Group-theoretical study of band nodes and the emanating nodal structures in crystalline materials

Abstract

Topological materials usually possess protected gapless states in either the boundary or bulk, exhibiting various properties such as spin-momentum locking, Klein tunneling, Fermi arcs and so on. Database searches using symmetry data at high-symmetry points have catalogued thousands of topological materials revealing a magnitude of band nodes (BNs) at high-symmetry points or lying within high-symmetry lines/planes. A complete mapping from symmetry data (namely, representation of little group) in any BN to the $k\cdot p$ model characterizing low-energy Hamiltonian around the BN (and from the $k\cdot p$ model to concrete BN, inversely), is expected to complete the characterization of all BNs and gapless states. Here we first review recent progress on classifying BNs by systematically and automatically constructing $k\cdot p$ models based on recently completed tabulation of all irreducible (co-)representation matrices of little groups of the 1651 magnetic space groups. As one indispensable input in constructing a symmetry-allowed and generic $k\cdot p$ model, the expansion order, has been carefully and systematically truncated for any BN to a reasonable nonzero integer, by comparing the emanating nodal structure (ENS, including nodal point, nodal line and nodal surface) near the BN obtained by the explicitly constructed $k\cdot p$ model and that by pure symmetry analysis using compatibility relations (CRs). Owing to the progress, we are able to summarize all 25 different configurations of ENS near BN required by CRs, provide a complete mapping from $k\cdot p$ model to its realization around BN, and the corresponding ENS by CRs in an accessible file, and also reveal the protection mechanism of additional nodal lines that escape conventional analysis by CRs and is only predictable by constructing $k\cdot p$ model. The symmetry-based classification results on all BNs could facilitate large-scale materials prediction and hold promise for realizing topological semimetals suitable for device applications.

1 Introduction

Various gapless low-energy excitations appear around topologically protected band node (BN) within the boundary or bulk Brillouin zone (BZ) in topological materials [1–13], such as strong topological insulators protected by time-reversal symmetry (TRS) [14–17], topological crystalline insulators [18] Weyl semimetals [19–22] and Dirac semimetals [23–26]. Space group (SG) has played a vital role in introducing fruitful topological phases such as mirror protected topological crystalline insulator [27], nonsymmorphic Dirac insulator with wallpaper fermions [28], high-order topological insulator [29–32], type II Weyl semimetals [33], type II Dirac semimetals [34–36], semimetals with unconventional multi-fold-degenerate excitations [37–42], semimetals with hourglass-shape band connectivity [43–48], nodal line [49–53] and nodal surface [54–58] semimetals, nodal chain semimetals [59], Hopf-link semimetals [60–65] and so on. The first-principles calculations combined with symmetry analysis and constructing effective models based on SGs have facilitated the experimental discovery of topological materials, e.g. Dirac semimetal Na₃Bi [66], Weyl semimetal TaAs [67, 68] and charge-2 fermions in CoSi [69–71]. Mathematically, 230 SGs, generated by various translation symmetries and rotation/inversion/rotoinversion/screw/glide symmetries, have been enumerated to characterize all possible sets of symmetry operations of arrangements of atoms in any three-dimensional (3D) crystalline materials [72]. For nonmagnetic materials, an additional TRS exists. Onset of specific magnetic ordering (commensurate) leads to different kinds of combinations of time-reversal and spatial operations that are subject to group homomorphism theorem, resulting in 1421 magnetic space groups (MSGs) in total: 230/674/517 type I/III/IV MSGs have been enumerated around 70 years ago [72]. With the 230 SGs with the additional TRS as 230 type II MSGs included, there are in total 1651 MSGs, and their representations are thus applicable for all crystalline materials (nonmagnetic/magnetic) in nature: More than 180 thousand materials structures have been categorized according to SG in Inorganic Crystal Structure Database (ICSD) [73], of which around 70 thousand might be magnetic.

