CaTe: a new topological node-line and Dirac semimetal

Topological semimetals recently stimulate intense research activities. Combining first-principles calculations and effective model analysis, we predict that CaTe is topological node-line semimetal when spin-orbit coupling (SOC) is ignored. We also obtain the nearly flat surface state which has the drumhead characteristic. When SOC is included, three node lines evolve into a pair of Dirac points along the $M-R$ line. These Dirac points are robust and protected by $C_{4}$ rotation symmetry. Once this crystal symmetry is broken, the Dirac points will be eliminated, and the system becomes a strong topological insulator.

Unlike DSM and WSM whose band crossing points distribute at separate k points in the Brillouin zone (BZ), for the NLS, the crossing points around the Fermi level form a closed loop. Several compounds had been proposed as NLS included MTC [16], Bernal graphite [32], hyperhoneycomb lattices [33] and antiperovskite Cu 3 PdN [17,18] and Cu 3 NZn [18]. When SOC is neglected, for the system with band inversion, time reversal symmetry together with inversion symmetry or mirror symmetry will guarantee node line in 3D BZ [16-18, 26, 34, 35]. Same with TI and WSM, NLS also has an characteristic surface state, namely, drumhead like state [15][16][17][18]. Such 2D flat band surface state may become a route to achieve high temperature superconductivity [36,37].
In this article, based on first-principles calculations and effective model analysis, we propose that CaTe in CsCl-type structure is a NLS with drumhead like surface flat bands when SOC is ignored. As shown in Fig.  1(b), around the M point, there are three node-line rings, which is perpendicular to each other. When SOC is included, these three node-line rings evolve into two Dirac points along the M − R line. The Dirac points are robust and protected by the C 4 rotational symmetry. If the C 4 symmetry is broken, the system becomes a strong topological insulator with Z 2 indices (1;000). The 3D BZ and projected (001) two dimensional (2D) BZ of CaTe. Three dash circles are the scheme of the three line nodes around the M point. The blue circle is parallel to kz = 0 plane, red circle is parallel to kx = 0 plane and the green circle is parallel to ky = 0 plane.

CRYSTAL STRUCTURE AND METHOD
As one member of the alkaline-earth chalcogenides, CaTe have attracted tremendous interests because of its technological applications ranging from catalysis to luminescence [38][39][40][41][42]. CaTe undergoes a phase transition from NaCl-type structure at ambient conditions to CsCl-type structure at hydrostatic pressure about 33 GPa [38,39]. The structure of CaTe in CsCl-type is shown in Fig.1(a). The space group of this phase is P m3m (NO. 221). The electronic band structure calculations have been carried out using the full potential linearized augmented plane wave method as implemented in WIEN2K package [43]. To obtain accurate band inversion strength and band order, the modified Becke-Johnson exchange potential together with local-density approximation for the correlation potential (MBJLDA) has been applied [44]. The plane-wave cutoff parameter R MT K max is set to 7 and a 16 × 16 × 16 mesh was used for the BZ integral. The SOC interaction is included by using the second-order variational procedure.

