Nodal surface semimetals: Theory and material realization

Nodal surface semimetals: Theory and material realization

Weikang Wu Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore    Ying Liu Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore    Si Li Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore Beijing Key Laboratory of Nanophotonics and Ultrafine Optoelectronic Systems, School of Physics,
Beijing Institute of Technology, Beijing 100081, China
   Chengyong Zhong Institute for Quantum Information and Spintronics, School of Science,
Chongqing University of Posts and Telecommunications, Chongqing 400065, China
   Zhi-Ming Yu Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore    Xian-Lei Sheng Department of Physics, Key Laboratory of Micro-nano Measurement-Manipulation and Physics (Ministry of Education),
Beihang University, Beijing 100191, China
Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore
   Y. X. Zhao National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China    Shengyuan A. Yang Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore

We theoretically study the three-dimensional topological semimetals with nodal surfaces protected by crystalline symmetries. Different from the well-known nodal-point and nodal-line semimetals, in these materials, the conduction and valence bands cross on closed nodal surfaces in the Brillouin zone. We propose different classes of nodal surfaces, both in the absence and in the presence of spin-orbit coupling (SOC). In the absence of SOC, a class of nodal surfaces can be protected by spacetime inversion symmetry and sublattice symmetry and characterized by a index, while another class of nodal surfaces are guaranteed by a combination of nonsymmorphic two-fold screw-rotational symmetry and time-reversal symmetry. We show that the inclusion of SOC will destroy the former class of nodal surfaces but may preserve the latter provided that the inversion symmetry is broken. We further generalize the result to magnetically ordered systems and show that protected nodal surfaces can also exist in magnetic materials without and with SOC, given that certain magnetic group symmetry requirements are satisfied. Several concrete nodal-surface material examples are predicted via the first-principles calculations. The possibility of multi-nodal-surface materials is discussed.


I Introduction

Topological states of matter have been attracting great interest in recent physics research. Inspired by the previous studies on topological insulators Hasan and Kane (2010); Qi and Zhang (2011); Bansil et al. (2016), the research focus is now shifted towards topological semimetals Chiu et al. (2016); Burkov (2016); Yan and Felser (2017); Armitage et al. (2018). In these materials, the electronic band structures possess nontrivial topology and/or symmetry-protected band crossings close to the Fermi level, such that the low-energy electrons behave drastically different from the usual Schrödinger-type fermions. For example, in Weyl/Dirac semimetals Armitage et al. (2018); Wan et al. (2011); Murakami (2007); Young et al. (2012), the conduction and valence bands cross linearly at isolated points, around which the electrons resemble the massless Weyl/Dirac fermions from the standard model, making it possible to simulate intriguing high-energy and relativistic physics phenomena in table-top experiments Nielsen and Ninomiya (1983); Volovik (2003); Guan et al. (2017).

The nontrivial band crossings can be classified based on their dimensionalities. The crossings in the aforementioned Dirac/Weyl semimetals are isolated zero-dimensional (0D) points. Materials with 1D band crossings, known as nodal-line semimetals, have also been proposed and intensively studied in recent works Weng et al. (2015); Yang et al. (2014); Mullen et al. (2015); Yu et al. (2015); Kim et al. (2015); Chen et al. (2015); Fang et al. (2016); Li et al. (2017). For 3D materials, there is one remaining possibility: The band crossings may form a 2D nodal surface, namely each point on the surface is a crossing point between two bands whose dispersions are linear along the surface normal direction. Although both are 2D manifolds in the 3D Brillouin zone (BZ), we stress that the nodal surface is distinct from the ordinary Fermi surface, because the coarse-grained quasiparticles excited from a nodal surface have an intrinsic pseudospin degrees of freedom (representing the two crossing bands) Zhong et al. (2016), behaving effectively as 1D massless Dirac fermions along the surface normal direction, and therefore may have interesting physical properties. In addition, in contrast to the ordinary Fermi surface as a sphere, each nodal surface identified in this article is a torus with opposite edges attached on the periodic boundaries of the BZ.

To date, such nodal-surface semimetals (NSSMs) have only appeared in a few scattered theoretical studies. In Ref. Zhong et al. (2016), Zhong et al. proposed a family of stable graphene network materials, each exhibiting a pair of nodal surfaces close to the Fermi energy. Liang et al. Liang et al. (2016) studied the hexagonal Ba ( V, Nb, or Ta; S or Se) and found a single nodal surface in these materials when the spin-orbit coupling (SOC) can be neglected. From symmetry and topology analysis, Bzdušek and Sigrist discussed the possibility to stabilize nodal surfaces in centrosymmetric systems Bzdušek and Sigrist (2017). More recently, the nodal surface similar to Ref. Liang et al. (2016) was predicted for a proposed acoustic metamaterial MXi (), and the stability of nodal surfaces against perturbations was theoretically investigated Türker and Moroz (2018). Admittedly, our current understanding of NSSMs is still at the primitive stage with several important issues to be addressed. First, the nodal surfaces in the carbon allotropes were found through first-principles calculations, but its topological origin has not been clearly elucidated. Second, given that the nodal surfaces in the aforementioned works are all vulnerable against SOC, is it possible to have nodal surfaces robust under SOC? This important question is still waiting to be answered. Third, the carbon allotropes in Ref. Zhong et al. (2016) have yet to be synthesized, while the nodal surfaces in BaVS Liang et al. (2016) can only be maintained in the room-temperature structural phase. It is thus important to search for more realistic material candidates to facilitate the experimental investigation on NSSMs.

