Symmetry-enforced band crossings in trigonal materials: Accordion states and Weyl nodal lines

TL;DR

This paper classifies nonsymmorphic symmetry-enforced band crossings in trigonal materials with strong spin-orbit coupling, identifying Weyl points and nodal lines near the Fermi energy with potential experimental implications.

Contribution

It provides a comprehensive classification of nonsymmorphic band degeneracies in trigonal materials, highlighting two types of Weyl crossings and identifying real materials exhibiting these features.

Findings

Identification of accordion Weyl points in Cu2SrSnS4 and Te

Discovery of Weyl nodal lines in Te16Si38

Detailed ab-initio analysis of band structures and surface states

Abstract

Nonsymmoprhic symmetries, such as screw rotations or glide reflections, can enforce band crossings within high-symmetry lines or planes of the Brillouin zone. When these band degeneracies are close to the Fermi energy, they can give rise to a number of unusual phenomena: e.g., anomalous magnetoelectric responses, transverse Hall currents, and exotic surface states. In this paper, we present a comprehensive classification of such nonsymmorphic band crossings in trigonal materials with strong spin-orbit coupling. We find that in trigonal systems there are two different types of nonsymmorphic band degeneracies: (i) Weyl points protected by screw rotations with an accordion-like dispersion, and (ii) Weyl nodal lines protected by glide reflections. We report a number of existing materials, where these band crossings are realized near the Fermi energy. This includes Cu2SrSnS4 and elemental tellurium (Te), which exhibit accordion Weyl points; and the tellurium-silicon clathrate Te16Si38, which shows Weyl nodal lines. The ab-initio band structures and surface states of these materials are studied in detail, and implications for experiments are briefly discussed.

Tables

Table 1. Table 1: Classification of nonsymmorphic band crossings in trigonal materials with time-reversal symmetry and strong spin-orbit coupling. The first column lists the trigonal space groups that exhibit topological band crossings. The second and third columns indicate the high-symmetry lines and planes in which Weyl nodal points and Weyl nodal lines appear, respectively. The definition of the coordinate system and high-symmetry labels in the Brillouin zones are given in Fig.  1 (a) and Fig.  3 . The electron fillings ν 𝜈 \nu for which a band insulator is possible, are given in the fourth column, where m ​ ℕ 𝑚 ℕ m\mathbb{N} denotes the set { m , 2 ​ m , 3 ​ m , ⋯ } 𝑚 2 𝑚 3 𝑚 ⋯ \{m,2m,3m,\cdots\} . Example materials which realize the predicted band crossings are listed in the last column.

Space group	Weyl nodal points	Weyl nodal lines	Filling constraint	Materials
144  $\left(P{3}_1\right)$	$\Gamma$$\Delta$A (6)	–	6$\mathbb{N}$	Cu2SrSnS4
145  $\left(P{3}_2\right)$	$\Gamma$$\Delta$A (6)	–	6$\mathbb{N}$	Cu2SrGeS4
151  $\left(P{3}_{1}12\right)$	$\Gamma$$\Delta$A (6)	–	6$\mathbb{N}$	Ag2HPO4
152  $\left(P{3}_{1}21\right)$	$\Gamma$$\Delta$A (6)	–	6$\mathbb{N}$	Te, Se
153  $\left(P{3}_{2}12\right)$	$\Gamma$$\Delta$A (6)	–	6$\mathbb{N}$	–
154  $\left(P{3}_{2}21\right)$	$\Gamma$$\Delta$A (6)	–	6$\mathbb{N}$	–
158  $\left(P3c1\right)$	–	$k_x{k}_z$-plane	4$\mathbb{N}$	–
159  $\left(P31c\right)$	–	$k_x=\pi$	4$\mathbb{N}$	–
161  $\left(R3c\right)$	–	$k_x=-k_y$	4$\mathbb{N}$	Te16Si38

Table 2. Table 2: This table lists how the Bloch bands | ψ m ​ ( k → ) ⟩ ket subscript 𝜓 𝑚 → 𝑘 |\psi_{m}(\vec{k})\rangle pair up into Kramers partners at the A point of the BZ in materials with a three-fold screw rotation C 3 , p subscript 𝐶 3 𝑝 C_{3,p} . The pairing depends on p 𝑝 p , i.e., the fractional lattice translation part of C 3 , p subscript 𝐶 3 𝑝 C_{3,p} , see Eq. ( 2 ). The right column indicates the eigenvalue labels m 𝑚 m of the Kramers pairs at the A point.

$p$	Pairings of $m$
1	(0,0), (1,2)
2	(0,1), (2,2)

Table 3. Table 3: Double-valued irreps for SG No. 144 (P3 1 ) without time-reversal symmetry at the points Γ Γ \Gamma and A 𝐴 A and at the line Δ Δ \Delta . For the labelling of the irrpes we use the same convention as in Ref.  Elcoro  et al. ( 2017 ) .