The irreducible representations (irreps) or irreducible co-representations (co-irreps) of the 1651 MSGs are fundamental quantities generally applicable to any materials crystallized in the corresponding MSG, even without knowing any materials detail (such as lattice parameters, atomic species and positions), and can be utilized to derive generic results prior to realistic calculations or experimental measurements, thus expected to provide a unified understanding of seemingly diverse physical phenomena from the symmetry point of view. Powerful tool in materials predictions of a target property can be established using symmetry. Indeed, the complete classification of band topology based on all the 230 SGs/1651 MSGs in the frame of symmetry-indicator theory [74, 75] or topological quantum chemistry [76, 77], has been applied in developing non-target-oriented topological materials search schemes and subsequent database searches have revealed a great wealth of electronic materials with nontrivial band topology near the Fermi level [78–87]. Remarkably, a great portion of materials were shown to host band nodes (BN) (a point in the BZ where two branches of bands touch) whose characteristics (e.g. topological charge [88], the nodal structures such as nodal line [49–53] and nodal surface [54–58] that threads the BN) require a further identification. Such identification were usually case-by-case for one material or focusing on specific BN/symmetries (e.g. deriving two-dimensional chiral fermions around high-symmetry points (HSPs) by Maňes in the 230 SGs [89], unconventional fermions beyond high-energy classification by Bradlyn et al. [37] et al. using high-dimensional (3, 4, 6, 8-dimensional) irreps or co-irreps at HSPs), which is obviously unsuitable for a large-scale identification. In the beginning of 2021, through constructing symmetry-allowed $k\cdot p$ model around BN for all the type II 230 MSGs, Yu et al. [90] aggregated their results as a dictionary mapping the representation in any BN to the type and topological characteristics of the corresponding emergent particle, concluding all possible BNs that can be realized in nonmagnetic materials. However, the irreps/co-irreps for the 674 type III MSGs and 517 type IV MSGs have not been well-tabulated before 2021 [77, 91], thus hindering the exhaustive classification of BNs in these MSGs for magnetic materials.

Here we first review the recent progress aiming at characterization of all BNs that can occur in crystalline materials in the frame of group theory, which calls for a new algorithm or methodology suitable for constructing $k\cdot p$ models [92] in a large scale. In 2021, using the computed (co-)irrep matrices for all k points in the 1651 MSGs, the authors have proposed a method of exhaustively constructing any $k\cdot p$ model around any k point of the BZ [91]. The so-called elementary $k\cdot p$ blocks are proposed and all obtained for all the 1651 MSGs considering single-valued or double-valued representations (namely, in the spinless or spinful settings, respectively), thus reformulating any construction of symmetry-allowed $k\cdot p$ model which is originally equivalent with solving a set of constraint equations, to be simply a linear combination of the obtained elementary $k\cdot p$ blocks [91]. Then systematic construction of $k\cdot p$ models is feasible, and subsequently a complete classification of all BNs in the 1651 MSGs in both spinless and spinful settings was accomplished [93]. It is worth pointing out that Refs. [94, 95] also provided a complete tabulation of possible topological properties on the emergent particles around BNs by constructing $k\cdot p$ models based on 674 type III MSGs and 517 type IV MSGs, respectively. By these complete results, given a concrete BN whose representation can be known (e.g. from first-principles calculations), one only need to look up into the tabulations in Refs. [90, 91, 93–95] and quickly retrieve the $k\cdot p$ model and possible topological charge. Furthermore, owing to that the $k\cdot p$ models have been exhausted for all BNs, given a $k\cdot p$ model to realize for BN, the task is thus reformulated to be finding/designing materials/artificial structures crystallized in any of the MSGs which can realize this model, which might already be proposed to own interesting properties (such chiral anomaly [96] for Weyl fermions). Next we review the classification of all BNs in Sect. 2, starting from briefly describing the formation of all possible BNs in crystalline materials in Sect. 2.1, then describing the nodal structures that emanating from the BNs in Sect. 2.2 to constructing all $k\cdot p$ models up to reasonable expansion orders that give solutions repeating the emanating nodal structures (ENSs) required by compatibility relations (CRs) in Sect. 2.3. We also provide an example on realizing a target $k\cdot p$ model in Sect. 2.4. Then in Sect. 3, we showcase all different 25 configurations of ENSs for all possible BNs, and in Sect. 4, we use three $k\cdot p$ models to demonstrate the unstable/stable ENS, which can be gapped/unaffected by higher order $k\cdot p$ terms. In Sect. 5, we provide all $k\cdot p$ models that predict nodal line(s) that cannot be inferred by a purely symmetry analysis using CRs.