Motivated by the above questions and issues, in this work, we present a theoretical study of the 3D NSSMs both in the absence and in the presence of SOC. In the absence of SOC, we propose two different classes of symmetry-protected nodal surfaces corresponding to the two materials studied in Refs. Zhong et al. (2016); Liang et al. (2016). The first class of the nodal surfaces (as those in the graphene networks) are protected by spacetime inversion symmetry and sublattice symmetry, and are characterized by a topological index. The second class of nodal surfaces are guaranteed by a combination of two-fold screw-rotational symmetry and time-reversal symmetry. After the inclusion of SOC, the former class of nodal surfaces is destroyed generically, but the latter can still be protected with the requirement that the space-time inversion symmetry is violated. Furthermore, it is found that both in the absence and in the presence of SOC, the protection in the latter case persists even in the presence of magnetic orders perpendicular to the screw axis, where, although time reversal symmetry is violated, the system is still invariant under the combination of screw rotation and time reversal. Via first-principles calculations, we identify several candidate NSSM materials with relatively clean low-energy band structures, which support the theoretical analysis in this paper and will facilitate experimental studies on the novel physical properties associated with nodal surfaces.

It is noteworthy that the band structures of the two classes are essentially different due to their qualitatively distinct origins. Nodal surfaces in the first classes have the nontrivial topological charge, and therefore appear in pairs in the BZ conforming the Nielsen-Ninomiya no-go theorem Nielsen and Ninomiya (1981a, b), while those in the second class mainly due to the two-fold nonsymmorphic symmetry exist alone in the BZ, for which the topological essence was revealed in Ref. Zhao and Schnyder (2016). Particularly, each degenerate point on such a nodal surface belong to a twisted band structure analogous to the Möbius strip Zhao and Schnyder (2016), and the band structure as a whole is equivalent to a collection of Mobius strip parametrized by a torus (a 2D sub-BZ perpendicular to the screw axis).

This paper is organized as follows. In Sec. II, we discuss the two classes of nodal surfaces in the absence of SOC. In Sec. III, we propose nodal surfaces that can be protected in the presence of SOC. In Sec. IV, we generalize the discussion to magnetically ordered systems, predicting nodal surfaces in magnetic materials. In each section, we present corresponding material examples with band structures obtained from first-principles calculations. The details of our first-principles methods are presented in the Appendix. Discussion and conclusion are presented in Sec. V.

Ii Nodal surface in the absence of SOC

ii.1 Nodal surface with topological charge

Let us start with the first class (class I) of nodal surfaces, which require spacetime inversion symmetry with the relation and sublattice symmetry . Here and denote space inversion and time reversal, respectively, and the sublattice symmetry satisfies the anti-commutation relation with the Hamiltonian: . It is required that and are mutually independent, namely . The relation of can be realized by particles with integer spin, such as photonic crystals or phonons of mechanical systems and certain cold-atom systems and by electronic systems with negligible SOC, such as the carbon allotropes studied in Ref. Zhong et al. (2016). In Ref. Zhong et al. (2016), the symmetry/topology aspect was not fully exposed, and will be our focus in the following discussion.

We now address the topological classification of the class-I nodal surfaces under the constraint of and symmetries Zhao et al. (2016); Bzdušek and Sigrist (2017). Since both and inverse , operates trivially in momentum space. As is unitary but anti-unitary, we can choose the representation in momentum space that , which is unique up to a unitary transformation. Here is the complex conjugation. Therefore, a -symmetric Hamiltonian is real in this representation. On the other hand, if is represented by , where is the third Pauli matrix and the identity matrix with being the number of valence bands, then the Hamiltonian takes the block anti-diagonal form, i.e.,  Zhao et al. (2016). Because of , can always be converted to a real matrix by a unitary transformation. Thus, for any gapped momentum , the Hamiltonian (after being flattened) can be topologically regarded as a point in the space (with the assumption ), which is exactly the Hamiltonian space for the class BDI among the ten Altland-Zirnbauer (AZ) symmetry classes Altland and Zirnbauer (1997); Schnyder et al. (2008). The space of has two disconnected components, namely , and the sign of the determinant of specifies which component belongs to.

In the framework of topological classification of nodal surfaces, the spatial codimension of nodal surfaces in three dimensions is zero, namely a 0D sphere consisting of two points is chosen to surround a nodal surface in 3D BZ from its transverse dimensions Volovik (2003); Zhao and Wang (2013). We now consider two gapped points with , respectively. Then, any path connecting the two points in momentum space has to pass through at least a band-crossing point with as a result of the mean value theorem, and the band-crossing point has a two-fold degeneracy, which is of topological stability once and are preserved. These two-fold degenerate band-crossing points generically spread out a compact nodal surface in momentum space, namely a class-I nodal surface, which accordingly has a non-trivially topological charge.