Irreps/Sym. Element	$C_3$
${\overline{\Gamma }}_4$	$-1$
${\overline{\Gamma }}_5$	$e^{-i\pi /3}$
${\overline{\Gamma }}_6$	$e^{i\pi /3}$
${\overline{\Delta }}_4$	$e^{i{k}_z/3}$
${\overline{\Delta }}_5$	$e^{i\left(k_z-\pi \right)/3}$
${\overline{\Delta }}_6$	$e^{i\left(k_z+\pi \right)/3}$
${\overline{A}}_4$	$e^{-i2\pi /3}$
${\overline{A}}_5$	$+1$
${\overline{A}}_6$	$e^{i2\pi /3}$

Table 4. Table 4: Double-valued irreps of SG No. 158 ( P ​ 3 ​ c ​ 1 𝑃 3 𝑐 1 P3c1 ) without time-reversal symmetry at the TRIMs Γ Γ \Gamma , A 𝐴 A , M 𝑀 M , and L 𝐿 L , and the plane D 𝐷 D (i.e., the k x ​ k z subscript 𝑘 𝑥 subscript 𝑘 𝑧 k_{x}k_{z} -plane), see Fig.  3 (b). We use the same convention as in Ref.  Elcoro  et al. ( 2017 ) for labelling the irreps.

Irreps/Sym. Element	$M_{010}$
${\overline{\Gamma }}_4$	$-i$
${\overline{\Gamma }}_5$	$+i$
${\overline{\Gamma }}_6$
${\overline{A}}_4$	$+1$
${\overline{A}}_5$	$-1$
${\overline{A}}_6$
${\overline{M}}_3$	$-i$
${\overline{M}}_4$	$+i$
${\overline{D}}_3$	$-i{e}^{i{k}_z/2}$
${\overline{D}}_4$	$+i{e}^{i{k}_z/2}$
${\overline{L}}_3$	$+1$
${\overline{L}}_4$	$-1$

Equations

$$C_{3,p}:(x,y,z)\to (-y,x-y,z+\frac{p}{3})\otimes (\frac{1}{2}{\sigma }_0-\frac{i\sqrt{3}}{2}{\sigma }_z),$$

$$C_{3,p}|{\psi }_m(\vec{k})⟩=e^{i\pi (2m+1)/3}{e}^{-ip{k}_z/3}|{\psi }_m(\vec{k})⟩,$$

$$\chi (A_{\sigma })=\sum _{i=1}^n\chi (B_{i,\sigma }).$$

$$M|{\psi }_{\pm }(\vec{k})⟩=\pm i{e}^{-i{k}_z/2}|{\psi }_{\pm }(\vec{k})⟩.$$

$$M_{010}:(x,y,z)\to (x,x-y,z+\frac{1}{2})\otimes (-\frac{i\sqrt{3}}{2}{\sigma }_x-\frac{i}{2}{\sigma }_y),$$

$$M_{210}:(x,y,z)\to (-x,-x+y,z+\frac{1}{2})\otimes (\frac{i}{2}{\sigma }_x+\frac{i\sqrt{3}}{2}{\sigma }_y),$$

$$M_{110}:(x,y,z)\to (-y,-x,z+\frac{1}{2})\otimes (-\frac{i\sqrt{3}}{2}{\sigma }_x-\frac{i}{2}{\sigma }_y),$$

Full text

I Introduction

The famous non-crossing theorem by Wigner and von Neumann von Neumann and Wigner (1929) states that Bloch states with the same symmetry cannot be degenerate at a generic point in the Brillouin zone (BZ), which prevents the formation of band crossings. Instead, when two Bloch bands of the same symmetry approach each other at a generic momentum, they start to hybridize and undergo an avoided level crossing. However, the non-crossing theorem does not apply to bands with non-trivial wavefunction topology, which can form topologically protected band degeneracies Chiu et al. (2016); Volovik (2013); Armitage et al. (2017); Burkov (2018); Yang et al. (2017). When these band crossings are in the vicinity of the Fermi energy, they lead to a range of interesting phenomena, such as arc and drumhead surface states Wan et al. (2011); Burkov et al. (2011); Xu et al. (2015); Huang et al. (2015); Chan et al. (2016); Bian et al. (2016a, b), transverse topological currents Rui et al. (2018); Armitage et al. (2017); Burkov (2018), and anomalous magnetoelectric responses Bzdusek et al. (2016), which could potentially be utilized for device applications Shi et al. (2015); Schaibley et al. (2016).

There are two different types of topological band crossings, namely, accidental band crossings and symmetry-enforced band crossings. The former are protected by symmorphic crystal symmetries and are only perturbatively stable Chiu and Schnyder (2014); Shiozaki and Sato (2014); Zhao et al. (2016). That is, they can be adiabatically removed by large symmetry-preserving deformations of the Hamiltonian, for example, through pair annihilation. Examples of accidental band crossings include Dirac points and Dirac lines that are protected by space-time inversion, reflection, or rotation symmetry Castro Neto et al. (2009); Wang et al. (2012); Chiu and Schnyder (2015); Wang et al. (2013); Xie et al. (2015); Yamakage et al. (2016). Symmetry-enforced band crossings Young et al. (2012); Schoop et al. (2016); Bzdusek et al. (2016); Zhao and Schnyder (2016); Michel and Zak (1999); Young and Kane (2015); Alexandradinata et al. (2016); Furusaki (2017); Takahashi et al. (2017); Yang et al. (2017); Fang et al. (2015); Malard et al. (2018); Zhang et al. (2018), on the other hand, arise in the presence of nonsymmorphic symmetries and are globally stable, i.e., they cannot be removed even by large deformations of the Hamiltonian. In other words, these band crossings are required to exist due to nonsymmorphic symmetries alone, independent of the chemical composition and other details of the material. This fact allows us to construct the following strategy to discover new topological semimetals, which consists of three steps: (i) identify the space groups (SGs) whose nonsymmorphic symmetries enforce the desired band crossings, (ii) perform a database search for materials in these SGs, and (iii) compute the electronic band structure of these materials to check whether the band crossings are near the Fermi energy. Previously, we applied this strategy to discover new topological semimetals with hexagonal symmetries Zhang et al. (2018). Here, we extend this analysis to trigonal systems.