ELECTRONIC STRUCTURE
Firstly, we calculate the band structure of CaTe and show the result without SOC in Fig. 2(a). By checking the wave functions, we find that the valence bands and conduction bands are mainly contributed by 5p z (blue) state of Te and 3d z 2 (red) state of Ca, respectively, as shown in Fig. 2(a). The band inversion happened at M point where the energy of Te-5p z state is higher than the energy of Ca-3d z 2 state by about 0.75 eV. Interestingly, this kind of band inversion is not caused by the SOC, which is different from most topological materials [1,2]. We calculate the electronic structure of CaTe by applying tensile strain to check the origin of band inversion at M point. The energy difference between Te-5p z state and Ca-3d z 2 state decreases as increasing the tensile strain. We find that when a≥1.05a 0 , the band inversion at M point disappear. With the time reversal symmetry and inversion symmetry, the band inversion results in node lines as proved in Ref. [16].
The effective Hamiltonian of node line around M point can be established by using the ⇀ k · ⇀ p method. Considering the crystal symmetry at M point and time reversal symmetry, the effective Hamiltonian can be written as following form: where the σ x and σ z are Pauli matrices, σ 0 is unit ma- system has both of time reversal symmetry and inversion symmetry, thus, the component of σ y is zero [16]. The eigenvalues of this 2 × 2 effective Hamiltonian are and g z ( On the other hand, g x ( ⇀ k ) = 0 confine the node lines in three mutually perpendicular planes (namely, k x =0 plane, k y =0 plane and k z =0 plane ) as illustrated in Fig.   1(b). Due to the fact that g 0 ( ⇀ k ) does not equal to zero which breaks the electron-hole symmetry, consequently, the node lines have finite energy dispersion.
When SOC is considered, three node lines evolve into two Dirac points at M − R line as shown in Fig. 2(b). At M point, the two states near Fermi level belong to irreducible representation Γ − 7 and Γ + 6 , respectively. While along the M − X line, two bands have the same irreducible representation Γ 5 as shown in Fig. 2 Along the M − R line, which reserve the C 4 rotation symmetry, two states with Γ − 7 and Γ + 6 at M point evolve into Γ 7 and Γ 6 , thus the hybridization between these two bands is forbidden, there is a Dirac point as shown in Fig.2(b). When the C 4 rotational symmetry is broken, like by strain effect, the band crossing point will disappear, and this 3D DSM will become a strong TI with topological indices Z 2 to be(1;0,0,0). To understand the band inversion at M point and the topological property of this system, we derive a low energy effective Hamiltonian at M point based on the projection-operator method [13]. M point has D 4h symmetry and also time reversal symmetry. As discussed above, at M point, Γ − 7 symmetry state has angular momentum j z = ±3/2 and Γ + 6 symmetry state has angular momentum j z = ±1/2. Therefore using the basis of (|j z = − 1 2 d , |j z = + 1 2 d , |j z = − 3 2 p , |j z = + 3 2 p , the effective Hamiltonian around M point can be written as (see APPENDIX for detail.): where δk x,y,z are small displacement from ⇀ k c . In the vicinity of ⇀ k c , the band dispersion is a linear, thus our effective Hamiltonian is nothing but 3D massless Dirac fermions.
The band inversion at M point and the Dirac nodes in CaTe suggest the existence of topological nontrivial surface state. To study the surface states in CaTe we use a 200-unit-cells-thick (001) slab with top (bottom) surface terminated by Ca (Te) atoms. The surface state is then calculated by using the tight-binding method. The hopping parameters are determined from a maximally localized Wannier functions (MLWFs) [45], which are projected from the Bloch state derived from first-principles calculations. Fig. 3(a)/(b) shows the surface state of CaTe (001) surface without/with SOC, respectively. When SOC is ignored, the system is a NLS, and possess nearly flat sur- face band around the Fermi energy. As shown in Fig.  3(a), our numerical results find that the nearly flat surface "drumhead" state appears in the interiors of the projected nodal line rings on the (001) surface around the M point. Since the slab we used has two surfaces, there are two surface states as shown in the red lines in the Fig.  3(a). The particle-hole symmetry is broken by nonzero term g 0 ( ⇀ k ), thus these two surface bands are not perfect flat with about 70 meV bandwidth. This type of 2D flat bands are proposed as a novel route to achieve high temperature superconductivity [36,37].
When the SOC is included, three node lines are gapped out and become a pair of Dirac points along the M − R line, thus the NLS become a 3D DSM. There is bulk Dirac node projected on M point (the red dot) as shown in Fig. 3(b). Along the M − X, there is also a projected bulk Dirac node, which locate near the X point denoted by red dots. Fig. 3(b) is clearly shows that the gapped bulk states along the Γ − X direction and the existence of surface Dirac cones (in the blue circle) due to the topologically nontrivial Z 2 indices, like the same case in Na 3 Bi [11] and Cd 3 As 2 [12].

CONCLUSION
In summary, based on first-principles calculations and effective model analysis, we suggest that CaTe in CsCltype structure is a NLS when SOC is ignored. There are three node-line rings which are perpendicular to each other around the M point. With band inversion at M point, this NLS is protected by the time reversal symmetry and inversion symmetry. When the SOC is included, three node-line rings become a pair of Dirac points. These Dirac nodes are robust and protected by the C 4 crystal symmetry and the system become a DSM.
Under the operation of crystal symmetry and time reversal symmetry, the Hamiltonian should be invariant.
This requires the function f i ( ⇀ k ) and the associated Γ i matrices belong to the same irreducible representation. Thus the key problem is to determine the irreducible representation for f i ( ⇀ k ) and Γ i matrices, which can be done by the projection-operator method.
Because the SOC is included, we use the double space group. Under the projection-operator method, we present the irreducible representation of Dirac Γ matrices and polynomials of ⇀ k , and their transformation under time reversal in Table I  Γ matrices representation T Γ1,Γ13 x − k 2 y }kz R11 -(kx, ky), (k 3 x , k 3 y ), (k 2 x ky, k 2 y kx), (kxk 2 z , kyk 2 z ) R12 -