Figure 1: (a) Lattice structure and (b) unit cell of the carbon allotrope QGN(1,2). (a) shows a supercell, in which the unit cell is marked by the black box. In (b), the red-colored carbon atoms have a hybridization character (which are labeled from site 1 to site 8), whereas the blue-colored carbon atoms have a hybridization character. (c) Corresponding Brillouin zone, where the two nodal surfaces are schematically shown by the shaded surfaces. (d) Calculated band structure for QGN(1,2) without SOC.

The class-I nodal surface can be realized by the carbon allotropes, which has been studied in Ref. Zhong et al. (2016) with other emphases. This family of materials share both and symmetry with negligible SOC, and therefore have the desired symmetry with . The chiral symmetry also emerges in the low-energy effective theory by appropriately tuning the chemical potential of each site. To be concrete, we take one particular example in this family, namely QGN(1,2), where QGN stands for the quadrilateral graphene network, as shown in Fig. 1. The crystal structure of QGN(1,2) has the space group symmetry of (No. 131). Its unit cell contains two kinds of graphene nanorbbon-like structural motifs: The edge atoms show a strong hybridization character, whereas the remaining atoms exhibit hybridization character and form four zigzag chains running along the -axis. From atomic orbital projection, one finds that the low-energy states are dominated by the orbitals on the eight carbon atomic sites which are marked in red color in Fig. 1(b). Thus, it is emerged a sublattice symmetry in low-energy regime, which relates electrons at odd sites to holes at their even nearest neighbours and vice versa (see the detailed discussion below), given that the chemical potential is appropriately tuned for each orbital.

The band structure of QGN(1,2) obtained from the first-principles calculations is plotted in Fig. 1(d). The band structures with and without SOC have little difference, due to the negligible SOC strength of carbon atoms. One observes the linear-type band crossing along the -Z and the M-A paths. As mentioned in Ref. Zhong et al. (2016), these crossing points form nodal surfaces and there is a pair of such surfaces related by or in the BZ, as illustrated in Fig. 1(c). Each point on the surfaces is a linear crossing point between two bands along the surface normal direction (approximately the direction). And either nodal surface is quite flat with very small energy variation, and is close to the Fermi level.

To reveal the topological nature of the class-I nodal surface of QGN(1,2), we first work out the tight-binding (TB) model corresponding to its low-energy physics. At low energies, the two crossing bands are dominated by the orbitals from the eight carbon atoms in a unit cell, which are numbered clockwisely from 1 to 8 as shown in Fig. 1(b), and the eight-band TB model is explicitly given in Appendix A. The sublattice symmetry relates to with , and therefore is represented as . Here and hereafter, we express the matrices using the Kronecker products of the Pauli matrices (, ) and the identity matrices. The chiral symmetry operator can be converted to by the unitary transformation that exchanges and as well as and , so that the transformed Hamiltonian is block diagonalized with the upper right block denoted by . The operators are now ordered as . From Fig. 1(b), maps to , to for odd orbitals, and to , to for even orbitals, namely with being the first Pauli matrix. Since is simply complex conjugation, it is found that , which commutes with . The relation implies there exists a unitary transformation converting to , which turns out to be . After this unitary transformation, the upper right block becomes a real matrix . Now the topological charge of a nodal surface can be defined as


where are two points on two sides of the nodal surface, respectively. It is verified in Appendix A that both nodal surfaces illustrated in Fig. 1(c) have the nontrivial topological charge.

We have two remarks before proceeding. First, in Ref. Bzdušek and Sigrist (2017), Bzdušek and Sigrist developed a theory for the centrosymmetric extension of the ten AZ classes. The example discussed here would fit into the BDI class discussed in their work. Second, the similar symmetry (known as the chiral symmetry) also naturally emerge in the low-energy excitation spectrum for superconductors. Hence, nodal surfaces may also appear in certain centrosymmetric superconductors, like that discussed in Refs. Agterberg et al. (2017); Bzdušek and Sigrist (2017).

ii.2 Essential nodal surface dictated by nonsymmorphic symmetry

Next, we consider the second class (class-II) of nodal surfaces existing in the absence of SOC. Such surface is a result of the combination of a two-fold screw symmetry and time reversal symmetry. It is essential in the sense that its presence and location are solely dictated by the symmetries, as we discuss below.

Consider the two-fold screw rotation , which is a nonsymmorphic symmetry involving a half translation along the rotation axis. In momentum space inverses and , and preserves . Without SOC, one finds that , where is the translation along the -direction by a lattice constant. is anti-unitary and inverses with the relation . Thus, the combination is anti-unitary and only inverses . Since , satisfies


We now regard the system under consideration as a collection of D -subsystems parametrized by . Each D subsystem with given effectively has the same non-symmorphic magnetic symmetry with the relation (2). Hence for a two-band theory, it has a twisted band structure of Möbius strip with a single band-crossing at the boundary of the 1D sub-BZ, which is enforced by the topological nature of the two-fold non-symmorphic symmetry, as shown in Refs. Zhao and Schnyder (2016); Michel and Zak (1999); Young and Kane (2015), where the momentum-dependent symmetry operator is also associated with a unit winding number. The Möbius-strip type band structure has a nontrivial topological charge, which means that, given the symmetry being preserved, a single band-crossing is topologically stable, and can only be gapped in pair in four-band theories Zhao and Schnyder (2016). Since a linear crossing point exists for every 1D subsystem at the sub-BZ boundary, the collection of crossings form a nodal surface on the boundary plane of the 3D BZ with .