There has been a number of previous works, which considered specific types of nonsymmorphic band crossings Bzdusek et al. (2016); Furusaki (2017); Takahashi et al. (2017); Yang et al. (2017) and proposed candidate materials, where these are realized Chen et al. (2017); Bradlyn et al. (2017); Schoop et al. (2018); Topp et al. (2016). However, a comprehensive classification of all possible nonsymmorphic band-crossings in trigonal materials has not been performed before. We find that among the 25 trigonal SGs, six support Weyl points with accordion-like dispersions and three support Weyl nodal lines (see Table 1). We identify several materials, where these topological band crossings are realized. This includes elemental tellurium (Te) with accordion Weyl points, and the trigonal high-temperature form of Te16Si38 with Weyl nodal lines. As a byproduct of our analysis, we also obtain the tight filling constraints for the existence of band insulators, which are in agreement with the recent literature Parameswaran et al. (2013); Watanabe et al. (2016). Furthermore, we find that in trigonal materials, as opposed to hexagonal materials Zhang et al. (2018), there exist no band crossings protected by off-centered symmetries Yang et al. (2017); Fang et al. (2015); Malard et al. (2018). We note that our analysis takes into account the full connectivity of the bands in the entire BZ, which goes beyond the results obtained through symmetry indicators Zhang et al. (2019); Vergniory et al. (2019); Tang et al. (2019).

The remainder of this paper is organized as follows. In Sec. II we classify Weyl points protected by screw rotation symmetries and discuss how symmetries constrain the multiplicities of the Weyl points. Section III is devoted to the study of Weyl nodal lines protected by glide reflection. In Sec. IV we discuss the filling constraints imposed by the nonsymmorphic symmetries. In Sec. V we present the example materials that realize the predicted band crossings. Our summary and conclusions are given in Sec. VI. In Appendix A we present additional figures of the band structure calculations. The calculated surface states of tellurium are shown in Appendix B.

II Non-symmorphic nodal points

In this section we show how three-fold screw-rotation symmetries in trigonal space groups lead to protected Weyl points. To derive this we employ two different approaches: (i) Using symmetry analysis we show how the evolution of eigenvalues of the symmetry operator along a line in the BZ enforces band crossings (Sec. II.1). (ii) Using crystallographic group theory we show how the compatibility relations between irreducible representations (irreps) at different high-symmetry points of the BZ Michel and Zak (2001); Kruthoff et al. (2017); Elcoro et al. (2017) lead to enforced band crossings (Sec. II.2). We also give a brief discussion on how time-reversal and SG symmetries lead to higher multiplicities of the Weyl points (Sec. II.3).

II.1 Symmetry eigenvalues

In general, a symmetry operator $G=\{g|\vec{t}\}$ of a given space group is a combination of a point group symmetry $g$ and a lattice translation $\vec{t}$. In case of a nonsymmorphic symmetry the lattice translation $\vec{t}$ is some fraction of the Bravais-lattice vector. Here, we are interested in the three-fold screw rotation, which has the following form:

[TABLE]

such that $p\in\{1,2\}$. Within the $g$-invariant space of the BZ, i.e. where $g\vec{k}=\vec{k}$, the Bloch states $|\psi_{m}(\vec{k})\rangle$ can be constructed to be simultaneous eigenfunctions of G and the Hamiltonian, satisfying the following eigenvalue equation:

[TABLE]

with $m\in\{0,1,2\}$. The above equation follows from the fact that $C_{3,p}^{3}=-p\vec{R}$, where $\vec{R}$ is a Bravais lattice vector (here $\vec{R}=\hat{z}$) and the minus sign arises due to $2\pi$ rotation of the electron spin, relevant for spin-orbit coupling. From Eq. (2), we observe that the eigenvalues of $C_{3,p}$ evolve along the line segment $\vec{k}=(0,0,k_{z})$ with $k_{z}\in[0,\pi]$, i.e., along the path $\Gamma-\Delta-A$ in Fig. 1, which is invariant under $C_{3}$ rotation.

Note that the points $\Gamma$ and $A$ are time-reversal invariant momenta, where in the presence of time-reversal symmetry bands pair-up with their respective Kramers partner. In terms of the eigenvalues in Eq. (2) this means that a band with real eigenvalue pairs with itself, while a band with complex eigenvalue pairs with the band whose eigenvalue is the complex conjugate. Using the label $m$ to identify the bands and corresponding eigenvalues, we find the following pairing scenario: At the $\Gamma$ point ($k_{z}=0$) there are two Kramers pairs of bands, namely, $(1,1)$ and $(0,2)$, irrespective of the value of $p$. At the $A$ point ($k_{z}=\pi$), on the other hand, Kramers pairs depend on the value of $p$ and pair up as indicated in Table 2. Since the bands pair up differently at $\Gamma$ and A, they show, in the absence of additional symmetries, a nontrivial connectivity along the path $\Gamma-\Delta-A$, as displayed in Fig. 1(b). We find that for both $p=1$ and $p=2$ there are six bands forming a connected group, which must cross at least two times, leading to two Weyl point degeneracies with an accordion-like dispersion.

