \hfill {~{h}}_{33}\hfill \end{matrix}\right)(\mathbf{k})$ and defining k_± = k_x ± ik_y , symmetry conditions require that for ${~{\mathcal{C}}}_{3,{\Gamma}}$

$$\begin{equation}\left(\begin{matrix}\hfill {~{h}}_{11}\hfill & \hfill {~{h}}_{12}\hfill & \hfill {~{h}}_{13}\hfill \\ \hfill \hfill & \hfill {~{h}}_{22}\hfill & \hfill {~{h}}_{23}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill {~{h}}_{33}\hfill \end{matrix}\right)\left({\text{e}}^{\text{i}\enspace \frac{2\pi }{3}}{k}_{+},{\text{e}}^{-\text{i}\enspace \frac{2\pi }{3}}{k}_{-},{k}_{z}\right)=\left(\begin{matrix}\hfill {~{h}}_{11}\hfill & \hfill {\text{e}}^{-\text{i}\enspace \frac{2\pi }{3}}{~{h}}_{12}\hfill & \hfill {\text{e}}^{\text{i}\enspace \frac{2\pi }{3}}{~{h}}_{13}\hfill \\ \hfill \hfill & \hfill {~{h}}_{22}\hfill & \hfill {\text{e}}^{-\text{i}\enspace \frac{2\pi }{3}}{~{h}}_{23}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill {~{h}}_{33}\hfill \end{matrix}\right)({k}_{+},{k}_{-},{k}_{z}),\end{equation} \tag{ A20 }$$

for ${~{\mathcal{I}}}_{{\Gamma}}$

$$\begin{equation}\left(\begin{matrix}\hfill {~{h}}_{11}\hfill & \hfill {~{h}}_{12}\hfill & \hfill {~{h}}_{13}\hfill \\ \hfill \hfill & \hfill {~{h}}_{22}\hfill & \hfill {~{h}}_{23}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill {~{h}}_{33}\hfill \end{matrix}\right)(-{k}_{+},-{k}_{-},-{k}_{z})=\left(\begin{matrix}\hfill {~{h}}_{11}\hfill & \hfill {~{h}}_{12}\hfill & \hfill -{~{h}}_{13}\hfill \\ \hfill \hfill & \hfill {~{h}}_{22}\hfill & \hfill -{~{h}}_{23}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill {~{h}}_{33}\hfill \end{matrix}\right)({k}_{+},{k}_{-},{k}_{z}),\end{equation} \tag{ A21 }$$

for ${~{\mathcal{T}}}_{{\Gamma}}{~{\mathcal{I}}}_{{\Gamma}}$

$$\begin{equation}\left(\begin{matrix}\hfill {~{h}}_{11}\hfill & \hfill {~{h}}_{12}\hfill & \hfill {~{h}}_{13}\hfill \\ \hfill \hfill & \hfill {~{h}}_{22}\hfill & \hfill {~{h}}_{23}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill {~{h}}_{33}\hfill \end{matrix}\right)({k}_{+},{k}_{-},{k}_{z})=\left(\begin{matrix}\hfill {~{h}}_{22}\hfill & \hfill {~{h}}_{12}\hfill & \hfill -{~{h}}_{23}^{{\ast}}\hfill \\ \hfill \hfill & \hfill {~{h}}_{11}\hfill & \hfill -{~{h}}_{13}^{{\ast}}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill {~{h}}_{33}\hfill \end{matrix}\right)({k}_{+},{k}_{-},{k}_{z}),\end{equation} \tag{ A22 }$$

and for ${~{\mathcal{M}}}_{{\Gamma}}$

$$\begin{equation}\left(\begin{matrix}\hfill {~{h}}_{11}\hfill & \hfill {~{h}}_{12}\hfill & \hfill {~{h}}_{13}\hfill \\ \hfill \hfill & \hfill {~{h}}_{22}\hfill & \hfill {~{h}}_{23}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill {~{h}}_{33}\hfill \end{matrix}\right)(-{k}_{-},-{k}_{+},{k}_{z})=\left(\begin{matrix}\hfill {~{h}}_{22}\hfill & \hfill {~{h}}_{12}^{{\ast}}\hfill & \hfill {~{h}}_{23}\hfill \\ \hfill \hfill & \hfill {~{h}}_{11}\hfill & \hfill {~{h}}_{13}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill {~{h}}_{33}\hfill \end{matrix}\right)({k}_{+},{k}_{-},{k}_{z}),\end{equation} \tag{ A23 }$$

respectively. The Hamiltonian up to second order is:

$$\begin{equation}{~{\mathcal{H}}}_{{\Gamma}}(\mathbf{k})=\left(\begin{matrix}\hfill a{k}_{{\Vert}}^{2}+{a}^{\prime }{k}_{z}^{2}+{\varepsilon }_{1}\hfill & \hfill ic{k}_{-}{k}_{z}+{c}^{\prime }{k}_{+}^{2}\hfill & \hfill d{k}_{+}\hfill \\ \hfill -ic{k}_{+}{k}_{z}+{c}^{\prime }{k}_{-}^{2}\hfill & \hfill a{k}_{{\Vert}}^{2}+{a}^{\prime }{k}_{z}^{2}+{\varepsilon }_{1}\hfill & \hfill -d{k}_{-}\hfill \\ \hfill d{k}_{-}\hfill & \hfill -d{k}_{+}\hfill & \hfill b{k}_{{\Vert}}^{2}+{b}^{\prime }{k}_{z}^{2}+{\varepsilon }_{2}\hfill \end{matrix}\right).\end{equation} \tag{ A24 }$$

