Abstract
Node line bandtouchings protected by mirror symmetry (named as mNLs), the product of inversion and time reversal symmetry S = PT (named as sNLs), or nonsymmorphic symmetry are nontrivial topological objects of topological semimetals in the Brillouin Zone. In this work, we screened a family of MgSrSitype crystals using first principles calculations, and discovered that more than 70 members are nodeline semimetals. A new type of multiloop structure was found in AsRhTi that a sNL touches robustly with a mNL at some “nexus point”, and in the meanwhile a second mNL crosses with the sNL to form a Hopflink. Unlike the previously proposed Hopflink formed by two sNLs or two mNLs, a Hopflink formed by a sNL and a mNL requires a minimal threeband model to characterize its essential electronic structure. The associated topological surface states on different surfaces of AsRhTi crystal were also obtained. Even more complicated and exotic multiloop structure of NLs were predicted in AsFeNb and PNiNb. Our work may shed light on search for exotic multiloop nodeline semimetals in real materials.
Introduction
The band crossings of the conduction and valence bands in a topological semimetal are interesting topological objects of Brillouin Zone (BZ) which bring about unique electronic structures and electrical properties, such as giant magnetoresistance, parity anomaly and “drumhead” states at material’s surfaces.^{1,2} Depending on the dimensionality of band crossings, topological semimetals are classified into three categories, the Weyl semimetals (WSMs)^{3,4,5,6,7,8} or Dirac semimetals (DSMs),^{9,10,11} nodeline semimetals (NLSMs),^{12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41} and nodesurface semimetals (NSSMs).^{42,43} Unlike the DSMs and WSMs whose band crossings take place at discrete points in the BZ, the band crossings of NLSMs form closed loops. When circling around these loops, an electron picks up an nontrivial Berry phase π in its wave function, whose effect can be detected by transporting measurements. Though having been extensively proposed in graphene networks,^{13} antiperovskites,^{14,15} SrIrO_{3},^{20} TlTaS_{2},^{17} BaTaS,^{42} HfC,^{29} CaP_{3}/CaAs_{3}^{30,31}, and Co_{2}MnGa,^{35} etc, the direct evidence of existence of nodeline (NL) states in real materials is rare.^{18,25,26,27} Finding new materials with clean and robust NL band crossing around the fermi level is still a demanding task in the field of condensed matter physics.
Three types of NLs have been discovered based on their protecting symmetry.^{2} The first type of NLs is protected by mirror symmetry, which is named as mNL in this work and shown schematically in Fig. 1a. Due to the mirror symmetry, the mNL is pinned to the invariant plane of the mirror symmetry. The second type of NLs is protected by the combination of timereversal symmetry T and inversion symmetry P, i.e., S=PT. This type of NLs, named as sNL here, can present at any region of the BZ as shown at the right of Fig. 1a. The last type of NLs is protected by nonsymmorphic symmetries and usually appears at the boundary of BZ.^{20,42}
Recently, there rises a new trend of investigating NLSMs with multiple NL loops.^{28,29,33,34,35,36,38} In those NLSMs, NL loops may intersect with each other and entangle into a variety of structures, such as nodenet,^{38} nodechain^{28}, and Hopflink,^{32,33,34,35,36} etc. For example, two mNLs will be stuck together at some points dubbed as “nexus points” on the crossline of two invariant planes of mirror symmetries (see in Fig. 1b).^{44} In the case of two sNLs, the sNLs can be separated, touched or crossed with unrestricted locations in the BZ (see in Fig. 1c). The crossed sNLs are also called Hopflink due to their topological invariant being the Hopflink number.^{33} While the existence of multipleloop NLs has been realized in photonic lattice,^{45} their existence in fermionic systems has not been identified thus far.
In this work, using first principles calculations, we screened the family of MgSrSitype crystals which consists of 660 members, and found more than 70 compounds are NLSMs showing a variety of NL structures. The NLs are protected by the mirror symmetry or the S symmetry contained in the Pnma space group of MgSrSitype crystals. Importantly, in contrast to a previous report in which the member AsRhTi was predicted to be a NLSM with a single NL band crossing,^{22} we found in this material a new type of multiloop NL structure as shown in Fig. 1d, where a sNL sticks to a mNL at some “nexus point” (denoted by O) and penetrates the invariant plane of mNL at some general point (denoted by E in Fig. (1d)). Interestingly, we also found a third mNL crosses the sNL and a Hopf link is formed. Unlike the case of two crossed mNLs or two crossed sNLs, this novel multiloop NL structure requires a minimal threeband model to describe its essential electronic structure. Even more exotic multiloop NL structures were further uncovered in AsFeNb and PNiNb. Some of the NLSMs show very clean band structures at the fermi level without other trivial bands. Our work therefore provides a promising platform for the material realization of new topological semimetals with exotic NL structures.