We note that the presence of additional symmetries such as inversion or mirror, which either commute or anti-commute with $C_{3,p}$, may lead to additional degeneracies at the high-symmetry points in the BZ. This can remove the above discussed Weyl points. In particular, out of the eighteen trigonal space groups with hexagonal lattice system (P-trigonal) there are only six space groups without inversion symmetry (SGs with Nos. 144, 145, 151, 152, 153, and 154). These are the only SGs which host symmetry-enforced Weyl points with accordion-like dispersion, see Table 1.

In passing we note that the seven trigonal space groups with rhombohedral lattice system (R-trigonal) do not have a nonsymmorphic three-fold screw-rotation symmetry, but just three-fold symmorphic rotation symmetries (i.e. $p=0$ in $C_{3,p}$). Consequently, the eigenvalues of these symmorphic rotation symmetries are momentum independent and hence there are no symmetry-enforced Weyl points for R-trigonal space groups.

II.2 Compatibility relations between irreps

Next, we explain how the existence of symmetry-enforced Weyl points can be derived form the compatibility relations of irreducible representations (irreps) at different high-symmetry points/lines of the BZ Michel and Zak (2001); Kruthoff et al. (2017); Elcoro et al. (2017). Here, we give a brief outline of this derivation for SG No. 144 (P31). Additional details can be found in Ref. Zhang et al. (2018).

Bloch bands $|\psi_{m}(\vec{k})\rangle$ restricted to a high-symmetry line (or point) can be labeled by the irreducible representations (irreps) of the symmetry operators that leave this line (or point) invariant. An irrep of a given symmetry group can be viewed as a set of matrices which form a group that is homomorphic to the symmetry group, with the property that there does not exist a unitary transformation that brings all these matrices into block diagonal form. While these sets of matrices are not unique, their trace (i.e., their character) is unchanged under any unitary transformation. Therefore, for a given high-symmetry line (or point) in the BZ one can define a character table, which lists the irrpes and their characters for all the symmetry operations that leave this line (or point) invariant.

Now, as we move from a high-symmetry point to a high-symmetry line in the BZ, the symmetry is lowered, and hence, an irrep $A_{\sigma}$ of the high-symmetry point is, in general, no longer irreducible for the high-symmetry line. That is the irrep $A_{\sigma}$ of the high-symmetry point decomposes, in general, into smaller irreps $B_{i,\sigma}$ of the high-symmetry line. However, under such a decomposition the character $\chi$ of each relevant symmetry must be preserved, i.e.,

[TABLE]

The above equation defines the compatibility relations between the irreps $A_{\sigma}$ at a high-symmetry point and the irreps $B_{i,\sigma}$ at a high-symmetry line Bradley and Cracknell (1972); Miller and Love (1967).

We shall now use these compatibility relations to study how the irreps decompose in SG No. 144 (P31) for the path $\Gamma-\Delta-A$, which is left invariant under the screw rotation ${C}_{3,1}$. Since we are interested in electronic band structures with spin-orbit coupling, we need to consider the double-valued irreps of SG No. 144 (P31) (see Table 3), as the identity operator is not equivalent to a rotation by $2\pi$ but rather by $4\pi$ on the Bloch sphere. At the time-reversal invariant momenta (TRIMs), Bloch bands form Kramers pairs due to time-reversal symmetry. This means that at the TRIMs pairs of complex conjugate irreps combine together to form a time-reversal symmetric irrep Bradley and Cracknell (1972); Miller and Love (1967). For the $1d$ irreps of Table 3 we can construct these time-reversal symmetric irreps by inspection. At the $\Gamma$ point we find that the time-reversal symmetric irreps are given by the direct product of $\bar{\Gamma}_{5}$ with $\bar{\Gamma}_{6}$ and $\bar{\Gamma}_{4}$ with itself. Similarly, at the $A$ point the time-reversal symmetric irreps are given by the direct product of $\bar{A}_{4}$ with $\bar{A}_{6}$ and $\bar{A}_{5}$ with itself, see Fig. 1(c). As we move from the TRIMs $\Gamma$ and $A$ to a point on the high-symmetry line $\Delta$, the two-dimensional time-reversal symmetric irreps decompose into one-dimensional irreps, according to Eq. (3), see left part of Fig. 1(c). Connecting the respective irreps that split away from the points $\Gamma$ and $A$ we find that the bands connect in a non-trivial way, with a minimum number of two Weyl point crossings, as shown in the right part of Fig. 1(c). This is in exact agreement with the symmetry analysis presented above. Similar analysis can be done for the remaining five space groups of Table 1 and we find perfect agreement therein.

II.3 Weyl point multiplicities

So far we have only discussed the appearance of Weyl points on the $\Gamma-\Delta-A$ line. However, it follows from symmetry arguments together with the fermion doubling theorem Nielsen and Ninomiya (1981) that these cannot be the only Weyl points in the BZ. This is because (i) there has to be another Weyl point away from the $\Gamma-\Delta-A$ line with opposite chirality, since the chiralities of all Weyl points formed by one pair of bands must cancel out to zero Nielsen and Ninomiya (1981); and (ii) time-reversal and SG symmetries lead to additional copies of the Weyl points that are located away from the $\Gamma-\Delta-A$ line. Using these arguments, we now determine for each SG the minimum number of Weyl points that is compatible with the symmetries and the fermion doubling theorem. To do so, we first note that we can divide the BZ into two parts, an upper part with $k_{z}\in[0,\pi]$ and a lower part with $k_{z}\in[-\pi,0]$, which are related by time-reversal symmetry. Hence, it is sufficient to analyze the Weyl point configuration in the upper part; the configuration in the lower part can be obtained by applying time-reversal symmetry (see Fig. 2). For concreteness, we assume in the following that the Weyl points at the $\Gamma-\Delta-A$ line have positive chirality $+1$.

