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

The charge density wave (CDW) in 1T-TiSe2 accompanied by the periodic lattice distortion has a nontrivial symmetry configuration. The symmetry is important as an indication of the mechanism and a cue for experimental probes. We examine the symmetry of the system and clear up the connection of electronic structures between the normal and the CDW states. Especially, we unravel the consequential irreducible representations (IRs) of electronic states and, more ahead, those of gap functions among bands when the CDW occurs. Normally symmetry-related topology will be achieved directly, so we assert that the theory is valuable and practical for the search of topological CDW insulators.

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][25][26][27] . With this motivation, we will elucidate the symmetry properties of the CDW state in 1T -TiSe 2 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 Hamilto-nianĤ 0 with symmetry group G 0 . The symmetry elements R i of G 0 will of course commute with the Hamiltonian, i.e. R i ,Ĥ 0 = 0. When the CDW order develops at low temperatures, the CDW Hamiltonian becomeŝ H CDW =Ĥ 0 +∆, where∆ is to capture the CDW order. As a result, some symmetriesR i will be broken when R i ,Ĥ CDW = R i ,∆ = 0. Most obvious is the breaking of the translation symmetry T since T ,∆ = 0. [For a commensurate CDW the translation symmetry can be restored by enlarging the unit cell:T CDW =T M for some integer M such that T CDW ,∆ = 0.] So, the symmetry group will be reduced to a subgroup G of G 0 which will contain symmetry elements isomorphic to those that are left invariant under the CDW distortion in 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 onedimensional (1D) IR of the little group G q 0 , a subgroup of 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 symmetryR j ∈ G q 0 with eigenvalue either +1 or −1, i.e.R j∆R † j = ±∆. Note that the broken symmetries in the CDW state also include those associated with eigenvalues of −1, which are arXiv:1807.11107v2 [cond-mat.mtrl-sci] 3 Sep 2021 at the point L. The CDW OP will be one of the IRs L ± 1,2 . To avoid confusion, the conduction band at L is labeled with the IR given in parentheses in column 1.
related to the ordering vectors for the direction of the atomic displacements 28 .

II. SYMMETRY OF 1T -TISE2
Pristine 1T -TiSe 2 crystallizes in a hexagonal lattice structure with point group D 3d and space group G 0 = P3m1. Its low-temperature phase is the triple-q CDW state with ordering vectors , which connect Γ and the three inequivalent L points (L 1,2,3 ), see Fig.  1(a). The little group of L 2 contains point-group elements {E, 2 [100] , m [100] ,1} as the group C 2h , which has four 1D IRs A g , B g , A u , and B u (Table I). With 3 [001] we obtain conjugate representations for L 2 and L 3 . Including translation T of the Bravais lattice, IRs of the OP can be written as We describe the OP for a given IR by the vector φ = (φ 1 , φ 2 , φ 3 ) ∈ R 3 where the φ accompanying the IR determines the symmetry group (isotropy group). The energy is better for the maximal isotropy group 29,30 so we will take φ ∝ (1, 1, 1).
Referring to the symmetries of OPs in Table I, 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 a 3 translation produces a complementary −1. Similarly states with the parity eigenvalue −1 respect the combined symmetry T a3 i (= {1|001}). As , there is inversion symmetry with respect the center at 0, 0, 1 2 . We thus conclude that the space group for L + 1 and L − 2 is P3m1 (No. 164) and for L + 2 and L − 1 is P3c1 (No. 165).

III. HAMILTONIAN
The normal state band structure of TiSe 2 using the generalized gradient approximation (GGA) without spinorbit coupling (SOC) is shown in Fig. 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 aŝ where the basis The prime on the summation in Eq. (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 H ΓL : where H Γ and H L account for the normal-state band structure around Γ and L's, respectively. In ψ † Γk = (ψ † Γ1k , ψ † Γ2k , ψ † Γ3k ), the former two operators stand for the E g states and the last one is for A 2u . In ψ Lk = (ψ † L1k , ψ † L2k , ψ † L3k ), the three bands at the three L's are included with k being relative to the corresponding L's. The CDW gap ∆ ΓL is a 3 × 3 matrix, where the matrix element ∆ ij refers to the CDW potential between ψ † Γi and ψ Lj . We will mainly discuss H ΓL and H AM . The off-diagonal element V, which is induced by the second secondary OP, is expected to be weak.
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: 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 Eq. (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 ΓK line and the other along the ΓM 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 Fig. 2(c). Band structures for different k z values are given in Appendix D.