Results and discussion
Multiloop NLs in AsRhTi
The group of MgSrSitype crystal is consisted of 660 members as documented in ICSD.^{46,47,48,49,50,51,52,53,54,55,56} It takes a Pnma space group which contains a mirror plane m_{y}, two glide mirror planes \(\tilde m_x\) and \(\tilde m_z\) and the space inversion P. On the k_{x} = 0 and k_{z} = 0 invariant planes, the glide planes \(\tilde m_x\) and \(\tilde m_z\) act in the same way as the normal mirror planes,^{20} while on the boundary of BZ, they should be treated differently since their fractional translations may lead to additional band degeneracy. If the time reversal T is also a symmetry, the compounds become symmetrical under the composed operation S=PT. Both conditions of existence of the sNL and mNLs are therefore fulfilled in MgSrSitype crystals. In Fig. (2a) we plotted the crystal structure of a prototype compound, AsRhTi and the two key symmetries for NLs, a mirror plane m_{y} and inversion symmetry P are highlighted. It can be found that in the unit cell of AsRhTi it contains two layers of atoms and the plane of the atom layers is overlapped with the mirror plane of m_{y}. The inversion center, on the other hand, is off the atom layers and locates at the middle of two neighboring layers. The corresponding Brillouin Zone and highsymmetry paths are also shown below the crystal structure of AsRhTi.
Let us show here that the valence and conduction bands of AsRhTi do cross and produce a multiloop NL structure. We plotted the GGA band structure of AsRhTi in Fig. 2b where one can readily find that the valence and conduction bands cross at the intermediate points of ΓX and ΓZ, indicating a NL lying in the invariant plane k_{y} = 0 of the mirror symmetry m_{y}. In order to demonstrate the NL structure more clear, we further plotted in Fig. 2c the 2D band structure of AsRhTi on the k_{y} = 0 plane with varying k_{x} and k_{z}. Since the band structure is symmetrical under the transformations k_{x} → −k_{x} and k_{z} → −k_{z}, only the region of k_{x} > 0 and k_{z} > 0 is used for simplicity. From the profile of the energy difference of three bands that has been projected on the bottom of Fig. 2c, one sees a central NL α surrounding the Γ point. Outside the NL α is a mNL β. Outmost is a third mNL γ that encloses the U point. It is the m_{y} provides the needed protection for the three mNLs. Interestingly, besides these inplane mNLs, we also find an isolate bandtouching point E outside the NL γ. This bandtouching point E is more obviously seen in the 2D energy band structure where it is highlighted by a dotted circle in Fig. 2c.
The detail of the NLs near the point E is revealed by a 3D profile of energy differences of bands in Fig. 2d, where a denser discretion of BZ is adopted to obtain the energy bands with the tightbinding hamiltonian constructed by the MLWF method. In this bandcrossing profile, an extra NL δ vertical to k_{y} = 0 plane is discovered. Since NL δ does not lie in any high symmetry path or plane, it must be an sNL that is protected by symmetry S. More detailly, the sNL δ is found to stick to the mNL β on a nexus point and the point E is the very point that NL δ penetrates through the k_{y} = 0 plane. Another interesting feature in Fig. 2d is that the mNL γ crosses the sNL and the two form a Hopflink. Previously Hopflinks made of two mNLs or two sNLs have been already proposed and it is argued that for the first one needs a minimal fourband effective model to describe the electronic structure of the Hopflink,^{35} while for the later one only needs a minimal twoband model.^{33,34} Here we demonstrate that a Hopflink can be made of a sNL and a mNL, and its corresponding electronic structure is correctly described by a minimal threeband model given below.