2 Review on classifying all BNs in the 1651 MSGs

2.1 Formation of various BNs

In the beginning, we first overview various mechanisms of BN formation. One BN should be formed by at least two neighboring bands, which touch at the BN and these bands should split in at least one direction in the vicinity of the BN. The number of these bands thus defines the degeneracy of the BN (denoted by d). Hence, in general, the BN can be formed from one degenerate (co-)irrep, or two different/identical (co-)irreps ((co-)irrep denotes representation of the little group of the BN). We use irrep/co-irrep once the little group contains no/at least one antiunitary operation, respectively. Symmetries might constrain the energy bands to touch, it can also constrain the low-energy Hamiltonian around the BN (and constrain the dispersion [97–99]), by the transformation matrices of all elements in the little group from the representation in the BN. The constraints leave a number of parameters allowed to be nonvanishing, which can be determined by fitting with first-principles/experimental results. Any other MSG operation not belonging to the little group would relate the BN with another BN, and thus only independent BNs in the irreducible BZ need to be studied. Let’s denote the single-electron spectrum by $E_{n\vec{k}}$ whose wavefunction is $\psi _{n\vec{k}}=e^{\mathrm{i}\vec{k}\cdot \vec{r}}u_{n\vec{k}}$. The band index n can be categorized by the irreps (or co-irreps) of the little group of k⃗, $G(\vec{k})$ according the Schur’s lemma [72]: Each degenerate energy level at k⃗ (corresponding to several band indices) corresponds to one (co-)irrep without accidental degeneracy. Such degeneracy might contribute to one BN once the bands split near k⃗, as shown in Fig. 1(a) where the BN is formed at one HSP (d is thus equal to the degeneracy of the energy level). Besides, two branches of bands along high-symmetry plane (HSPL)/high-symmetry line (HSL) with two different (co-)irreps (whose dimensions are denoted by $d_{1}$ and $d_{2}$, respectively) can touch ($d=d_{1}+d_{2}$). This can be understood as follows. The Hamiltonian along the HSPL/HSL involving the two branches of bands can be written in the form as $\left [ \begin{array}{c@{\quad}c} a_{1} I_{1}&0 \\ 0&a_{2} I_{2}\end{array} \right ]$ where $a_{1}$ and $a_{2}$ are real functions and $I_{1}$ and $I_{2}$ are $d_{1}\times d_{1}$ and $d_{2}\times d_{2}$ identity matrices, respectively. Note that such form of Hamiltonian is required by symmetry constraints of little group of the HSPL/HSL. Hence, there is only one equation left: $a_{1}=a_{2}$: For HSPL, the solution of the equation is thus a nodal line within the HSPL, as shown in Fig. 1(b). In this case, any point in the nodal line can be thought as a BN and the nodal line is the ENS (defined to nodal line/surface or both threading the BN owning the same degeneracy as the BN). For HSL, one solution might be found for the equation $a_{1}=a_{2}$ corresponding to one point in the HSL which acts as the BN, and the such BN might be a nodal point (the ENS is simply the BN) or lie in a nodal line/several nodal lines within one/several neighboring HSPL/HSPLs containing the HSL, which can be determined by CRs as shown below.

Figure 1

(a) We schematically show the BN (red sphere) formed by two branches of bands touching at one HSP. Since HSP has no variables (e.g. $(0,0,0)$, rather than HSL/HSPL, e.g. $(0,0,w)/(u,v,0)$ with 1/2 variables), such BN is only be formed from one degenerate (co-)irrep. (b) Two branches of bands cross with opposite mirror/glide eigenvalues and a nodal line (in blue) is then formed in an HSPL ($k_{x}$-$k_{y}$ plane)

It is worth pointing out that in generic point (GP) and HSPL, two identical (co-)irreps can touch understandable as follows. In this case, the Hamiltonian involving the two branches of bands can be written in the form as $\left [ \begin{array}{c@{\quad}c} \epsilon _{a}&V \\ V^{\dagger}&\epsilon _{b}\end{array} \right ]$ where $\epsilon _{a}$ and $\epsilon _{b}$ are two identity matrices of the same size, and V represents the symmetry-allowed term as the hybridization of the two branches of bands. Obviously, we need to solve at least 2 equations ($\epsilon _{a}=\epsilon _{b}, V=0$, where $V=0$ might correspond to more than one equations) for the BN (this then rules out the possibility that such BN can be found in an HSL, which has only one free variable along the HSL to be determined). For the BN to appear by two identicial (co-)irreps, it can be known that the degeneracy of the BN can only be two, otherwise there would be more than 3 equations left: It can be located at GP (when PT symmetry is broken: P is spatial inversion and T is time-reveral operation) [19, 100], or when PT symmetry is present in the spinless setting, or lies in an HSPL when the HSPL owns $C_{2}T$ symmetry ($C_{2}$ denotes the two-fold rotation around the normal direction of the HSPL). The corresponding $k\cdot p$ models can be easily constructed as follows.