Here ${k}_{{\Vert}}={k}_{x}^{2}+{k}_{y}^{2}$ and all parameters a, a', b, b', c, c', d, ɛ₁, and ɛ₂ are real. Finally, we transform ${~{\mathcal{H}}}_{{\Gamma}}$ back to ${\mathcal{H}}_{{\Gamma}}$ by

$$\begin{equation}{\mathcal{H}}_{{\Gamma}}(\mathbf{k})={\mathcal{U}~{\mathcal{H}}}_{{\Gamma}}(\mathbf{k}){\mathcal{U}}^{-1}\end{equation} \tag{ A25 }$$

$$\begin{equation*}=\left(\begin{matrix}\hfill a{k}_{{\Vert}}^{2}+{a}^{\prime }{k}_{z}^{2}+{\varepsilon }_{1}+c{k}_{y}{k}_{z}+{c}^{\prime }({k}_{x}^{2}-{k}_{y}^{2})\hfill & \hfill -c{k}_{x}{k}_{z}-2{c}^{\prime }{k}_{x}{k}_{y}\hfill & \hfill i\sqrt{2}d{k}_{y}\hfill \\ \hfill -c{k}_{x}{k}_{z}-2{c}^{\prime }{k}_{x}{k}_{y}\hfill & \hfill a{k}_{{\Vert}}^{2}+{a}^{\prime }{k}_{z}^{2}+{\varepsilon }_{1}-c{k}_{y}{k}_{z}-{c}^{\prime }({k}_{x}^{2}-{k}_{y}^{2})\hfill & \hfill i\sqrt{2}d{k}_{x}\hfill \\ \hfill -i\sqrt{2}d{k}_{y}\hfill & \hfill -i\sqrt{2}d{k}_{x}\hfill & \hfill b{k}_{{\Vert}}^{2}+{b}^{\prime }{k}_{z}^{2}+{\varepsilon }_{2}\hfill \end{matrix}\right).\end{equation*}$$

A.1.2. Hamiltonians at L and M

The Hamiltonian at L's ${\mathcal{H}}_{L}$ is identical to ${\mathcal{H}}_{M}$, and can be written as

$$\begin{equation}{\mathcal{H}}_{L}(\mathbf{k})=\left(\begin{matrix}\hfill {\xi }_{1}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\xi }_{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_{3}\hfill \end{matrix}\right)(\mathbf{k}).\end{equation} \tag{ A26 }$$

Here we assume that within the low-energy region, the three bands are independent. The symmetry operators are then given by

$$\begin{equation}\begin{aligned}{\mathcal{C}}_{3,L}=\left(\begin{matrix}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{matrix}\right)\text{,}\;\qquad {\mathcal{M}}_{L}=\left(\begin{matrix}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{matrix}\right)\text{,}\;\\ {\mathcal{I}}_{L}=\left(\begin{matrix}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{matrix}\right)\text{,}\;\qquad {\mathcal{T}}_{L}=\left(\begin{matrix}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{matrix}\right)K.\end{aligned}\end{equation} \tag{ A27 }$$

Similar as before, we conclude that

$$\begin{equation}{\xi }_{2}(\mathbf{k})=A{k}_{x}^{2}+B{k}_{y}^{2}+C{k}_{z}^{2}+D{k}_{y}{k}_{z}+{\varepsilon }_{3},\end{equation} \tag{ A28 }$$

$$\begin{equation}{\xi }_{1}(\mathbf{k})={\xi }_{2}(\mathfrak{R}\mathbf{k}),\enspace {\xi }_{3}(\mathbf{k})={\xi }_{2}({\mathfrak{R}}^{-1}\mathbf{k}).\end{equation} \tag{ A29 }$$

A.1.3. Hamiltonians at A

Although the conduction band (A_g ) extends monotonically along L–M, in some calculations the band structure shows a band inversion along Γ–A, where the E_g valence bands switch with the E_u conduction bands. We consider this case in which the valence band becomes IR E_u at A.