SG Nos. 144, 145, 151, and 153.—

Band structures in SG Nos. 144 and 145 possess only two symmetries: the three-fold screw rotation along the $k_{z}$ axis $C_{3,p}$ and time-reversal symmetry. In order to cancel the $+1$ chirality of the Weyl point at the $\Gamma-\Delta-A$ line, we need to place a negative chirality Weyl point somewhere else in the BZ. If we put it at an arbitrary position, the screw rotation $C_{3,p}$ would generate two more copies with the same chirality, leading to an overcompensation. Instead, in order to construct a Weyl point configuration with minimal multiplicity, we need to put the negative chirality Weyl point at one of the other two $C_{3,p}$-invariant lines, i.e., either on $K-P-H$ or on $K^{\prime}-P^{\prime}-H^{\prime}$. With this, the chiralities cancel out and we obtain a minimal Weyl point multiplicity of four, see Fig. 2(a).

SG Nos. 151 and 153 possess an additional two-fold screw rotation along the $k_{y}$ axis, which however does not affect the above argument. Hence, also in these SGs the minimal Weyl point multiplicity is four.

SG Nos. 152 and 154.—

SG Nos. 152 and 154 contain, besides the three-fold screw rotation $C_{3,p}$, a two-fold rotation $C_{2}$ along the (110) axis, which maps $(k_{x},k_{y},k_{z})\to(k_{x},-k_{y},-k_{z})$. This $C_{2}$ rotation puts extra constraints on the multiplicity of the Weyl points. To find these constraints we combine the $C_{2}$ rotation with time-reversal symmetry, yielding the mirror symmetry $(k_{x},k_{y},k_{z})\to(-k_{x},k_{y},k_{z})$ [dashed lines in Fig. 2(b)]. If we now put a negative chirality Weyl point on the $K-P-H$ line, the mirror symmetry would generate another negative chirality Weyl point at the $K^{\prime}-P^{\prime}-H^{\prime}$ line, leading to an overcancellation. Instead we place positive chirality Weyl points both at $K-P-H$ and $K^{\prime}-P^{\prime}-H^{\prime}$, which we then compensate by negative chirality Weyl points on the three mirror planes, as shown in Fig. 2(b). This configuration of twelve Weyl points has minimal multiplicity, satisfies all the symmetries, and has zero total chirality.

III Non-symmorphic nodal lines

Using the techniques discussed in the previous section we will now prove that glide-reflection symmetries lead to symmetry-enforced nodal lines in materials with trigonal space groups. As before, this derivation is base on (i) the momentum-dependence of the symmetry eigenvalues and (ii) the compatibility relations between irreps.

III.1 Symmetry eigenvalues

We consider glide mirror symmetries $M$ with a fractional shift of one half of the lattice constant along the $z$ axis. Applying these symmetries twice we obtain $M^{2}=-\hat{z}$, where the minus sign is due to the $2\pi$ rotation of the electron spin. From this it follows that the eigenvalues of the Bloch wavefunctions in the mirror planes are given by

[TABLE]

The mirror planes contain either two or four TRIMs at which the Bloch bands are Kramers degenerate (blue and red dots in Fig. 3). It follows from Eq. (4) that the $M$ eigenvalues of the Bloch bands at the TRIMs $A$, $L$, and $Z$ are $\pm 1$, while at the TRIMs $\Gamma$, $M$, and $F$ they are $\pm i$. Due to the anti-unitary time-reversal symmetry, at $\Gamma$, $M$, and $F$ the Kramers pairs are formed between bands with opposite $M$ eigenvalues (blue dots), while at $A$, $L$, and $Z$ they are formed between bands with the same eigenvalues (red dots). Hence, as we move from a blue TRIM to a red TRIM the Kramers pairs must switch partners. In the absence of additional symmetries this leads to a non-trivial band connectivity with a symmetry-enforced band crossing [see Fig. 3(a)]. Since this is true for any path connecting a blue TRIM to a red TRIM, the Weyl band crossings occur along a one-dimensional line separating the blue TRIMs from the red TRIMs [see Figs. 3(b)-(d)]. We note that the presence of additional symmetries, in particular inversion, leads to additional degeneracies, thereby invalidating the above argument. Thus, symmetry-enforced nodal lines only occur in SGs with a mirror-glide symmetry but no inversion symmetry. Among the trigonal SGs there are only three SGs satisfying these criteria, namely Nos. 158, 159, and 161.

SG No. 158 ($P3c1$) has hexagonal lattice system (P-trigonal) and contains the glide-mirror symmetry

[TABLE]

which leaves the $k_{x}k_{z}$-plane invariant. Following the above arguments we find that this glide-mirror symmetry leads to two nodal loops in the $k_{x}k_{z}$-plane, which separate the red TRIMs from the blue TRIMs, as shown in Fig. 3(b). If the irrep at the A point has dimension four (instead of two), these nodal lines touch the A point [dashed curve in Fig. 3(b)]. We note that these nodal loops must be symmetric under time-reversal symmetry and the point-group operations of SG No. 158, but are otherwise free to move within the mirror plane.