The crossings on the surface can also be understood as a result of the Kramers degeneracy. One notes that any point on the plane is invariant under , but from Eq. (2), the anti-unitary symmetry satisfies


on the whole plane. Thus, the two-fold Kramers degeneracy arises at every point on the plane. This degeneracy is generically lifted away from this plane due to the loss of symmetry protection, so that a nodal surface is formed at the plane. The above argument was presented in Ref. Liang et al. (2016), where BaVS was identified as a candidate material with this type of nodal surfaces.

As we have mentioned, the condition for the presence of the class-I nodal surface is quite stringent: Besides the symmetries and , its appearance also requires regions in BZ with inverted band orderings (different indices). In contrast, the presence of the class-II nodal surface discussed here is solely guaranteed by symmetry (hence can be regarded as an essential band-crossing), and its location is fixed at (if the screw axis is along ). This makes it easier for search for candidate materials by analyzing the space groups.

We emphasize that although the existence of class-II nodal surface is determined by symmetry, its energy is not determined and depends on the specific material. In addition, there could be strong energy variation from point to point on the nodal surface. In order for the nodal surface to manifest in physical properties, a “good” candidate NSSM needs to satisfy the following requirements: (i) the nodal surface should have relatively small energy variation; (ii) its energy should be close to Fermi level; and (iii) it is desired that no other extraneous band is present at low energy.

Figure 2: (a) Crystal structure of KYO and (b) the corresponding Brillouin zone. In (b), R and E are the mid-points of the paths A-L and -M, respectively. (c) Band structure of KYO without SOC. The red arrows indicate the band degeneracy along the paths on the nodal surface.

Here we identify example materials which possess the class-II nodal surfaces. The first example is the compound KYO with the structure in the space group (No. 186), as illustrated in Fig. 2(a). This material is predicted in Materials Project Jain et al. (2013). In the crystal structure, each yttrium atom is surrounded by a tetrahedron of O atoms and is located at the positions: and with , while the K atoms fill the space between the tetrahedra, forming two types of triangles with different sizes. The material is nonmagnetic and has a two-fold screw axis along the -direction exists, hence the condition for a class-II nodal surface is satisfied. To be noted, the material does not have the inversion symmetry.

The calculated band structure of KYO in the absence of SOC is displayed in Fig. 2(c). Since the material is composed of light elements, the SOC has negligible effect on the band structure, which has been checked by our density functional theory (DFT) calculation. From the result, one observes that a nodal surface is indeed present in the plane, where the two low-energy bands cross linearly. In Fig. 2(c), we purposely show the dispersion along a generic -path E-R, which is not a high-symmetry path [see Fig. 2(b)] and on which the linear crossing can be clearly observed. This is in agreement with the above argument that the anti-unitary symmetry guarantees a nodal surface at the plane. In addition, the energy variation of the nodal surface at the plane is small ( eV), the linear dispersion range is relatively large (above 0.5 eV), and there is no other extraneous band close to the Fermi level. These features are desired for a NSSM.

Figure 3: (a) Perspective view and (b) top view of the crystal structure of the Mo family materials, where (Na, K, Rb, In, Tl) and (S, Se, Te). The insert in (b) shows the face-sharing Mo octahedra chains. (c) Band structure of RbMoS without SOC. The red arrows indicate the band degeneracy along the paths on the nodal surface. The Brillouin zone here has the same shape as that in Fig. 2(b).

The second example is a family of materials with the formula Mo, where (Na, K, Rb, In, Tl) and (S, Se, Te). These Mo compounds have been synthesized in experiment Hönle et al. (1980); Huster et al. (1983). They share the same crystalline structure with the space group (No. 176), as shown in Figs. 3(a) and 3(b). The structure features close-packed one-dimensional columns, consisting of face-sharing Mo octahedra surrounded by atoms. The columns are oriented along the -direction, arranged in a trigonal lattice in the - plane, and related to each other by a screw rotation. The atoms are intercalated in the large holes between the columns. The space group contains the and symmetries. And it should be noted that the structure preserves the inversion symmetry . This point will be important for the later discussion in Sec. III when we deal with the case with SOC included.

Let us consider one specific example RbMoS in this family, for which the effect of SOC on the low-energy bands is small. The calculated band structure for RbMoS in the absence of SOC is shown in Fig. 3(c). Similarly to KYO, one observes a nodal surface located at the plane, guaranteed by the anti-unitary symmetry . The linear dispersion range is above 1 eV, even larger than that of KYO. The nodal surface is very flat in energy, and it sits almost exactly at the Fermi level. Such a band structure would be ideal for studying the class-II nodal surfaces.

Before proceeding, we mention that the structure of the Mo family compounds have strong quasi-1D character, which may induce structural instabilities towards Peierls distortion. First-principles calculations indicated that except for TlMoTe and RbMoTe, other members in this family would be prone to a Peierls distortion that breaks the symmetry Liu and Zunger (2017), hence destroying the nodal surface. However, such distortion has not been detected in experiment Potel et al. (1980); Chevrel and Sergent (1986), and future studies are needed to clarify this issue.