The E_u bands are two-fold degenerate at A, where they are described by the same Hamiltonian (with different parameters) as the upper 2 × 2 block of ${\mathcal{H}}_{{\Gamma}}$ in equation (A25). In addition, we show the coupling Hamiltonian between the E_g and the E_u bands in $\mathcal{V}$. In the basis where the symmetry operators for the E_u bands take the form

$$\begin{equation}{\mathcal{I}}_{{\Gamma}}^{\prime }=\left(\begin{matrix}\hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{matrix}\right),\qquad {\mathcal{M}}_{{\Gamma}}^{\prime }=\left(\begin{matrix}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{matrix}\right),\qquad {\mathcal{C}}_{3,{\Gamma}}^{\prime }=\left(\begin{matrix}\hfill -\frac{1}{2}\hfill & \hfill \frac{\sqrt{3}}{2}\hfill \\ \hfill -\frac{\sqrt{3}}{2}\hfill & \hfill -\frac{1}{2}\hfill \end{matrix}\right),\end{equation} \tag{ A30 }$$

the k ⋅ p model will read

$$\begin{equation}{\mathcal{H}}_{{\Gamma}A}=\left(\begin{matrix}\hfill {t}_{1}{k}_{z}\hfill & \hfill i{t}_{2}{k}_{x}{k}_{y}{k}_{z}\hfill \\ \hfill -i{t}_{2}{k}_{x}{k}_{y}{k}_{z}\hfill & \hfill {t}_{1}{k}_{z}\hfill \end{matrix}\right)\end{equation} \tag{ A31 }$$

with real parameters t₁ and t₂. The coupling Hamiltonian indicates anticrossing among the E_g and E_u bands.

A.2. CDW gap functions

The gap functions in equations (5)–(7) follow

$$\begin{equation}{\mathcal{C}}_{3,{\Gamma}}\mathcal{D}(\mathbf{k}){\mathcal{C}}_{3,L}^{-1}=\mathcal{D}(\mathfrak{R}\mathbf{k}),\end{equation} \tag{ A32 }$$

$$\begin{equation}{\mathcal{I}}_{{\Gamma}}\mathcal{D}(\mathbf{k}){\mathcal{I}}_{L}^{-1}={\eta }_{I}\mathcal{D}(-\mathbf{k}),\end{equation} \tag{ A33 }$$

$$\begin{equation}{\mathcal{M}}_{{\Gamma}}\mathcal{D}(\mathbf{k}){\mathcal{M}}_{L}^{-1}={\eta }_{M}\mathcal{D}(\mathfrak{M}\mathbf{k}),\end{equation} \tag{ A34 }$$

where ${\eta }_{{C}_{2}},\enspace {\eta }_{M}$ are eigenvalues of the corresponding operations for the OP. The allowed values for η_I and η_M are ±1. The four IRs ${L}_{1}^{+},\enspace {L}_{2}^{+},\enspace {L}_{1}^{-},\enspace {L}_{2}^{-}$ have symmetries with (η_I , η_M ) = (1, 1), (1, −1), (−1, −1), (−1, 1), respectively. To obtain the gap functions, we have, through equations (A33) and (A34),

$$\begin{equation}\left(\begin{matrix}\hfill {{\Delta}}_{11}\hfill & \hfill {{\Delta}}_{12}\hfill & \hfill {{\Delta}}_{13}\hfill \\ \hfill {{\Delta}}_{21}\hfill & \hfill {{\Delta}}_{22}\hfill & \hfill {{\Delta}}_{23}\hfill \\ \hfill {{\Delta}}_{31}\hfill & \hfill {{\Delta}}_{32}\hfill & \hfill {{\Delta}}_{33}\hfill \end{matrix}\right)(\mathbf{k})={\eta }_{I}\left(\begin{matrix}\hfill {{\Delta}}_{11}\hfill & \hfill {{\Delta}}_{12}\hfill & \hfill {{\Delta}}_{13}\hfill \\ \hfill {{\Delta}}_{21}\hfill & \hfill {{\Delta}}_{22}\hfill & \hfill {{\Delta}}_{23}\hfill \\ \hfill -{{\Delta}}_{31}\hfill & \hfill -{{\Delta}}_{32}\hfill & \hfill -{{\Delta}}_{33}\hfill \end{matrix}\right)(-\mathbf{k})={\eta }_{M}\left(\begin{matrix}\hfill {{\Delta}}_{13}\hfill & \hfill {{\Delta}}_{12}\hfill & \hfill {{\Delta}}_{11}\hfill \\ \hfill -{{\Delta}}_{23}\hfill & \hfill -{{\Delta}}_{22}\hfill & \hfill -{{\Delta}}_{21}\hfill \\ \hfill {{\Delta}}_{33}\hfill & \hfill {{\Delta}}_{32}\hfill & \hfill {{\Delta}}_{31}\hfill \end{matrix}\right)(\mathfrak{M}\mathbf{k}).\end{equation} \tag{ A35 }$$