The reason that why we need a minimal threeband model to describe the multiloop NL structure of Fig. 1d and of Fig. 2d is obvious: The mNL is only produced by a pair of bands with opposite mirror parities. A robust and isolate band crossing point E on the invariant plane k_{y} = 0, however, is only possible when the crossing bands have equal mirror parities. Otherwise there would be a mNL passing through the isolate point.^{29} The general form of the threeband hamiltonian should be written as,
Since the system preserves the symmetry S, the imaginary part of the offdiagonal element H_{nm} (k) (n, m = 1, 2, 3, and n≠m) vanishes. On the other hand, The mirror symmetries, i.e., \(\tilde m_x\), \(m_y\), and \(\tilde m_z\), lay on the entries another constraint that the diagonal element H_{nn}(k) should be an even function of k_{x}, k_{y}, and k_{z}.^{15} For the offdiagonal entry H_{nm}(k) with n≠m, it becomes an even (odd) function of k_{i} (i = x, y, z) if the orbital n and m have the equal (opposite) mirror parities with respect to symmetry m_{i}.^{15} The above symmetry consideration helps us to reduce the hamiltonian of Eq. (1) to a simpler form up to a second order of k,
Here we supposed that the second and third orbitals have the equal mirror parities opposite to that of the first. A constant term \(H_{11}\hat I\) has been subtracted from the original Hamiltonian because of its irrelevance to the structure of NLs. The parameters of Eq. (2) are chosen dimensionless for simplicity. By choosing suitable values of parameters ε_{i}, a_{i}, b_{i}, c_{i}, and λ_{j} (i = 1, 2, 3 and j = 1, 2), the main features of multiloop NL structure of Fig. 2d are well reproduced as shown in Fig. 2e.
The nontrivial electronic structure of a NLSM is revealed by its topological surface states (SSs). In a slab, the projection of a NL in the bulk BZ onto the 2D BZ will divide it into regions of different topological orders characterized by Z_{2} topological charge v,^{57}
where n, k> denotes the Bloch eigenstate and k_{⊥} is the component of momentum normal to the slab. In the regions of v = 1, there exists ingap topological surface states at each k point, forming the so called 2D “drumhead” states.^{57} In Fig. 3a, b we have shown the surface band structures of AsRhTi on the (010) and (001) surfaces, respectively. The corresponding 2D profiles of density of state (DOS) at fixed energies of −30 and 0 meV are also plotted in Fig. 3c, d. In Fig. 3a and Fig. 3c, one finds SSs spread throughout the inner region enclosed by the projection of mNL β of Fig. 2d. Since the sNLs are normal to surface (010) (see in Fig. 2d), no SS is found at (010) surface for the sNLs. In contrast, on the (001) surface the projection of sNLs form two ellipses on ΓX and SSs link the two ellipses across the boundary of BZ (see in Fig. 3d). It is seen from Fig. 3d that the fermi surface also cuts some trivial bands, producing extra carrier pockets above and below the k_{y} = 0 plane as can be seen in Fig. 3d.
In the above discussions, we have not included the spinorbit coupling (SOC). The inclusion of SOC induces small gaps on the NLs at the scale of several meV and thus its effect can be ignored at room temperature. For compounds containing heavier elements below, such as SiIrTa, the SOC gap is not small that the NLSMs eventually are turned into topological insulators. We also checked the effect of electron interactions by adopting the hybrid density functional approximation (HSE06)^{58} and find the HSE06 result reproduces GGA result very well (see in the Fig. 2b). One finds that HSE06 calculation reproduces GGA band structure and makes the fermi surface even clearer by pushing down the valence band at Γ and U.
Diverse node lines structures in MgSrSitype crystals
The MgSrSitype crystals is a large family of binary and ternary crystals which contains more than 660 members. As expected that isostructure crystals may have similar electronic structure, we thus screened all 660 compounds to discover new NLSMs. More than 70 NLSMs are readily found and listed in Table 1, where the NLSMs are divided into several groups based on their chemical compositions. A variety of NLs structures were discovered. The NL structures for each group is similar. In Fig. 4 we plotted 5 representative NL structures and their corresponding fermi surfaces. From Fig. 4a, one sees PPtSc has a single NL loop lying in the k_{y} = 0 plane and its fermi surface takes a distorted toruslike shape (see in Fig. 4a). For SiNiZr, the NL extends across the boundary of BZ and one can see from Fig. 4b that some portions of the NL outside the BZ is folded back. The single NL of SiCoV shown in Fig. 4c, contrarily, lies on the k_{z} = 0 plane unlike those of PPtSc and SiNiZr in the k_{y} = 0 plane, indicating its protecting symmetry being \(\tilde m_z\). Both fermi surfaces of PPtSc and SiCoV take the simple toruslike shapes and show very clean fermi surfaces, a promising property for the experimental detection of their nontrivial electronic structures.
Interestingly, even more exotic NL geometries are found in AsFeNb and PNiNb. There exist multiple mNLs in the k_{x} = 0, k_{y} = 0, and k_{z} = 0 planes, together with sNLs sticking to the mNLs (see in Fig. 4d, e). For AsFeNb, its NLs form a novel cagelike structure. However, its fermi surface is dirty which is messed up by some trivial bands. For PNiNb, one finds an isolated mNL lies in the invariant plane k_{y} = 0, and off the plane NLs protected by \(\tilde m_x\) and \(\tilde m_z\) are found touching near Y point. Around the planes of k_{x} = ±k_{z}, there exist eight segments of sNLs sticking to the mNLs of k_{x} = 0 plane. Luckily the fermi surface of PNiNb, shown in Fig. 4j, is very clean and quite similar to the NLs structure of Fig. 4e. Therefore PNiNb can be a promising compound for exploring new NLSMs with exotic multiloop NL structure.