IV. GINZBURG-LANDAU THEORY
We turn now to discuss the Ginzburg-Landau theory for the L − 1 CDW state. Images of the primary OP (L − under various symmetry operations are: so that the free-energy density which satisfies symmetries of the pristine state to the fourth order becomes: Here α < 0 when T < T c for nonzero φ and β > 0 for stability. Therefore, we seek a secondary OP that is linearly coupled to the primary OP φ, i.e. we look for coincidence of the primary and secondary OPs (Appendix B). Symmetry requirements suggest that the secondary OP ζ will also have three components with the coupling energy where ζ follows the rules in Eq. (B2) except that m [100] , I, and T a3 are without the minus signs. The symmetry conditions dictate that the components of the secondary OP take ordering vectors Q 1 = 1 2 a * 1 , Q 2 = 1 2 a * 2 , and Q 3 = 1 2 (a * 1 + a * 2 ) and that they belong to IR M + 1 . In addition to the terms similar to Eq. (B5), the free energy for the secondary OP can contain the cubic term ζ 1 ζ 2 ζ 3 . Besides, coexistence of L − 1 and M + 1 OPs will induce a third (second secondary) OP ϕ in higher order as where ϕ, which we call IR A − 2 , is parity-odd, mirror-odd and belongs to the ordering vector 1 2 a * 3 . 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 ⊗ 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.

V. IRS OF BANDS
Although the gap functions in Eq. (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: 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 Fig. 2 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, Fig. 1(b).

VI. TOPOLOGY
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 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 Fig. 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 Eq. (19).

VII. CONCLUSION AND DISCUSSION
We have presented an in-depth analysis of the symmetries as well as the topology of the electronic structure of bulk 1T -TiSe 2 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 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 timereversal and inversion symmetries are preserved, inclusion of the SOC gaps band crossings for out-of-plane nfold (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 -TiSe 2 is a semimetal [40][41][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 -TiSe 2 to exchange-correlation functional, Hubbard U , and van der Waals interactions has been explored extensively in the literature [44][45][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 -TiSe 2 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 -TiSe 2 . 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. Here we discuss the construction of our mean-field 2 × 2 × 2 charge density wave (CDW) Hamiltonian without spin-orbit coupling for bulk 1T -TiSe 2 . It is written aŝ where the prime indicates that k runs over states near the Fermi surfaces and . 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 Fig. 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 II). The bands at A have the IR E u whereas at the L and M , they have A g IR (Table I). For ψ † Γk , which contains three creation operators for the hole pockets around Γ, we assign E g to the first two operators and A 2u to the last operator. For ψ † Ak , we denote ψ † Ak = (ψ † A1k , ψ † A2k ). As for ψ Lk , it contains three creation operators for the electron pockets around the three L points, ψ † Lk = (ψ † L1k , ψ † L2k , ψ † L3k ). Replacing L by M , we obtain ψ † M k . 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 where H Γ and H L account for the normal-state band structure around Γ and L's, respectively. In these expression, H Γ , H L and ∆ ΓL are all 3 × 3 matrices written as The change in lattice parameters and associated hydrostatic pressure are marked. Color regions identify the CDW gapped (white), CDW semimetal (green), and normal metal (gray) states. The system evolves from a large gap CDW insulator to a CDW semimetal for c/a = 0.97c0/a0) and a non-CDW metallic state for c/a = 0.96c0/a0. a0 and c0 (a and c) define the lattice constants at zero (finite) pressure.
∆ 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: We choose Note that here we adopt a representation in which I L , I L , M Γ and M A matrices are diagonal, where the diagonal elements are the eigenvalues of the corresponding symmetries at various symmetry points. In this basis, the C 3,Γ and 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 |ψ L1k → |ψ L2Rk under 3 [001] ; while under m [100] , which leaves L 2 invariant and interchanges L 1 and L 3 , |ψ L2k → |ψ L2Mk and |ψ L1k ↔ |ψ L3Mk .