Iii Nodal surface in the presence of SOC

Compared to the case without SOC, the inclusion of SOC makes at least two important differences in terms of symmetry properties: (i) The time reversal operator now satisfies because the time reversal operation reverses the electron spin and (ii) all rotation operations need to operate on the spin in addition to the spatial degree of freedom.

For class-I nodal surfaces, the protection mechanism discussed in Sec. II.1 no longer holds, because of point (i). Hence introducing SOC will generally destroy class-I nodal surfaces. We explicitly verify this point by artificially increasing the SOC strength in QGN(1,2), and indeed find that the nodal surface is gapped out by SOC (see Appendix C).

For class-II nodal surfaces, a more involved analysis is needed. It turns out that the inversion symmetry plays an important role in this case. In order to ensure the boundary Kramers degeneracies due to are protecting a nodal surface, symmetry has to be violated. First, Eq. (3) still holds with SOC. Although , performing twice also rotates spins, and we have


where denotes the -rotation on spin, contributing a factor of . As , Eq. (2) still holds, so does Eq. (3). However, as long as inversion symmetry is present, the Kramers degeneracies on the BZ boundary with can extend to the whole bulk of BZ (along any generic path) because of the spacetime inversion symmetry . In contrast to the case without SOC, , hence there is actually the Kramers degeneracy at every point in the BZ due to , which acts locally in momentum space. Consequently, the degeneracy at the plane in this case does not represent a nodal surface according to our definition.

Figure 4: Calculated band structures of TlMoTe (a) without SOC and (b) with SOC. Panel (c) shows the Brillouin zone. R and E are the mid-points of the paths A-L and -M, respectively. (d) Enlarged view of the low-energy band dispersion around the R and H points, showing that the original band-crossing at the nodal surface is gapped by SOC.

To explicitly demonstrate the above point, we perform the first-principles calculation on the material TlMoTe in the Mo family with SOC included. Its crystal structure has been shown in Fig. 3(a) and 3(b). As mentioned before, the lattice structure of TlMoTe preserves the inversion symmetry , and unlike RbMoS, the SOC in TlMoTe has appreciable effects on the band structure and needs to be taken into account. In Figs. 4(a) and 4(b), we plot the band structures of TlMoTe without and with SOC. In the absence of SOC, a nodal surface is present at the plane, similar to that for RbMoS. After including SOC, band splitting is observed at the original nodal surface. One notes that the band dispersion along the K-H and E-R no longer shows the band crossing [see Figs. 4(b) and 4(d)], hence the two-fold degeneracy by in the plane does not lead to a nodal surface. This demonstrates that when is preserved, can no longer guarantee a nodal surface in the presence of SOC. Meanwhile, it is noted that in Fig. 4(b), the crossing is maintained along -A path at the A point, which is enabled by the six-fold rotational symmetry on this path. A recent study shows that the crossing point at A is actually a cubic Dirac point with linear dispersion along and cubic dispersion in the plane Liu and Zunger (2017).

Figure 5: (a) Side view and (b) top view of the crystal structure for TaTeI. The inset in (b) shows the cluster unit of the triangular lattice. (c) Calculated band structure for TaTeI in the presence of SOC. The right panel shows the enlarged view of the band structure near the Fermi level along a generic path E-R-H, where E-R is perpendicular to the plane and R-H is in the plane. The Brillouin zone and the labeled points are the same as in Fig. 2(b).

Hence, from the above argument, a necessary condition for a -protected nodal surface would be that the symmetry needs to be broken (here we consider nonmagnetic materials where is preserved). This will generally lift the two-fold degeneracy away from the plane, thus pairs of non-degenerate bands along a generic path have to cross at the plane, making the doubly-degenerate plane a nodal surface. Here, it should be mentioned that for systems without SOC (those in Sec. II.2), spin is a dummy degree of freedom. Hence, if the spin degeneracy is counted, then the nodal surface there would be four-fold degenerate. In comparison, in the presence of SOC, the obtained nodal surfaces are two-fold degenerate essentially due to a Kramers-like degeneracy, and there is no protection for a four-fold degenerate surface (at least by the symmetries discussed in this work).

This picture is indeed confirmed by our calculation on an example material TaTeI. Its crystal structure has the space group (No. 186), which does not possess an inversion center. TaTeI is a member of the material family ( Nb, Ta; S, Se, Te; and Cl, Br, I) Smith and Miller (1996, 1998). The crystal structure consists of ordered, close-packed layers of I and Te atoms, interleaved by Ta atoms, as shown in Fig. 5(a) and 5(b). Figure 5(b) displays the top view of a single TaTeI layer, which is similar to the structure of 1T’-MoS. This layered material have several types of stacking. Here we consider the bulk structure with the layer stacking pattern. The calculated band structure with SOC included is shown in Fig. 5(c). The material is a semiconductor with a bandgap about 0.48 eV. Although it is not a semimetal, by examining the band dispersion around the plane, one finds that there is indeed a nodal surface in that plane consisting of linear band-crossing points. This is more easily observed in the right panel of Fig. 5(c), which shows a zoom-in image for the dispersion along the generic path E-R. Along the path, the bands are non-degenerate (due to the broken ), and a pair of bands must cross at R in the plane, hence a nodal surface is formed in that plane.