For the BN at GP, once PT is not present, the $k\cdot p$ model is subjected to only translation group symmetry and can be written as $\vec{A}\cdot \vec{q}\sigma _{0}+\sum _{i,j}R_{ij}\sigma _{i}q_{j}$ to the 1st order in q⃗, where $i,j=x,y,z$, $\vec{A}, R_{ij}$ are real parameters and $\sigma _{i}$ and $\sigma _{0}$ are Pauli matrices (q⃗ is measured from the BN). Note that this $k\cdot p$ model is the most generic form of two-band model to the 1st order of which the 12 parameters in total are not constrained by any point symmetry, and the ENS can be found by solving for q⃗ satisfying the linear equations $\sum _{j} R_{i,j}q_{j}=0, i=x,y,z$. It is then easy to known that the solution is $\vec{q}=(0,0,0)$ as along as $|R|\ne 0$: In this case, the ENS is a nodal point, which is simply the BN itself. One might expect that ENS of higher dimensionality (e.g. nodal line/surface) to be found once $|R|=0$, which actually requires more symmetry constraints or meticulous fine tuning of parameters: The former requirement renders the BN to be at high-symmetric k point (not at GP). For the latter requirement, even $|R|=0$ can be tuned to occur accidentally, $|R|$ would be ≠0 by any weak perturbation so that the ENS of higher dimensionality is unstable. Note that higher-order $k\cdot p$ terms can be allowed for such Weyl point at GP, that can be neglected insofar as only the low-energy behavior is concerned (q⃗ is restricted within a neighboring vicinity). Furthermore, though higher order terms can be allowed by symmetry, other bands might be involved since the higher order terms dominate against the linear order term in the region relatively far from the BN. However, higher order terms might give rise to interesting consequences such as larger topological charge [101], which deserves a future study. Other than the Weyl point at GP, nodal line can also appear in GP (namely, all the points constituting the nodal line are GPs): Spinless PT symmetry (namely, PT squares to identity) constrains the $k\cdot p$ model to be $\vec{A}\cdot \vec{q}\sigma _{0}+\sum _{i\ne 2,j}R_{ij}\sigma _{i}q_{j}$ (we choose the basis so that PT is represented by $\sigma _{0}\mathcal{K}$, $\mathcal{K}$: complex conjugate), and there definitely exists a nodal line solution for the ENS given by $R_{xx}q_{x}+R_{xy}q_{y}+R_{xz}q_{z}=0,R_{zx}q_{x}+R_{zy}q_{y}+R_{zz}q_{z}=0$, as the PT protected nodal line in the spinless setting [50]. Note that solution of nodal surface might appear if $\frac{R_{xx}}{R_{zx}}=\frac{R_{xy}}{R_{zy}}=\frac{R_{xz}}{R_{zz}}$, but it needs a fine tuning obviously and should be protected to be table against perturbations by additional symmetry. For the BN in an HSPL protected by $C_{2}T$ symmetry in both spinless and spinful settings, the $k\cdot p$ model around the BN can be written as $(A_{x} q_{x}+A_{y} q_{y})\sigma _{0}+ (R_{xx}q_{x}+R_{xy}q_{y}) \sigma _{x}+(R_{yz}q_{z})\sigma _{y}+(R_{zx}q_{x}+R_{zy}q_{y})\sigma _{z}$ (we choose a basis so that $C_{2}T$ is represented by $\sigma _{0}\mathcal{K}$ noting that the HSPL should contain only one 1 dimensional co-irrep and the $k\cdot p$ model is expanded to the 1st order) for which the ENS is simply the BN itself. Or when PT is a symmetry in the spinless setting in addition (note that spinful PT symmetry requires bands are at least two-fold degenerate), the $k\cdot p$ model then becomes $(A_{x} q_{x}+A_{y} q_{y})\sigma _{0}+ (R_{xx}q_{x}+R_{xy}q_{y}) \sigma _{x}+(R_{zx}q_{x}+R_{zy}q_{y})\sigma _{z}$ for which the ENS is a nodal line which is constrained within the HSPL.

All in all, BN can be diagnosed either by computing the representation of energy bands (the BN corresponds to degenerate (co-)irrep of HSP/HSL/HSPL, or two different (co-)irreps of HSL/HSPL) or computing Berry phase/Chern number [88] (the BN can be located at GP/HSPL with two identical (co-)irreps). The later cases have been exhausted above. Hence, a complete classification of all BNs is then reduced to find all possible degenerate (co-)irreps of HSPs/HSLs/HSPLs (attention should be paid that when the spinful PT symmetry ($(PT)^{2}=-1$) exists, the degeneracy of BN should be >2), and all combinations of two different (co-)irreps of HSLs and HSPLs. Since the (co-)irreps of little groups, $G(\vec{k})$, for all the 1651 MSGs have been tabulated [77, 91], and thus all BNs can be collected for all the 1651 MSGs considering both spinless and spinful settings. Note that in Ref. [93], we obtained 62717 and 36253 spinless and spinful BNs, respectively. These BNs are counted for all independent HSPs/HSL segments/HSPL polygons. Here, we further reduce the numbers of BNs to be 46855 and 26556 for the spinless setting and spinful setting, respectively, by removing the coinciding HSL segments/HSPL polygons. Once knowing the (co-)irreps in any BN, one can then construct a $k\cdot p$ model expanded to any order of q⃗ in principle which has been made simple and routine owing to Ref. [91]. On the other hand, the CRs between the wavefunctions at k⃗ and its vicinity $\vec{k}+\vec{q}$ constrain the band connectivity, and thus can tell how bands split near the BN (we denote the band splitting patter as CR-BSP) so that we can know along which direction(s), the band touching can be maintained as the BN (thus contributing to nodal line(s) in the ENS) or along which plane(s), the band touching can be maintained (thus contributing to nodal surface(s) in the ENS). Through a systematic investigation, all possible ENSs required by CRs near BNs can be found: Next, we will overview the ten CR-BSPs as listed in [93], based on which all possible configurations of CR-BSPs will be described in Sect. 3 and schematically shown in Figs. A1-A5 of Appendix.