a. Hamiltonian at Γ
We will write down the Hamiltonian at Γ H Γ . Here it is easier to use a basis that makes the three-fold rotation operator diagonal. The basis is unitarily transformed as ψ † Γk = ψ † Γk U and the symmetry operators transform, for instance, asC 3,Γ = U −1 C 3,Γ U, where the unitary matrix In this "tilde" basis, the symmetry operators arẽ where the last operator is the time-reversal operator with complex conjugation K. This tilde Hamiltonian follows the symmetry constraints    (k) and defining respectively. The Hamiltonian up to second order is: Here k || = k 2 x + k 2 y and all parameters a, a , b, b , c, c , d, ε 1 , and ε 2 are real. Finally, we transformH Γ back to H Γ by Hamiltonians at L and M The Hamiltonian at L's H L is identical to H M , and can be written as Here we assume that within the low-energy region, the three bands are independent. The symmetry operators are then given by Similar as before, we conclude that c. 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 H Γ in Eq. (A25). In addition, we show the coupling Hamiltonian between the E g and the E u bands in V. In the basis where the symmetry operators for the E u bands take the form the k · p model will read with real parameters t 1 and t 2 . The coupling Hamiltonian indicates anticrossing among the E g and E u bands.

CDW gap functions
The gap functions in Eqs. (5)(6)(7) follow In particular, the gap functions associated with L 2 , which is invariant under both inversion and the mirror symmetry, are: So the gap functions for the four CDW states to lowest order in k are: 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 Eq. (A32), Therefore, 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: where the momentum k is relative to the A point and the subscript 4, 5 are used to denote the two E u bands.
Appendix B: Ginzburg-Landau theory for the two order parameters We examine the Ginzburg-Landau theory for primary and secondary order parameters (OPs). The primary OP φ = (φ 1 , φ 2 , φ 3 ) is taken to belong to the L − 1 irreducible representation (IR), which is delineated by where A u is an IR in the point group C 2h (little co-group at L) and e −iQi·T stand for translational representations. The 3D ordering vectors are defined by Q 1 = 1 2 (a * 1 + a * 3 ), Q 2 = 1 2 (a * 2 + a * 3 ), and Q 3 = 1 2 (a * 1 + a * 2 + a * 3 ). The images of φ under symmetry operations in the space group are 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 φ as an ordinary coordinate vector x = (x, y, z), 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 φ 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] andĪ. Their image groups are also T h .
(B12) The expansion of the free energy to the fourth degree is sufficient for our discussion. Because the vector space of ζ has group T symmetry, a cubic term is observed, and the lowest-degree coupling form reads as 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 = φ 1 = φ 2 = φ 3 and ζ = ζ 1 = ζ 2 = ζ 3 , omit the sixth degree terms and obtain the total free energy as The optimal values of φ and ζ (φ andζ) are obtained from ∂F ∂φ |φ ,ζ = ∂F ∂ζ |φ ,ζ = 0 that givē β ζ 3 + γ ζ 2 + α ζ + λφ 2 = 0.
Based on the fact that the secondary OP is induced by the primary OP, we demand thatζ = 0 whenφ = 0, suggesting that α > 0 as well as γ 2 − 4α β < 0.
For nonzeroφ, it happens when α + 2λζ < 0, which indicates that T c is modified by the nonzero of the secondary OP. Replacingφ in Eq. (B16) by Eq. (B15) yields the cubic equation, 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,ζ ∝φ 2 and thusψ ∝φ 3 .