In particular, the gap functions associated with L₂, which is invariant under both inversion and the mirror symmetry, are:

$$\begin{align}{{\Delta}}_{12}(-\mathbf{k})& ={\eta }_{I}{{\Delta}}_{12}(\mathbf{k}),\qquad {{\Delta}}_{22}(-\mathbf{k})={\eta }_{I}{{\Delta}}_{22}(\mathbf{k}),\enspace \\ {{\Delta}}_{32}(-\mathbf{k})& =-{\eta }_{I}{{\Delta}}_{32}(\mathbf{k}),\end{align} \tag{ A36 }$$

$$\begin{align}{{\Delta}}_{12}(\mathfrak{M}\mathbf{k})& ={\eta }_{M}{{\Delta}}_{12}(\mathbf{k}),\qquad {{\Delta}}_{22}(\mathfrak{M}\mathbf{k})=-{\eta }_{M}{{\Delta}}_{22}(\mathbf{k}),\enspace \\ {{\Delta}}_{32}(\mathfrak{M}\mathbf{k})& ={\eta }_{M}{{\Delta}}_{32}(\mathbf{k}).\end{align} \tag{ A37 }$$

So the gap functions for the four CDW states to lowest order in k are:

$$\begin{equation}{L}_{1}^{+}:\enspace {{\Delta}}_{12}(\mathbf{k})\sim \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.,\qquad {{\Delta}}_{22}(\mathbf{k})\sim \left\{{k}_{x}{k}_{y},{k}_{x}{k}_{z}\right\},\qquad {{\Delta}}_{32}(\mathbf{k})\sim \left\{{k}_{y},{k}_{z}\right\},\enspace \end{equation} \tag{ A38 }$$

$$\begin{equation}{L}_{2}^{+}:\enspace {{\Delta}}_{12}(\mathbf{k})\sim \left\{{k}_{x}{k}_{y},{k}_{x}{k}_{z}\right\},\qquad {{\Delta}}_{22}(\mathbf{k})\sim \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.,\qquad {{\Delta}}_{32}(\mathbf{k})\sim {k}_{x},\end{equation} \tag{ A39 }$$

$$\begin{equation}{L}_{1}^{-}:\enspace {{\Delta}}_{12}(\mathbf{k})\sim {k}_{x},\qquad {{\Delta}}_{22}(\mathbf{k})\sim \left\{{k}_{y},{k}_{z}\right\},\qquad {{\Delta}}_{32}(\mathbf{k})\sim \left\{{k}_{x}{k}_{y},{k}_{x}{k}_{z}\right\},\enspace \end{equation} \tag{ A40 }$$

$$\begin{equation}{L}_{2}^{-}:\enspace {{\Delta}}_{12}(\mathbf{k})\sim \left\{{k}_{y},{k}_{z}\right\},\qquad {{\Delta}}_{22}(\mathbf{k})\sim {k}_{x},\qquad {{\Delta}}_{32}(\mathbf{k})\sim \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.\end{equation} \tag{ A41 }$$

Two functions in the brackets are assumed to combine linearly. The remaining gap functions can be obtained via the three-fold rotations as follows. From equation (A32),

$$\begin{equation}\left(\begin{matrix}\hfill {{\Delta}}_{11}\hfill & \hfill {{\Delta}}_{12}\hfill & \hfill {{\Delta}}_{13}\hfill \\ \hfill {{\Delta}}_{21}\hfill & \hfill {{\Delta}}_{22}\hfill & \hfill {{\Delta}}_{23}\hfill \\ \hfill {{\Delta}}_{31}\hfill & \hfill {{\Delta}}_{32}\hfill & \hfill {{\Delta}}_{33}\hfill \end{matrix}\right)(\mathfrak{R}\mathbf{k})=\left(\begin{matrix}\hfill -\frac{1}{2}{{\Delta}}_{13}+\frac{\sqrt{3}}{2}{{\Delta}}_{23}\hfill & \hfill -\frac{1}{2}{{\Delta}}_{11}+\frac{\sqrt{3}}{2}{{\Delta}}_{21}\hfill & \hfill -\frac{1}{2}{{\Delta}}_{12}+\frac{\sqrt{3}}{2}{{\Delta}}_{22}\hfill \\ \hfill -\frac{1}{2}{{\Delta}}_{23}-\frac{\sqrt{3}}{2}{{\Delta}}_{13}\hfill & \hfill -\frac{1}{2}{{\Delta}}_{21}-\frac{\sqrt{3}}{2}{{\Delta}}_{11}\hfill & \hfill -\frac{1}{2}{{\Delta}}_{22}-\frac{\sqrt{3}}{2}{{\Delta}}_{12}\hfill \\ \hfill {{\Delta}}_{33}\hfill & \hfill {{\Delta}}_{31}\hfill & \hfill {{\Delta}}_{32}\hfill \end{matrix}\right)(\mathbf{k}).\end{equation} \tag{ A42 }$$