From the summarized result in Table 1, we find that the electron counting is a useful indicator for the search of NLSMs. The number of electrons, being totally 32 electrons/unit cell according to the counting scheme of Landrum et al.,^{54} are all the same for the AsRhTi^{G}, PFeV^{G}, PNiV^{G}, SiCoV^{G}, and SiNiTi^{G} groups, while the PPtSc^{G} has 40 electrons/unit cell. In other words, all the NLSMs listed in Tab. 1 has an count of 8 electrons per formula unit. According to a simple 8N rules of Zintl, one may expect that a NiSiTi should be an insulator or semiconductor.^{59} It is the strong bonding of transition metal atoms of SiNiTi leads to the semiemtal state rather than an insulating one.^{54} Here our highthroughput calculations and screening of NLSMs in MgSrSi series teach us that semimetals with filled octet orbitals could be good candidate of topological SMs.
It is worthy to note that many MgSrSitype crystals, though containing magnetic atoms, are actually nonmagnetic. For example, the ternary MMʹX (M = transition metal, Mʹ = late transition metal, X = main group element) compounds exhibit paramagnetic behaviors for M = Sc, Ti and V.^{54} Previously, TiCoP, ZrCoP, and VCoSi have already been characterized to be paramagnetic metallic conductors. The trends of nonmagnetic ground state of MgSrSitype crystal with openshell magnetic atoms is attributed to the strong bonding of MMʹ by Goodenough that the MMʹ bonding energy of the MgSrSitype ternary may dominate over the intraatomic Coulomb interaction which favors magnetic state in the freeatom limit.^{60}
In conclusion, we screen the MgSrSitype crystals and more than 70 compounds are found to host node line band crossing in their band structures. Due to the coexistence of reflection symmetry and inversion symmetry in the space group, AsRhTi is found to take a novel multiloop NL structure, in which a sNL protected by PT symmetry touches robustly with a mNL protected by mirror symmetry at some “nexus point”, and in the meanwhile a second mNL crosses with the sNL to form a Hopflink. A essential threeband k p model is provided to give an effective description of the low energy electrons on the Fermi surface. Topological surface states exhibiting the nontrivial NL structure is also demonstrated. Even more exotic multiloop NL structures are uncovered in AsFeNb and PNiNb subgroups. All the found NLSMs are consistent with 8N valence rules of Zintl, indicating the valence rules should be an useful indicator for the searching of topological SMs.
Methods
The first principles calculations were performed by the Vienna ab initio simulation package (VASP)^{61} and the projected augmentedwave (PAW) potential is adopted.^{62,63} The exchangecorrelation functional introduced by Perdew, Burke, and Ernzerhof (PBE)^{64} within generalized gradient approximation (GGA) is applied in the calculations. The energy cutoff for the planewave basis is set as 520 eV and the forces are relaxed less than 0.01 eV/Å. The positions of atoms are allowed to relax while the lattice constants of the unit cells are fixed to the experimental values documented in the Inorganic Crystal Structure Database (ICSD). The bandcrossings are calculated from tightbinding models which are constructed by using the Maximally Localized Wannier Functions (MLWF) method coded in WANNIER90.^{65} To give a more accurate description of electron interaction, hybrid density functional approximation is further adopted.^{58}
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
The work is supported by National Natural Science Foundation of China (NSFC) (Grants No. 11574215, No. 11575116, No. 11274359, and No. 11422428). H. M. W is also supported by the National 973 program of China (Grants No. 2018YFA0305700 and No. 2013CB921700), and the “Strategic Priority Research Program (B)” of the Chinese Academy of Sciences (Grant No. XDB07020100). The calculations in this work were performed on the supercomputers of Shanghai supercomputer Center and of the high performance computing center of Nanjing University.
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Q.F.L., R.Y., and H.M.W. conceived the study. J.L. and L.Y. wrote the necessary code, J.L., L.Y., and Q.F.L. carried out the calculations. Q.F.L. analyzed the data. All four authors contributed to writing the manuscript. J.L. and L.Y. contributed equally to this work.
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Correspondence to QiFeng Liang or Rui Yu.
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Further reading

Quantum oscillations and nontrivial topological state in a compensated semimetal TaP2
Physical Review B (2019)