2.2 Ten CR-BSPs

As stated above, the ENS near BN can be inferred from a pure symmetry analysis by CRs (in most cases, see Sect. 5 for exceptions) with no need of constructing an explicit $k\cdot p$ model. This itself avoids possible unstable ENS obtained by $k\cdot p$ model since the ENS from an explicit $k\cdot p$ model up to a unreasonable order could give rise to ENS that is gapped by including higher-order $k\cdot p$ terms. On the contrary, the ENS derived from CRs always exist for the $k\cdot p$ model up to any expansion order. A criterion of choosing a reasonable $k\cdot p$ expansion order, a cutoff order, for a BN is then established: Any larger expansion order than the cutoff order would not reduce the number and dimensionality of ENS any more while a less expansion order definitely gives redundant ENS (thus the cutoff order and any large orders are the reasonable expansion orders). Before we summarize the results of $k\cdot p$ models all up to reasonable orders, we firstly describe ten CR-BSPs [93]: We name the CR-BSP in the form of X-Y, where X describes the location of the BN, which can be P, L and PL, standing for HSP, HSL and HSPL, respectively; Y describes the ENS which can be nodal point, nodal line(s) along neighboring HSL(s)/lying within neighboring HSPL(s), nodal surface(s) along neighboring HSPL(s) or NSL for coexisting nodal line(s) and nodal surface(s) along neighboring HSL(s) and HSPL(s), respectively. We summarize the positions of BNs (such as HSP/HSL/HSPL) and dimensionality of ENSs for all ten CR-BSPs in Table 1.

Table 1 Summary of properties of ten CR-BSPs. The second row contains the location of each CR-BSP, which can also be identified by the letter before “-” in the name of each CR-BSP: P, L, PL for the location being HSP, HSL, HSPL, respectively. The third row contains the number of (co-)irreps in the BN for each CR-BSP. In the fourth row, we provide the dimensionality for the ENS, 1 for nodal line and 2 for nodal surface. In the last row, typical CR is given to guarantee the corresponding ENS, whose location and dimensionality can be identified by the CRs. Here, D denotes the (co-)irrep or (co-)irreps in the BN. For P-NP/L-NP, CR to any direction/plane shows a splitting pattern as $D\rightarrow D'+D''+\cdots$ . Note that for L-NP, the CR to the HSL where the BN is located, takes the form as $D\rightarrow D'+D''$ where $D'$ and $D''$ are two different (co-)irreps of this HSL composing the BN.$D_{\textrm{L}}$/$D_{\textrm{PL}}$ denotes one (co-)irrep in some HSL/HSPL composing the ENS, respectively. Note that for L-sNL/PL-NS, $D_{\textrm{L}}$/$D_{\textrm{PL}}$ is simply one (co-)irrep of the HSL/HSPL where the BN is located. For L-NL, $D'_{\textrm{PL}}$ and $D''_{\textrm{PL}}$ are two different (co-)irreps for one HSPL, containing the HSL where the BN is located, and the ENS is a nodal line lying in the HSPL composed by the two different (co-)irreps. For PL-NL, $D'_{\textrm{PL}}$ and $D''_{\textrm{PL}}$ are two different (co-)irreps for the HSPL where the BN is located, and the ENS is a nodal line lying in the HSPL composed by the two different (co-)irreps