Therefore,

$$\begin{equation}\begin{aligned}\left(\begin{matrix}\hfill {{\Delta}}_{11}\hfill \\ \hfill {{\Delta}}_{21}\hfill \end{matrix}\right)(\mathbf{k})& =\left(\begin{matrix}\hfill -\frac{1}{2}\hfill & \hfill -\frac{\sqrt{3}}{2}\hfill \\ \hfill \frac{\sqrt{3}}{2}\hfill & \hfill -\frac{1}{2}\hfill \end{matrix}\right)\left(\begin{matrix}\hfill {{\Delta}}_{12}\hfill \\ \hfill {{\Delta}}_{22}\hfill \end{matrix}\right)(\mathfrak{R}\mathbf{k}),\\ \left(\begin{matrix}\hfill {{\Delta}}_{13}\hfill \\ \hfill {{\Delta}}_{23}\hfill \end{matrix}\right)(\mathbf{k})& =\left(\begin{matrix}\hfill -\frac{1}{2}\hfill & \hfill \frac{\sqrt{3}}{2}\hfill \\ \hfill -\frac{\sqrt{3}}{2}\hfill & \hfill -\frac{1}{2}\hfill \end{matrix}\right)\left(\begin{matrix}\hfill {{\Delta}}_{12}\hfill \\ \hfill {{\Delta}}_{22}\hfill \end{matrix}\right)({\mathfrak{R}}^{-1}\mathbf{k}),\\ {{\Delta}}_{31}(\mathbf{k})& ={{\Delta}}_{32}(\mathfrak{R}\mathbf{k}),\enspace {{\Delta}}_{33}(\mathbf{k})={{\Delta}}_{32}({\mathfrak{R}}^{-1}\mathbf{k}).\end{aligned}\end{equation} \tag{ A43 }$$

For the CDW gap functions among the E_u valence bands at A and the conduction bands at M's, an analysis along the preceding lines yields:

$$\begin{equation}{L}_{1}^{+}:\enspace {{\Delta}}_{42}(\mathbf{k})\sim \left\{{k}_{y},{k}_{z}\right\},\qquad {{\Delta}}_{52}(\mathbf{k})\sim {k}_{x},\enspace \end{equation} \tag{ A44 }$$

$$\begin{equation}{L}_{2}^{+}:\enspace {{\Delta}}_{42}(\mathbf{k})\sim {k}_{x},\qquad {{\Delta}}_{52}(\mathbf{k})\sim ,\left\{{k}_{y},{k}_{z}\right\}\end{equation} \tag{ A45 }$$

$$\begin{equation}{L}_{1}^{-}:\enspace {{\Delta}}_{42}(\mathbf{k})\sim \left\{{k}_{x}{k}_{y},{k}_{x}{k}_{z}\right\},\qquad {{\Delta}}_{52}(\mathbf{k})\sim \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.,\enspace \end{equation} \tag{ A46 }$$

$$\begin{equation}{L}_{2}^{-}:\enspace {{\Delta}}_{42}(\mathbf{k})\sim \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.,\qquad {{\Delta}}_{52}(\mathbf{k})\sim \left\{{k}_{x}{k}_{y},{k}_{x}{k}_{z}\right\}.\end{equation} \tag{ A47 }$$

where the momentum k is relative to the A point and the subscript 4, 5 are used to denote the two E_u bands.

We examine the Ginzburg–Landau theory for primary and secondary OPs. The primary OP $\overrightarrow {\phi }=\left({\phi }_{1},{\phi }_{2},{\phi }_{3}\right)$ is taken to belong to the ${L}_{1}^{-}$ IR (IR), which is delineated by

$$\begin{equation}{L}_{1}^{-}={A}_{u}\otimes \left({\text{e}}^{-\text{i}{\mathbf{Q}}_{1}\cdot \mathbf{T}},{\text{e}}^{-\text{i}{\mathbf{Q}}_{2}\cdot \mathbf{T}},{\text{e}}^{-\text{i}{\mathbf{Q}}_{3}\cdot \mathbf{T}}\right),\end{equation} \tag{ B1 }$$

where A_u is an IR in the point group C_2h (little co-group at L) and ${\text{e}}^{-\text{i}{\mathbf{Q}}_{i}\cdot \mathbf{T}}$ stand for translational representations. The 3D ordering vectors are defined by ${\mathbf{Q}}_{1}=\frac{1}{2}({\mathbf{a}}_{1}^{{\ast}}+{\mathbf{a}}_{3}^{{\ast}})$, ${\mathbf{Q}}_{2}=\frac{1}{2}({\mathbf{a}}_{2}^{{\ast}}+{\mathbf{a}}_{3}^{{\ast}})$, and ${\mathbf{Q}}_{3}=\frac{1}{2}({\mathbf{a}}_{1}^{{\ast}}+{\mathbf{a}}_{2}^{{\ast}}+{\mathbf{a}}_{3}^{{\ast}})$. The images of $\overrightarrow {\phi }$ under symmetry operations in the space group are

$$\begin{equation}\begin{aligned}{3}_{[001]}:& \left({\phi }_{1},{\phi }_{2},{\phi }_{3}\right)\to \left({\phi }_{2},{\phi }_{3},{\phi }_{1}\right),\\ {m}_{[100]}:& \left({\phi }_{1},{\phi }_{2},{\phi }_{3}\right)\to \left(-{\phi }_{3},-{\phi }_{2},-{\phi }_{1}\right),\\ \overline{I}:& \left({\phi }_{1},{\phi }_{2},{\phi }_{3}\right)\to \left(-{\phi }_{1},-{\phi }_{2},-{\phi }_{3}\right),\\ {T}_{{\mathbf{a}}_{1}}:& \left({\phi }_{1},{\phi }_{2},{\phi }_{3}\right)\to \left(-{\phi }_{1},{\phi }_{2},-{\phi }_{3}\right),\\ {T}_{{\mathbf{a}}_{2}}:& \left({\phi }_{1},{\phi }_{2},{\phi }_{3}\right)\to \left({\phi }_{1},-{\phi }_{2},-{\phi }_{3}\right),\\ {T}_{{\mathbf{a}}_{3}}:& \left({\phi }_{1},{\phi }_{2},{\phi }_{3}\right)\to \left(-{\phi }_{1},-{\phi }_{2},-{\phi }_{3}\right).\end{aligned}\end{equation} \tag{ B2 }$$