Iv Nodal surface in magnetic materials

The discussion in the previous sections are about nonmagnetic materials, such that is preserved. We have seen that plays an important role in stabilizing the nodal surfaces. Then the question arises: Is it possible to realize protected nodal surfaces also in magnetic materials?

Here we show that this is indeed possible, and the result also depends on whether the SOC in the system can be neglected or not.

First, consider the case in the absence of SOC. Without SOC, the spin and the orbital degrees of freedom are independent and can be considered as different subspaces. The spins can be oriented in any direction without affecting the orbital part of the wave-function. With a chosen spin polarization axis, the two spin channels are decoupled, and hence the bands for each spin species can be effectively regarded as for a spinless system. Therefore, for the states of one spin, all the crystalline symmetries are preserved Wang et al. (2016); Chang et al. (2017). Consequently, the analysis can be reduced to those in Sec. II, which means that it is in principle possible to have class-I as well as class-II nodal surfaces in a magnetic system, if the symmetry requirements presented in Sec. II are satisfied. Particularly, if the crystal structure possesses a two-fold screw axis , then there must exist a nodal surface (for each spin species) at .

Figure 6: (a) Side view and (b) top view of the crystal structure for CsCrI. (c) Calculated band structure for CsCrI in the absence of SOC, showing a ferromagnetic state. The red and blue lines represent spin-up and spin-down bands, respectively. In the absence of SOC, the nodal surface is preserved, and the band structure does not depend on the direction of the magnetic moments.

We demonstrate the above point using a concrete material example. We consider the material CsCrI, which is a member of the material family CsCr ( Cl, Br, I) Guen et al. (1979). The material takes the BaVS-type structure, with space group (No. 194), as shown in Fig. 6(a) and 6(b). The Cr atoms are surrounded by octahedra of atoms, forming one-dimensional chains along the -axis, with octahedra sharing common faces. Those chains are arranged into a trigonal lattice in the - plane, with the Cs atoms intercalated between the chains. Without considering the spin polarization, the lattice structure has a two-fold screw axis . The material assumes a ferromagnetic ground state, for which the magnetic moments are provided by the Cr atoms. The band structure for CsCrI without SOC is plotted in Fig. 6(c). One observes that the system is a half metal: The Fermi level crosses only the spin-up bands, whereas the spin-down bands are high in energy. Focusing on the spin-up bands that are near the Fermi level, one finds that there is indeed a nodal surface in the plane, as consistent with our above argument for the case without SOC. Due to the absence of SOC, orienting the magnetic moments along different directions will not affect the band structure, and hence will not affect the nodal surface. This is confirmed by our DFT results: We have repeated the calculation with different magnetic moment orientations (e.g., along the -direction and the -direction), and the obtained results are exactly the same as in Fig. 6(c).

Next, we consider the case in the presence of SOC. The spin and orbital degrees of freedom are tied with each other by SOC. It is generally not possible to label the bands as spin-up/spin-down. Hence, the symmetry operations need to act on both orbital and spin as a whole. Nevertheless, we note that the arguments in Sec. III rely on the existence of the composite symmetry , which can still be preserved although the individual or symmetry may be broken. If so, then the previous arguments in Sec. III still apply and a nodal surface can be guaranteed. Explicitly, this means that: in the presence of SOC, a nodal surface in the plane can be protected by the combined symmetry in conjunction with the absence of the symmetry in the system (note that may be preserved in certain antiferromagnet).

Figure 7: Calculated band structures for CsCrI in the presence of SOC, with magnetic moments along (a) the direction and (b) the direction. The nodal surface is preserved in (a) but not in (b).

We still take CsCrI as an example and now include SOC in the DFT calculation. It is important to note that: (i) if the magnetic moments are aligned in a direction perpendicular to the direction, then the combined symmetry is preserved (since the spins aligning in the - plane are flipped by both and , and therefore are intact by ); and (ii) for other directions (e.g., along the direction), the symmetry is broken. The symmetry is broken for both cases. Then according to our previous analysis, the nodal surface in the plane should be preserved in case (i) but not in case (ii). This is indeed confirmed by our DFT results. Figure 7(a) shows that result with the moments aligned along the -direction, where a nodal surface is observed in the plane. Figure 7(b) shows that result when the moments are aligned along the -direction, and one can see that the nodal surface is destroyed.

V Discussion and conclusion

We have discussed two classes of nodal surfaces in the absence of SOC. They are essentially different, namely the class-I surfaces are topologically charged, and two patches on the surfaces related by have opposite charges to satisfy the Nielsen-Ninomiya no-go theorem Nielsen and Ninomiya (1981a, b); however, the class-II ones are mainly originated from the topological features of screw symmetry, and therefore each degenerate point belongs to a Möbius-strip band structure, all of which spread out as a torus (a - sub-BZ). The class-I surfaces are not essential, requiring band inversions in part of the BZ; whereas the class-II surfaces are essential and guaranteed to appear solely by the symmetry. In addition, the shape and location of the class-I surfaces can vary in the BZ, and can be tuned, e.g. by lattice strain; whereas the the location of the class-II surface is fixed at the BZ boundary plane as a result of the combination of time reversal and screw rotation. The nodal surfaces in the examples discussed in Sec. III and IV are generalizations of the class-II surfaces, so they share similar characteristic features of the class-II surfaces listed above.