Consider X=P first. The CR-BSP can be P-NP, P-NL, P-NS and P-NSL. Once for all neighboring HSLs/HSPLs, the CRs take the form: $D\rightarrow D'+D''\ldots$ , meaning that the bands away from the BN (whose (co-)irrep is denoted by D) should splits into several branches of bands whose (co-)irreps are denoted by $D',D'',\ldots$ , the CR-BSP is thus named P-NP ($D', D''$ denote that (co-)irreps of the neighboring HSL/HSPL). On the contrary, the CRs from the BN to neighboring HSL/HSPL might take the form as $D\rightarrow D'$, meaning that the bands along the HSL/HSPL correspond to the (co-)irrep $D'$ (whose dimension is the same as D, the degeneracy of the BN). Once only HSL can allow such kind of CR, the CR-BSP is then P-NL for which the corresponding ENS is composed of the only nodal line along the corresponding HSL. On the other hand, once only HSPL can allow such kind of CR ($D\rightarrow D'$), the CR-BSP is then P-NS with the ENS being the nodal surface along the corresponding HSPL. For P-NSL, both HSL and HSPL in the vicinity can allow the CR of the form $D\rightarrow D'$. Note that coexisting emanating nodal lines and nodal surfaces as the ENS can only occur for BN at HSP. Then consider X=L, namely, the BN is located in an HSL. When only one (co-)irrep is considered for the BN, the CR-BSP can be L-NS and L-sNL. When the CR from the BN to a neighboring HSPL is in the form of $D\rightarrow D'$, this HSPL could become the ENS and the CR-BSP is named L-NS, otherwise the CR-BSP is L-sNL since the CR from the BN to the HSL hosting BN is naturally in the form of $D\rightarrow D'$ and the ENS is composed of a straight nodal line (coinciding with the HSL). Note that in order to distinguish from another kind of emanating nodal line for BN in HSL as described immediately below, we use “sNL” to emphasize that this nodal line is straight and simply along the HSL where the BN is located. Besides, when two different (co-)irreps ($D_{1}$ and $D_{2}$) are considered, the CRs from the BN to this HSL hosting the BN definitely require that the bands split along this HSL. When the CR from the BN to some neighboring HSPL containing the HSL takes the form as: $D_{1}\rightarrow D_{1}', D_{2}\rightarrow D_{2}'$ where $D_{1}'\ne D_{2}'$, the corresponding CR-BSP is named L-NL, for which the emanating nodal line lies within this neighboring HSPL as the ENS. On the contrary, when none of the neighboring HSPLs can host nodal line as the ENS, the CR-BSP is thus L-NP and the ENS is thus a nodal point, namely, the BN itself. Finally consider X=PL, namely, the BN is located within an HSPL. When only one (co-)irrep is considered, the CR-BSP is then PL-NS, and the emanating nodal surface simply coincides with this HSPL. On the other hand, when two different (co-)irreps are considered, the CR-BSP is then PL-NL and the emanating nodal line lies in this HSPL as the ENS for the BN.

2.3 Systematic construction of $k\cdot p$ models around BNs

Other than the ENS predicted purely from CRs (the ten CR-BSPs), constructing $k\cdot p$ model around BN could provide more quantitative results such as the energy dispersion (e.g. linear or quadratic) and topological charge. Denote the $k\cdot p$ model by $\mathcal{H}(\vec{q})$ (q⃗ is the wave vector measured from the BN at k⃗). Its form can be found by all solutions of the symmetry constraint equations: $D(R)\mathcal{H}(R^{-1}\vec{q})D(R)^{\dagger}=\mathcal{H}(\vec{q})$ where $R\in G(\vec{k})$ (little group of k⃗) and $D(R)$ is the representation matrix for the basis wavefunctions at k⃗ in the BN. The expansion order of the $k\cdot p$ model should be designated firstly, and the CR-required ENS can be applied to guide choosing a reasonable expansion order of $k\cdot p$ model around BN, as stated above. For example, when the CR-BSP is L-NL, it requires that the ENS should be nodal line(s) lying within the neighboring HSPL(s). Since the $k\cdot p$ model up to any order should respect the CRs, the solution for ENS by the explicit $k\cdot p$ model should contains the ENS by CRs. However, redundant ENS (which is unstable and can be gapped by higher order $k\cdot p$ terms) might be found. Any larger expansion order than the cutoff one would not eliminate the CR-required ENS (at most slightly deforming the emanating nodal line lying in HSPL, for CR-BSPs L-NL and PL-NL). Here we collect 2574 $k\cdot p$ models (denoted by $\mathcal{H}_{i}, i=1,2,\ldots ,2574$) with different forms from Ref. [93] to describe the 73411 BNs in total (see Table 2 and Table 3 for the statistics for the BN degeneracy and the corresponding $k\cdot p$ expansion cutoff order, respectively). We have also packaged all these 2574 $k\cdot p$ models and all of the realizations in the MSGs for both spinless and spinful settings in a file named “BN_2574kp.mx”, which can be imported by Mathematica. Figures A6, A7 of Appendix demonstrate the results for $\mathcal{H}_{1201}$ and Figs. A8, A9 of Appendix demonstrate the results for $\mathcal{H}_{2421}$. From a $k\cdot p$ model, $\mathcal{H}_{i}$, the realization of the model in BN (MSG, k⃗ point and setting (spinless or spinful)) can be retrieved easily. By “BN_2574kp.mx”, we also provide the ENS for each $k\cdot p$ model by listing the direction(s)/normal direction(s) of the neighboring HSL(s)/HSPL(s) that can be the ENS or accommodate the ENS. Furthermore, in Table 4, we list the emergent particles classified in Ref. [90] that can appear around HSPs in the 230 SGs (with an additional TRS) and the corresponding $k\cdot p$ models we constructed as a comparison. These explicit $k\cdot p$ models can be utilized to predict the magnitude of band gaps along unstable ENS as we demonstrate in Sect. 4.