Translations by a_1,2,3 will produce minus signs due to the oscillating nature of the CDW. From the collection of these images, one realizes that by treating $\overrightarrow {\phi }$ as an ordinary coordinate vector $\overrightarrow {x}=\left(x,y,z\right)$, the transformations correspond to those in the point group T_h . In other words, this is a homomorphism: the space group generates in the vector space of $\overrightarrow {\phi }$ the point group T_h . For other three IRs (${L}_{1}^{+},{L}_{2}^{+},{L}_{2}^{-}$), it is straightforward to obtain the associated images, which are similar to those of ${L}_{1}^{-}$ with modifications at m_[100] and $\overline{I}$. Their image groups are also T_h .

The secondary OP $\overrightarrow {\zeta }$ also transforms as the ${M}_{1}^{+}$ IR:

$$\begin{equation}{M}_{1}^{+}={A}_{g}\otimes \left({\text{e}}^{-\text{i}{\mathbf{Q}}_{1}^{\prime }\cdot \mathbf{T}},{\text{e}}^{-\text{i}{\mathbf{Q}}_{2}^{\prime }\cdot \mathbf{T}},{\text{e}}^{-\text{i}{\mathbf{Q}}_{3}^{\prime }\cdot \mathbf{T}}\right).\end{equation} \tag{ B3 }$$

The secondary OP has the A_g symmetry and importantly it describes a 2D CDW with ordering vectors ${\mathbf{Q}}_{1}^{\prime }=\frac{1}{2}{\mathbf{a}}_{1}^{{\ast}}$, ${\mathbf{Q}}_{2}^{\prime }=\frac{1}{2}{\mathbf{a}}_{2}^{{\ast}}$, and ${\mathbf{Q}}_{3}^{\prime }=\frac{1}{2}({\mathbf{a}}_{1}^{{\ast}}+{\mathbf{a}}_{2}^{{\ast}})$. The images of $\overrightarrow {\zeta }$ under the symmetry operations are

$$\begin{equation}\begin{aligned}{3}_{[001]}:& \left({\zeta }_{1},{\zeta }_{2},{\zeta }_{3}\right)\to \left({\zeta }_{2},{\zeta }_{3},{\zeta }_{1}\right),\\ {m}_{[100]}:& \left({\zeta }_{1},{\zeta }_{2},{\zeta }_{3}\right)\to \left({\zeta }_{3},{\zeta }_{2},{\zeta }_{1}\right),\\ \overline{I}:& \left({\zeta }_{1},{\zeta }_{2},{\zeta }_{3}\right)\to \left({\zeta }_{1},{\zeta }_{2},{\zeta }_{3}\right),\\ {T}_{{\mathbf{a}}_{1}}:& \left({\zeta }_{1},{\zeta }_{2},{\zeta }_{3}\right)\to \left(-{\zeta }_{1},{\zeta }_{2},-{\zeta }_{3}\right),\\ {T}_{{\mathbf{a}}_{2}}:& \left({\zeta }_{1},{\zeta }_{2},{\zeta }_{3}\right)\to \left({\zeta }_{1},-{\zeta }_{2},-{\zeta }_{3}\right),\\ {T}_{{\mathbf{a}}_{3}}:& \left({\zeta }_{1},{\zeta }_{2},{\zeta }_{3}\right)\to \left({\zeta }_{1},{\zeta }_{2},{\zeta }_{3}\right).\end{aligned}\end{equation} \tag{ B4 }$$

Note that a translation in a₃ has no effect due to the 2D character. Now the space group generates the point group T for $\overrightarrow {\zeta }$. Incidentally, the image group for ${M}_{1}^{+}$ IR is group T and is T_h for both ${M}_{1}^{-}$ and ${M}_{2}^{-}$ IRs.