SG No. 159 ($P31c$) has also hexagonal lattice system (P-trigonal) and contains the glide-mirror symmetry

[TABLE]

which leaves both the $k_{x}=0$ and the $k_{x}=\pi$ planes invariant, see Fig. 3(c). Both mirror planes have two TRIMs, between which the Kramers pairs change partners. Hence, according to the above arguments, there should be nodal lines in both mirror planes, which encircle either the $\Gamma$/$M$ or $A$/$L$ points. However, at the $A$ point there exist four-dimensional irreps, leading to a four-fold band degeneracy. For this reason, symmetry-enforced nodal lines generically occur only in the $k_{x}=\pi$ plane, but not in the $k_{x}=0$ plane 111We note that at the A point there exist both two-dimensional and four-dimensional irreps. Hence, nodal lines are also possible in the $k_{x}=0$ plane, but they are not guaranteed to exist by symmetry alone..

SG No. 161 ($R3c$) has rhombohedral lattice system (R-trigonal) and contains the glide mirror symmetry

[TABLE]

which leaves the plane $k_{x}=-k_{y}$ invariant, see Fig. 3(d). As in SG No. 158, this glide-mirror symmetry gives rise to two nodal lines separating the two red TRIMs from the two blue TRIMs. If the bands transform at the $Z$ point under an irrep with dimension four (instead of two), then the nodal lines touch the $Z$ point [dashed curve in Fig. 3(d)].

III.2 Compatibility relations between irreps

The existence of symmetry-enforced nodal lines can also be derived from the compatibility relations between irreps. Here, we present this derivation for SG No. 158 ($P3c1$). The derivation for SG Nos. 159 and 161 proceeds in a very similar way.

For SG No. 158 the relevant high-symmetry plane is the plane $D$ (i.e., the $k_{x}k_{z}$-plane), which is left invariant under the glide-mirror symmetry $M_{010}$, Eq. (5). This mirror plane contains the four TRIMs $\Gamma$, $A$, $L$, and $M$. To determine the connectivity of the bands we first need to derive the irreps at these four TRIMs and at the $D$ plane. The double-valued irreps without time-reversal symmetry are listed in Table 4. We find that these irreps are all complex with the exception of $\bar{\Gamma}_{6}$, which is pseudoreal. Pseudoreal irreps are time-reversal invariant by themselves, while complex irreps need to be paired up to construct time-reversal invariant irreps Bradley and Cracknell (1972); Miller and Love (1967) (cf. Fig. 4). Note that at the $A$ point there are both two-dimensional and four-dimensional time-reversal invariant irreps. The decomposition of these time-reversal invariant irreps, as we move from a TRIM to a nearby point in the mirror plane $D$, can be inferred from Eq. (3), or alternatively from the program DCOMPREL on the Bilbao Crystallographic Server Elcoro et al. (2017). The resulting compatibility relations are depicted in Figs. 4(a), 4(c), and 4(e). Using these compatibility relations, we can now construct the band connectivity diagrams for a path within the $D$ plane that connects two TRIMs, see Figs. 4(b), 4(d), and 4(f). We observe that for any path connecting $\Gamma$/$M$ to $L$, there must be at least one band crossing. The same is true for paths connecting $\Gamma$ to $A$ or $M$ to $A$, albeit here the band crossings can be pinned at the $A$ point, if the corresponding irrep is four dimensional. Hence, the $D$ plane must contain at least two Weyl nodal lines enclosing the $\Gamma$ and $M$ points, as shown in Fig. 3(b).

IV Filling constraints

According to band theory, a noninteracting band insulator can only exist if the electron filling $\nu$ is an even integer, i.e., $\nu\in 2\mathbbm{N}$. Vice versa, materials with $\nu\notin 2\mathbbm{N}$ must necessarily by (semi-)metals. In materials with nonsymmorphic symmetries, however, these filling constraints are more rigid. This is because nonsymmorphic symmetries enforce band crossings, that lead to groups of more than two connected bands. Thus, using the analysis form Secs. II and III we can derive the more rigid filling constraints. For example, for SG No. 144 (P31) we observe that along the $\Gamma-\Delta-A$ line six bands form a connected group. Thus, in an insulator with SG No. 144 this group of six bands must be fully filled, i.e., the electron filling $\nu$ must be an element of $6\mathbbm{N}$. Using these arguments we have derived the filling constraints for all the nonsymmorphic trigonal space groups, see fourth column of Table 1. These results are in full agreement with the analysis of Ref. Watanabe et al. (2016).

V Example Materials

Having identified the SGs whose nonsymmorphic symmetries enforce Weyl points and Weyl nodal lines, we can now perform a database search for materials in these SGs. We look for suitable compounds in the ICSD database ICS , the AFLOW database Curtarolo et al. (2012), and the Materials Project database Jain et al. (2013); mat , which yields four materials with accordion Weyl points and one material with Weyl nodal lines.

For each of these example materials we perform electronic band structure calculations with the Vienna ab initio simulation package (VASP) Kresse and Furthmüller (1996a, b), using the projector augmented wave (PAW) method Blöchl (1994); Kresse and Joubert (1999) and the PBE Perdew et al. (1996) and mBJ Becke and Johnson (2006); Tran and Blaha (2009) exchange-correlation functionals. As input for the ab initio calculations we use the experimental crystal structures of Refs. Teske (1976, 1979); Tordjman et al. (1978); Adenis et al. (1989); Jaussaud et al. (2004). In the following we discuss the band structures along those high-symmetry lines/planes where the predicted band crossings occur. The full band structures along all high-symmetry directions of the BZ are presented in Appendix A. For two examples we also compute surface states using tight-binding models constructed from Wannier functions Mostofi et al. (2008).