Table 2 Statistics on the number of $k\cdot p$ models $\mathcal{H}$ for each possible degeneracy of band node

Table 3 Statistics on the number of $k\cdot p$ models $\mathcal{H}$ for each $k\cdot p$ cutoff order

Table 4 28 types of emergent particles around HSPs for the 230 SGs with an additional TRS. The abbreviated and full name of each emergent particle [90] is provided in the 1st and 2nd column respectively. We provide the realization by $\mathcal{H}_{i}$ (with i shown in the table) for each emergent particle in spinless setting or spinful setting in the 3th or 4th column. Note that not all possible $\mathcal{H}_{i}$’s are provided due to the limited space. “-” means that realization is not allowed at HSP for a given emergent particle in the corresponding setting

2.4 Realization of charge-4 Weyl point

The logic of classifying all BNs based on MSG is schematically shown in Fig. 2: Given a concrete material structure, one can find concrete symmetry operations that constitute a symmetry group (isomorphic with an MSG) and then enumerate all possible BNs by the symmetry group for this material. Such enumeration is not limited to the given material, but general to any material belonging to this symmetry group/MSG. Note that MSGs can be listed [72] mathematically even without knowing concrete materials structures, and thus all possible BNs can be listed by the 1651 MSGs in total and then characterized [90, 91, 93–95]. In this part, we demonstrate how to utilize the results of all BNs to find realistic materials realizations of a target low-energy excitation. Suppose one might be interested in realizing a BN which is two-fold degenerate with $k\cdot p$ model expanded up to at least 3rd order: Expanding to lower orders would lead to additional ENS, along which tiny gaps should be opened $\sim q^{3}$ and large Berry curvatures are anticipated to result in a large topological charge (= Chern number obtained by integrating Berry flux in a sphere surrounding and close to the BN). Of the 2574 $k\cdot p$ models, there are 28 $2\times 2$ models expanded up to 3rd order whose CR-BSP is P-NP or L-NP. By Tables 10 and 9, half of these models exhibit additional nodal line threading the corresponding BN and the rest carries topological charge 3 or 4, as listed in Table 5.

Figure 2

Logic of exploiting MSG to classify and predict BNs. From a given material structure, one can obtain its MSG and the corresponding BN classification based on the MSG. Exhausting all possible BNs that can be realized in this MSG (not limited to this materials structure) is suitable for predicting a target BN in any material crystallized in this MSG. In fact, the BN classification can be done directly based on all possible abstract MSGs [90, 91, 93–95], and a category of all possible BNs realized in crystalline material can be constructed full of Weyl point, Dirac point, BN with Weyl nodal line and so on. To realize a BN, one can search for materials belonging to any MSG that can realize the BN

Table 5 All the 28 $2\times 2$ $k\cdot p$ models $\mathcal{H}_{i}$’s corresponding P-NP/L-NP with cutoff order being 3. Half of them allows additional nodal line (see Tables 9 and 10) and thus the topological charge from Chern number around the BN cannot be well-defined. For the rest models, the charge is found to be 3 or 4