The Ginzburg–Landau free energy for the primary OP $\overrightarrow {\phi }$ is invariant for the group T_h and it can be written as:

$$\begin{equation}{F}_{\text{primary}}=\frac{1}{2}\alpha { \overrightarrow {\phi }}^{2}+\frac{1}{4}{\beta }_{1}{ \overrightarrow {\phi }}^{4}+\frac{1}{4}{\beta }_{2}\left({\phi }_{1}^{4}+{\phi }_{2}^{4}+{\phi }_{3}^{4}\right)+{\Delta}{F}_{\text{primary}},\end{equation} \tag{ B5 }$$

where ${ \overrightarrow {\phi }}^{2}={\phi }_{1}^{2}+{\phi }_{2}^{2}+{\phi }_{3}^{2}$ is defined. It is known that to have a conclusive phase diagram, sixth-degree terms,

$$\begin{equation}\begin{aligned}{\Delta}{F}_{\text{primary}}& =-\frac{1}{6}{\gamma }_{1}{ \overrightarrow {\phi }}^{6}+\frac{1}{2}{\gamma }_{2}{ \overrightarrow {\phi }}^{2}\left({\phi }_{1}^{4}+{\phi }_{2}^{4}+{\phi }_{3}^{4}\right)\\ & \quad +\frac{1}{2}{\gamma }_{3}{\phi }_{1}^{2}{\phi }_{2}^{2}{\phi }_{3}^{2}+\frac{1}{2}{\gamma }_{4}\left({\phi }_{1}^{2}-{\phi }_{2}^{2}\right)\left({\phi }_{1}^{2}-{\phi }_{3}^{2}\right)\left({\phi }_{3}^{2}-{\phi }_{1}^{2}\right),\end{aligned}\end{equation} \tag{ B6 }$$

are required [29, 30]. When T < T_c and α < 0, a CDW state for nonzero $\overrightarrow {\phi }$ becomes the ground state. Moreover, only the so-called first type of CDW phases are possible [29, 30]: either of ${ \overrightarrow {\phi }}_{\text{I}}\propto \left(1,1,1\right)$, ${ \overrightarrow {\phi }}_{\text{II}}\propto \left(1,1,0\right)$, or ${ \overrightarrow {\phi }}_{\text{III}}\propto \left(1,0,0\right)$. The free energies for these three phases are given by

$$\begin{equation}{F}_{\text{primary},M}=-\frac{1}{48{D}_{M}}{\left({s}_{M}-{t}_{M}\right)}^{2}\left({s}_{M}+2{t}_{M}\right),\end{equation} \tag{ B7 }$$

where M = I, II, III, and

$$\begin{equation}{D}_{\text{I}}={\gamma }_{3}+{\gamma }_{2}+\frac{1}{9}{\gamma }_{3},\qquad {t}_{\text{I}}={\beta }_{1}+\frac{1}{3}{\beta }_{2},\qquad {s}_{\text{I}}=\sqrt{{t}_{\text{I}}^{2}-4\alpha {D}_{\text{I}}},\end{equation} \tag{ B8 }$$

$$\begin{equation}{D}_{\text{II}}={\gamma }_{1}+3{\gamma }_{2},\qquad {t}_{\text{II}}={\beta }_{1}+{\beta }_{2},\qquad {s}_{\text{II}}=\sqrt{{t}_{\text{II}}^{2}-4\alpha {D}_{\text{II}}},\end{equation} \tag{ B9 }$$

$$\begin{equation}{D}_{\text{III}}={\gamma }_{1}+\frac{3}{2}{\gamma }_{2},\qquad {t}_{\text{III}}={\beta }_{1}+\frac{1}{2}{\beta }_{2},\qquad {s}_{\text{III}}=\sqrt{{t}_{\text{III}}^{2}-4\alpha {D}_{\text{III}}}.\end{equation} \tag{ B10 }$$

An analysis of the first-order transitions between these phases is outside of scope of this paper. We will assume that the M = I phase is the favorable one for given parameters.

The free energy for the secondary OP is

$$\begin{equation}{F}_{\text{secondary}}=\frac{1}{2}{\alpha }^{\prime }{ \overrightarrow {\zeta }}^{2}+\frac{1}{4}{\beta }_{1}^{\prime }{ \overrightarrow {\zeta }}^{4}+\frac{1}{4}{\beta }_{2}^{\prime }\left({\zeta }_{1}^{4}+{\zeta }_{2}^{4}+{\zeta }_{3}^{4}\right)+{\gamma }_{1}^{\prime }{\zeta }_{1}{\zeta }_{2}{\zeta }_{3}.\end{equation} \tag{ B11 }$$

The expansion of the free energy to the fourth degree is sufficient for our discussion. Because the vector space of $\overrightarrow {\zeta }$ has group T symmetry, a cubic term is observed, and the lowest-degree coupling form reads as

$$\begin{equation}{F}_{\text{coupling}}=\frac{1}{3}\lambda \left({\phi }_{1}{\phi }_{2}{\zeta }_{3}+{\phi }_{2}{\phi }_{3}{\zeta }_{1}+{\phi }_{3}{\phi }_{1}{\zeta }_{2}\right),\end{equation} \tag{ B12 }$$

in which the primary OP is quadratic and the secondary OP is linear.