V.1 Materials with Weyl nodal points

Here, we present four materials which exhibit Weyl nodal points with accordion-like dispersions.

Cu2SrSnS4 and Cu2SrGeS4—

The chalcogenide Cu2SrSnS4, which crystallizes in SG No. 144 ($P3_{1}$) Teske (1976); Tong et al. (2019), is an example of a material with Weyl points along the $\Gamma$–$\Delta$–$A$ line, as discussed in Sec. II. The electronic band structure computed with the mBJ functional, see Figs. 5 and 10(a), shows a band gap of about 1.1 eV, which is somewhat smaller than the experimentally reported value of 1.78 eV Tong et al. (2019). We observe in Fig. 5 that there are groups of $6n$ connected bands between $\Gamma$ and $A$, which form a large number of Weyl points. This is in full agreement with the band connectivity diagrams of Figs. 1(b) and 1(c). The Weyl points of Cu2SrSnS4 are symmetry protected by the screw rotation $C_{3,1}$, Eq. (1). Their topological stability is ensured by quantized Chern numbers. Due to the bulk boundary correspondence, these Chern numbers lead to arc surface states. Unfortunately, due to the large band gap of $\sim 1.8$ eV, it will be very challenging to probe the bulk Weyl points or arc surface states of Cu2SrSnS4 using photoemission, tunneling spectroscopy, or transport experiments. Perhaps, one could try to grow Cu2SrSnS4 as a thin film on a conducting substrate, which might allow to probe the Weyl points using photoemission.

Cu2SrGeS4 is very similar to Cu2SrSnS4. It crystallizes in SG No. 145 ($P3_{2}$) Teske (1979) and its band structure shows a large number of Weyl points along the $\Gamma$–$\Delta$–$A$ line, see Fig. 10(b) in Appendix A.

Ag2HPO4—

Silver hydrogen phosphate with the chemical formula Ag2HPO4 is a salt, that crystallizes in SG No. 151 Tordjman et al. (1978). According to our analysis of Sec. II, band structures with this SG symmetry exhibit Weyl points along the $\Gamma$–$\Delta$–$A$ line. Figures 6 and 10(c) display the first-principles band structure of Ag2HPO4, computed with the mBJ functional. The band structure shows a band gap of about $2.8$ eV. Along the $\Gamma$–$\Delta$–$A$ direction we observe a group of 24 ($=6\times 4$) connected bands, with many crossings, similar to the band connectivity diagrams of Fig. 1. These band crossings are Weyl points, whose topology is characterized by a nonzero Chern number. The nontrivial topology of these Weyl points gives rise to arc states at the surface, which connect Weyl points with opposite chirality. In Fig. 7 we demonstrate this for the Weyl point marked by the gray circle in Fig. 6. At the (-110) surface we observe an arc state that emanates out from the projected Weyl point.

While the ab-initio band structure of Ag2HPO4 nicely illustrates the predicted accordion states, this material is rather unsuitable for experimental investigations. Its large band gap and the fact that it cannot be doped easily makes it experimentally impossible to probe the bulk Weyl points and arc surface states.

Te—

At ambient conditions, elemental tellurium (Te) has a trigonal crystal structure with SG No. 152 ($P3_{1}21$)  Cherin and Unger (1967a); Adenis et al. (1989). In this structure the Te atoms form helical chains along the $z$ direction, which are arranged in a hexagonal array 222Note that depending on the chirality of these chains Te can also be in SG No. 153 ($P3_{2}12$).. Recent first principles calculation have shown that the valence and conduction bands of Te at the $H$ point exhibit several Weyl points Hirayama et al. (2015); Nakayama et al. (2017). Moreover, it was found that hydrostatic or uniaxial strain leads to a band inversion at the $H$ point, thereby transforming Te into a strong topological insulator Agapito et al. (2013).

It follows from our analysis of Sec. II, that Te exhibits Weyl points not only at the $H$ point, but also along the $\Gamma$–$\Delta$–$A$ line, which is left invariant by the screw rotation $C_{3,1}$. This is clearly visible in the ab-initio band structure calculations of Figs. 8 and 10(d), which show Weyl points with accordion-like dispersions between $\Gamma$ and $A$. As before, we observe that the band connectivity fully agrees with the theoretical prediction of Fig. 1. By the bulk boundary correspondence, the Weyl points at the $H$ point and on the $\Gamma$–$\Delta$–$A$ line lead to numerous arc surface states, see Appendix B. These arc surface states should be readily observable in photoemission and scanning tunneling experiments.

In closing, we note that trigonal selenium (Se), which also crystalizes in SG No. 152 ($P3_{1}21$) Cherin and Unger (1967b), has a very similar band structure to tellurium, albeit the spin-orbit coupling is considerably smaller [Fig. 10(e)]. Therefore, the bands forming Weyl points along the $\Gamma$–$\Delta$–A line of Se are only separated by less than $\sim 50$ meV. This makes the experimental detection of the accordion Weyl points in Se more challenging than in Te.