We mention that both classes do not lead to any special boundary gapless modes. Although the first class has the nontrivial topological charge, similar to Weyl semimetals, the codimensionality of nodal surfaces or the dimensionality of sub topological insulators is zero, impossible to generate any boundary state. It may be noteworthy that each nodal surface considered here is a torus in the BZ, which cannot be continuously deformed to be a point, and hence should be distinguished from a nodal surface that is topologically a sphere deformable to a single point in the BZ Türker and Moroz (2018). (Note that for class-I surfaces, a pair of surfaces may merge into a single sphere when the system is strongly deformed.) We also mention that although the 0D charge does not produce any topological surface state, it is possible that additional 1D or 2D topological charges (having codimensions 1 and 2) may exist and lead to nontrivial surface states in these systems.

As we have mentioned, the nodal surfaces are distinct from the usual Fermi surfaces because it is formed by the crossing of two bands. As a result, the low-energy electrons acquire an intrinsic pseudo-spin degree of freedom, different from the usual Schrödinger fermions. Near a relatively flat nodal surface, the low-energy effective model may be written as


where is the Fermi velocity, measures the deviation from the nodal surface along the surface normal direction, and the Pauli matrix represents the pseudo-spin, corresponding to the two crossing bands. Equation (5) effectively describes 1D massless Dirac fermions, where the pseudospin is locked to the momentum. For transport along the -direction, we may expect interesting properties such as the enhanced mobility due to the suppression of back-scattering (for states around a single nodal surface) from scatterers that preserve the pseudospin.

Figure 8: (a) Crystal structure of CuSe and (b) the corresponding Brillouin zone. In (b), , X, Y, and M label the four -points that are above the points , X, Y, and M, and are located in the plane. (c) Band structure of CuSe (with SOC) along some generic paths. The red arrows indicate the band degeneracy along the paths on the two orthogonal nodal surfaces.

Regarding the class-II surfaces, if there are multiple two-fold screw axis, then it is possible to have multiple nodal surfaces on the BZ boundary planes. One example is shown in Fig. 8 for the material CuSe known as the mineral umangite, with the space group of (No. 113) Morimoto and Koto (1966); Chakrabarti and Laughlin (1981). As a typical intermetallic compounds, Cu atoms are bonded closely both with one another and with Se atoms. Two kinds of Cu atoms exist in the structure, with Cu(I) at the position and Cu(II) at the position with and . The material experiences a transition to an antiferromagnetic state at about 50 K Okamoto et al. (1969). Here, we focus on its paramagnetic phase above 50 K. The band structure of CuSe is plotted in Fig. 8(c). The space group contains two perpendicular screw axis and . With preserved , two mutually orthogonal nodal surfaces can be found in the and planes [as illustrated in Fig. 8(b)]. This is indeed observed in the band structure in Fig. 8(c), which shows that the non-degenerate bands along the generic paths -X and -Y cross at the two nodal surfaces. A material example with three nodal surfaces is also presented in Appendix C.

Finally, we mention that as a good NSSM material, the nodal surface should be close to the Fermi level and formed by the crossing between conduction and valence bands. This condition imposes constraints regarding the electron filling of the bands. The recent work by Watanabe et al. Watanabe et al. (2016) has studied the detailed filling constraints for realizing semimetal states for nonsymmorphic space groups, which will offer useful guidelines for searching NSSMs. In this work, we have identified several good candidate materials that satisfy the condition. However, for some cases, like the case in the presence of SOC, the examples we show are not ideal. For example, TaTeI discussed in Sec. III is actually a semiconductor. Nevertheless, it serves the purpose to illustrate that a nodal surface can indeed appear when the proposed symmetry requirements are satisfied. Based on our theory, we expect better NSSM materials to be discovered in future studies.

In conclusion, we have theoretically investigated the NSSMs which host robust nodal surfaces formed by the crossing of two bands close to the Fermi level. We clarify the symmetry/topology-protection of the nodal surfaces. In the absence of SOC, we identify two classes of nodal surfaces which are protected by different mechanisms. The class-I surfaces are protected by a combination of inversion, sublattice, and time reversal symmetries and are characterized by a index. The class-II surfaces are protected by a two-fold screw axis and the time reversal symmetry. The two classes differ in several aspects including the surface shape, location, and robustness against perturbation. The inclusion of SOC generally gaps the class-I surface, and we show that the class-II surface may be preserved provided that the inversion symmetry is broken. Furthermore, we generalize the analysis to magnetic materials. We find that class-II surfaces can exist in magnetic materials both without and with SOC, given that certain magnetic group symmetry requirements are satisfied. We have identified several concrete NSSM material examples, which will facilitate the experimental exploration of the intriguing properties of NSSMs.

The authors thank T. Bzdušek, X. C. Wu, and D. L. Deng for valuable discussions and comments on the manuscript. This work was supported by the Singapore Ministry of Education Academic Research Fund Tier 1 (SUTD-T1-2015004) and Tier 2 (MOE2015-T2-2-144). We acknowledge computational support from the Texas Advanced Computing Center and the National Supercomputing Centre Singapore.