The coupling with linear ζ indicates that the secondary OP will coincide with the primary OP. To see this, we reduce the problem by taking ϕ = ϕ₁ = ϕ₂ = ϕ₃ and ζ = ζ₁ = ζ₂ = ζ₃, omit the sixth degree terms and obtain the total free energy as

$$\begin{equation}F=\frac{1}{2}\alpha {\phi }^{2}+\frac{1}{4}\beta {\phi }^{4}+\frac{1}{2}{\alpha }^{\prime }{\zeta }^{2}+\frac{1}{3}{\gamma }^{\prime }{\zeta }^{3}+\frac{1}{4}{\beta }^{\prime }{\zeta }^{4}+\lambda \zeta {\phi }^{2}.\end{equation} \tag{ B13 }$$

The optimal values of ϕ and ζ ($\overline{\phi }$ and $\overline{\zeta }$) are obtained from $\frac{\partial F}{\partial \phi }{\vert }_{\overline{\phi },\overline{\zeta }}=\frac{\partial F}{\partial \zeta }{\vert }_{\overline{\phi },\overline{\zeta }}=0$ that give

$$\begin{equation}{\overline{\phi }}^{2}=\frac{1}{\beta }\left(-\alpha -2\lambda \overline{\zeta }\right),\end{equation} \tag{ B14 }$$

$$\begin{equation}{\beta }^{\prime }{\overline{\zeta }}^{3}+{\gamma }^{\prime }{\overline{\zeta }}^{2}+{\alpha }^{\prime }\overline{\zeta }+\lambda {\overline{\phi }}^{2}=0.\end{equation} \tag{ B15 }$$

Based on the fact that the secondary OP is induced by the primary OP, we demand that $\overline{\zeta }=0$ when $\overline{\phi }=0$, suggesting that α' > 0 as well as γ'² − 4α'β' < 0.

For nonzero $\overline{\phi }$, it happens when $\alpha +2\lambda \overline{\zeta }{< }0$, which indicates that T_c is modified by the nonzero of the secondary OP. Replacing $\overline{\phi }$ in equation (B15) by equation (B14) yields the cubic equation,

$$\begin{equation}{\beta }^{\prime }{\overline{\zeta }}^{3}+{\gamma }^{\prime }{\overline{\zeta }}^{2}+\left({\alpha }^{\prime }-\frac{2{\lambda }^{2}}{\beta }\right)\overline{\zeta }-\frac{\lambda \alpha }{\beta }=0.\end{equation} \tag{ B16 }$$

If the constant term λα/β is finite, the cubic equation always has a real root, explaining the coincidence of the primary and secondary OPs. Close to T_c when OPs are small, $\overline{\zeta }\propto {\overline{\phi }}^{2}$ and thus $\overline{\psi }\propto {\overline{\phi }}^{3}$.

The first-principles calculations were performed with the projector augmented wave method within the DFT [47] framework as implemented in the VASP code [48, 49]. The GGA with the Perdew–Burke–Ernzerhof parameterization and mBJ meta-GGA were used to include the exchange–correlation effects [50, 51]. We used a plane wave energy cut-off of 380 eV and a Γ-centered 12 × 12 × 8 k-mesh to sample the bulk BZ of 1T–TiSe₂. To access the nodal lines in the full BZ, we constructed a tight-binding model Hamiltonian by projecting first-principles results onto Wannier orbitals using the VASP2WANNIER90 interface [49, 52]. The Ti d and Se p states were included in the construction of Wannier functions.

We present the ab initio band structure of the CDW state on various k_z planes in figures 4(a)–(f). The band structure in figure 4(a) shows the states at the k_z = 0 plane, which is the same band structure that is presented in the main text. The bands are seen to cross at discrete points along the Γ–K and Γ–M directions at an energy ∼+0.1 eV above the Fermi level. With an increase in the k_z values, the band crossings start lowering their energies and finally merge into the Γ-point on the ${k}_{z}\sim 0.4\frac{2\pi }{c}$ plane at an energy ∼−0.05 eV, see figure 4(e). On further increasing the k_z value to $\frac{\pi }{c}$, the bands become gapped with a clear band gap in the band structure. A full BZ exploration of the band structure shows that the nodal points trace out the continuous lines which pass through the k_z = 0 plane as shown in figure 4(g). We find a total of 12 nodal lines which cross along the Γ–A direction in the BZ. These results are in accord with the theoretical analysis presented above and in the main text.

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Figure 5 shows the band structure of 1T–TiSe₂ 2 × 2 × 2 superlattice obtained with mBJ meta-GGA for various hydrostatic pressures. The band structure for c/a = c₀/a₀ [c and a (c₀ and a₀) are the lattice constants of pressurized (pristine) 2 × 2 × 2 1T–TiSe₂] is insulating with a well-defined gap between the valence and conduction bands. Due to this bandgap, the A_2u band at Γ does not show a band inversion. As we decrease c/a to apply a hydrostatic pressure, the bandgap decreases and finally vanishes for c/a = 0.97c₀/a₀, realizing a metallic state. There is a band inversion between valence and conduction bands, realizing multiple nodal crossings for c/a = 0.96c₀/a₀. With further lowering of the lattice parameters, the overlap between the valence and conduction bands increases without showing any band hybridizations. This implies that the system is metallic without any CDW order at high pressures.

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