V.2 Weyl nodal lines in Te16Si38

The tellurium-silicon clathrate Te16Si38 crystallizes in two different forms. While the low-temperature form is cubic, the high-temperature form is rhombohedral with trigonal SG No. 161 $(R3c)$ Jaussaud et al. (2004, 2005). According to our analysis of Sec. III, materials in this SG exhibit Weyl nodal lines that are protected by the glide mirror symmetry $M_{110}$. In Fig. 9 we present the first principles band structure (with PBE functional) of rhombohedral Te16Si38 along high-symmetry paths of the $k_{x}=-k_{y}$ plane, which is left invariant by $M_{110}$ [see also Fig. 10(f)]. We observe that along any path that connects a blue TRIM to a red TRIM in the BZ of Fig. 3(d), $4n$ bands form a connected group with many Weyl crossings. These Weyl crossings are part of Weyl nodal lines that separate the blue from the red TRIMs in the $k_{x}=-k_{y}$ plane. The stability of these Weyl nodal lines is guaranteed by a $\pi$-Berry phase, which by the bulk-boundary correspondence gives rise to drumhead surface states Chiu et al. (2018). In addition, the bands of rhombohedral Te16Si38 carry a nonzero Berry curvature which becomes especially large close to the Weyl nodal lines.

VI Conclusions

In summary, we have performed a systematic analysis of nonsymmorphic band crossings in nonmagnetic trigonal materials with strong spin-orbit coupling. We have found that among the 25 trigonal space groups, six support Weyl points, while three support Weyl nodal lines (see Table 1). The electronic bands at these Weyl points and Weyl lines are required to cross due to nonsymmorphic mirror or rotation symmetries alone, regardless of the chemical composition and other material details. Hence, these Weyl crossings occur in all bands of any material crystallizing in the space groups listed in Table 1. Using this insight, we have performed a database search and identified several existing materials that possess the predicted topological band crossings, see last column in Table 1. The nontrivial topology of the band crossings in these materials leads to a number of interesting and experimentally observable phenomena, e.g., arc and drumhead surface states, anomalous magneto-transport properties, and anomalous Hall effects. Particularly interesting are the nonsymmorphic band crossings and surface states of tellurium (Figs. 8 and 11). We hope that our findings will encourage experimentalists to study the topological properties of these compounds.

Acknowledgements.

The authors thank C. Ast, K. von Klitzing, A. Topp, M. G. Vergniory, and A. Yaresko for useful discussions. B.K. thanks the Max Planck Institute for Solid State Research in Stuttgart for financial support. C.-K.C. is supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant XDB28000000). Work at Princeton was supported by NSF through the Princeton Center for Complex Materials, a Materials Research Science and Engineering Center DMR-1420541.

Appendix A Additional band structure calculations

In this Appendix, we present additional band-structure calculations for the example materials studied in Sec. V of the main text. In Figs. 10(a), 10(b), and 10(c), we present the full band structures of the materials Cu2SrSnS4, Cu2SrGeS4, and Ag2HPO4, respectively, which exhibit Weyl points along $\Gamma-\Delta-A$. These Weyl points are symmetry enforced by the three-fold $C_{3,p}$ screw rotations. In the chalcogenides Cu2SrSnS4 and Cu2SrGeS4 the band gap is about 1.5 eV, while in the salt Ag2HPO4 it is about 2.8 eV. This makes it challenging to observe the Weyl points in photoemission or electron tunneling experiments. However, it might be perhaps possible to grow thin films of Cu2SrSnS4 and Cu2SrGeS4 on a conducting substrate to introduce charge carriers.

Figures 10(d) and 10(e) display the full band structures of trigonal tellurium and selenium. Along the $\Gamma-\Delta-A$ line there are groups of six connected bands with accordion dispersions. Since tellurium is metallic, it should be possible to observe its accordion dispersion and Weyl points using angle-resolved photoemission experiments. Tellurium has many Weyl points also away from the $\Gamma-\Delta-A$ line, for example, at the $H$ point Hirayama et al. (2015); Nakayama et al. (2017). These give rise to numerous arc surface states, see Appendix B.

Figure 10(f) shows the full band structure of the tellurium-silicon clathrate Te16Si38, which exhibits Weyl nodal lines in the $k_{x}=-k_{y}$ plane. We observe that along the paths $\Gamma-L$, $\Gamma-Z$, and $F-Z$, the bands form groups of $4n$ connected bands with Weyl crossings. These Weyl crossings are part of nodal lines that separate $\Gamma$ and $F$ from $Z$ and $L$ in the $k_{x}=-k_{y}$ plane of the rhombohedral BZ, see Fig. 4(d).

Appendix B Surface states of tellurium

Trigonal tellurium has many Weyl points and associated arc surface states. In Fig. 11 we present the surface state dispersion for the (-110) surface of Te 333Note that there is only one type of termination. The labels of the high-symmetry points in the (-110) surface BZ are given in Fig. 1(a). We observe that there are several surface states emanating from the Weyl points near the $H$ point. Remarkably, these surface states cross the entire surface BZ and lie within the bulk band gap, in between the valence and conduction bands. Since their energy is close to the Fermi level, they could play an important role for the (re-)interpretation of the unusual surface transport and photoemission measurements reported in Refs. von Klitzing and Landwehr (1971); Laude et al. (1972); Englert et al. (1977).

There are also arc surface states that are associated with the accordion bands discussed in Sec. V.1 and Fig. 8. For example, for the topmost Weyl point in the valence bands at the energy $E-E_{F}=-1.072$ (gray circle in Fig. 8), we observe an arc state that disperses towards the $\tilde{\Gamma}$ point.