Appendix A Tight-binding model and invariant for QGN(1,2)

As discussed in the main text, the low-energy physics mainly comes from electrons hopping among the eight sites, and therefore the Hamiltonian of the tight-binding model, with , describes an eight-band theory. After a gauge transformation to make the Hamiltonian periodic in and , we have


Here the wave-vectors are measured in the respective inverse lattice constants, and and are the real hopping amplitudes from site 1 to site 2 and from site 2 to site 3, respectively [see Fig. 1(b)].

Performing the combined unitary transformation with




the Hamiltonian becomes anti-diagonal [i.e., ] with the upper right lock being

Figure A1: Comparison of band structures of the eight-band tight binding model in Eq. A1 (red dashed lines) and the DFT result (blue solid lines). In the tight-binding model, the values eV and eV are obtained from fitting the low-energy DFT band structure.

Then it is found that . If , then there are two nodal surfaces with related by or symmetry. It is easy to check that because ( eV and eV, as obtained from the fitting of DFT result in Fig. A1); whereas . Thus, both nodal surfaces have the nontrivial topological charge according to the formula in Eq. (1).

Appendix B First-principles calculation method

The first-principle calculations are based on the DFT, as implemented in the Vienna ab initio simulation package Kresse and Hafner (1993); Kresse and Furthmüller (1996). The projector augmented wave method was adopted Blöchl (1994). The generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) realization Perdew et al. (1996) was adopted for the exchange-correlation potential. For all calculations, the energy and force convergence criteria were set to be  eV and  eV/Å, respectively. The BZ sampling was performed by using grids with a spacing of Å within a -centered sampling scheme. As the transition metal d orbitals may have notable correlation effects, we have validated our DFT results by the GGA+U method Dudarev et al. (1998) (see Appendix E). The key features are found to be qualitatively the same as the GGA results. Hence in the main text, we focus on the GGA results. For KYO, the optimized lattice parameters (, ) were used for the band structure calculation. For the other materials discussed here, the experimental values of their respective lattice parameters were used in the calculation.

Appendix C SOC effect on class-I surfaces

Here we consider the SOC effect on the class-I nodal surfaces. Take the QGN(1,2) discussed in Sec. II.1 as an example. Since the carbon is a light element with very weak SOC. The band structure with SOC shows negligible difference from that in Fig. 1(d) without SOC. In order to more clearly demonstrate that SOC in fact gaps the nodal surface, we artificially enhance the SOC strength in the DFT calculation by 30 times. The obtained band structure is plotted in Fig. A2. One clearly observes that a gap is opened by SOC at the original band-crossing point, destroying the class-I nodal surface.

Figure A2: (a) Band structure for QGN(1,2) with SOC. The SOC strength is artificially enhanced by 30 times, in order to show that the SOC gaps the original nodal surfaces. (b) Enlarged view around the original band crossing, corresponding to the region as indicated by the red arrow in (a).

Appendix D Material example with three nodal surfaces

Figure A3: (a) Crystal structure of RbSe and (b) the corresponding Brillouin zone. The insert in (a) shows the helical Se chain. In (b), is the body center of 1/8 Brillouin zone; X, Y, and Z are the face centers of the 1/8 Brillouin zone; U, T, and S are the mid-points of the paths U-R, T-R, and S-R, respectively. (c) Band structure of RbSe with SOC.

In Sec. V, we have shown that the compound CuSe with two perpendicular screw axis can host two orthogonal nodal surfaces on the BZ boundary planes. Here we consider another example material RbSe with three perpendicular nodal surfaces on the BZ boundary planes Boettcher (1979). Its crystal structure has the space group (No. 19), which has three two-fold screw axes perpendicular to each other: , , and . The time reversal symmetry is preserved due to the absence of magnetic ordering. There are four helical Se chains in the unit cell, and the Rb atoms are surrounded by the Se chains and related by the screw axis.

The calculated band structure for RbSe with SOC included is shown in Fig. A3(c) (only some generic paths are shown here). We find that the material is a semiconductor with a bandgap about 0.75 eV. Nevertheless, band crossings appear in both valence and conduction bands. By examining the band dispersion on the BZ boundary planes, one can find that the bands cross at these planes in pairs. For example, the non-degenerate bands along the three generic paths -X, -Y, -Z all cross at the BZ boundary planes. Hence there are three mutually orthogonal nodal surfaces as illustrated in Fig. A3(b).

Appendix E Hubbard U correction

Figure A4: Effects of the Hubbard U correction on band structures of (a) KYO (without SOC) with U=4.0 eV, (b) RbMoS (without SOC) with U=4.0 eV, (c) TlMoTe (SOC included) with U=4.0 eV, (d) TaTeI (SOC included) with U=2.0 eV, (e) CsCrI (without SOC) with U=5.0 eV, and (f) CuSe (SOC included) with U=5.0 eV.

To test the correlation effects of the transition metal d-orbitals, we performed the GGA+U calculations, by taking into account the on-site Coulomb interaction. The U values proper for each transition metal element have been tested. Figure A4 shows the representative results for the band structures of KYO, RbMoS, TlMoTe, TaTeI, CsCrI, and CuSe. One can observe that the results for all these compounds show little change in comparison with the GGA results. Hence, we focus on the GGA results in the main text.


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