Next step is to utilize the realizations in MSGs of these $k\cdot p$ models, and here we take the charge-4 Weyl point to illustrate: We list all corresponding HSPs of MSGs realizing the $k\cdot p$ models $\mathcal{H}_{i}, i=1201, 1215, 1257, 1302$ and 2374 in both spinless and spinful settings in Table 6. It can be quickly found that for type II MSGs (applicable for nonmagnetic materials), the charge-4 Weyl point can be realized only in the spinless setting as also predicted in Refs. [102, 103]. One natural choice is taking phonon as the platform to realize the spinless charge-4 Weyl point with an additional TRS. As a matter of fact, charge-4 Weyl point has been experimentally realized in artificial structures crystallized in SG 207 [104–106]. Ref. [107] performed calculations on the ICSD and catalogue all frequencies of all charge-4 Weyl points in phonon spectra of 128 nonmagnetic materials. On the other hand, TRS should be broken in the spinful setting for the charge-4 Weyl point, e.g. in the electronic band structures with spin-orbit coupling (SOC) included (type I/IV MSGs). As mentioned before, the $k\cdot p$ model for the charge-4 Weyl point need to take into account the 3rd (at least) order terms. Indeed, once only expanded to the 2nd order, the corresponding $k\cdot p$ model should exhibit nearly nodal lines along directions $\vec{e}_{4},\vec{e}_{5},\vec{e}_{6},\vec{e}_{7}$ (these directions can be found in Table 8) as shown in Fig. 3 where tiny gaps [108, 109] proportional to $q^{3}$ are anticipated by the $k\cdot p$ model, as also verified in concrete materials by first-principles calculations in Ref. [107]. These nodal lines can be viewed as protected by a fictitious PT symmetry, which is not genuinely present in the MSG, thus a quasi-quantized Berry phase ≈π can be found. Such PT symmetry can be identified once the $k\cdot p$ model is expanded to order less than 3, which applies in the zone very close to the charge-4 Weyl point. For the origin of the 3rd $k\cdot p$ term, one usually need a concrete analysis for realistic system. We provide a six-band toy model in Appendix, where the 3rd $k\cdot p$ term is related with the nearest neighbor hopping amplitude.

Figure 3

For charge-4 Weyl point, four straight and nearly nodal lines can be found along $\vec{e}_{i}, i=4,5,6,7$ to carrying quasi-quantized Berry phase ≈π. Take $\mathcal{H}_{1201}$ as an example which realizes charge-4 Weyl point. Setting $q_{x}^{2}=q_{y}^{2}=q_{z}^{2}$ (for the four straight lines) in the $k\cdot p$ model $\mathcal{H}_{1201}$ gives , which means that along the four lines, only tiny gaps $=2R_{4}q_{x}q_{y}q_{z}$ are opened

Table 6 Realization of charge-4 Weyl point in both spinless and spinful settings: The chagre-4 Weyl point can be formed by the BN at HSP. All the MSGs (in the form of “type,X.Y”) and the names and coordinates of the HSPs are provided, expected to facilitate materials predictions by matching the crystal structure symmetry with one MSG in this table

3 25 different ENS configurations

By exhaustively studying all ENSs for all BNs in the 1651 MSGs for both spinless and spinful settings based on CR-BSPs, it is concluded that there are only 25 different configurations of ENSs that are required by CRs. All possible configurations for the ten-CR-BSPs are schematically shown in Figs. A1, A2, A3, A4 and A5 of Appendix: These 31 configurations reduce to 25 ones after removing duplicates. For CR-BSP P-NL, the number of nodal lines, denoted as $n_{1}$, can take $1-7, 9, 13$ (see Table 7) and all possible high-symmetric directions are listed in Table 8. We also depict all the corresponding schematic diagrams in Fig. A1(b) and Figs. A4 and A5 of Appendix. The number of nodal surfaces for P-NS, denoted as $n_{2}$ can take 1,2,3 (see Fig. A1(d) and Figs. A2(a,b) of Appendix). For P-NSL, the numbers of the coexisting nodal lines and surfaces along the neighboring HSLs and HSPLs, $(n_{1},n_{2})$, can take three combinations: $(n_{1},n_{2})=(1,1), (2,2)$ and $(4,3)$ (see Figs. A2(d,e,f) of Appendix, respectively). For L-NS, see Figs. A1(e) and A2(c) of Appendix for the two only possible ENS configurations. Besides, Figs. A3(b-f) of Appendix display all possible ENS configurations for L-NL. The schematic diagrams for PL-NS and PL-NL can be found in Figs. A1(f) and A3(a) of Appendix. Note that the ENSs for CR-BSPs P-NL, P-NS, P-NSL, L-sNL, L-NS and PL-NS are all extended in the BZ since the ENSs are along HSL/HSPL. Interestingly, for P-NL, the maximal number of the straight and extended nodal lines along neighboring HSLs is found to be 13 (the directions are actually $\vec{e}_{1}$-$\vec{e}_{13}$ as shown in Table 8), which occurs in cubic systems. Besides, the maximal number of straight and extended nodal lines for P-NL that lie within the same plane is 6 for hexagonal system (along $\vec{e}_{1},\vec{e}_{2},\vec{e}_{14},\vec{e}_{15},\vec{e}_{16}, \vec{e}_{17}$ in Table 8). These 25 ENS configurations can be identified by directly computing the representation of the BN, expected to be applied to find materials realizations efficiently.

Table 7 Ten CR-BSPs and the number of possible neighboring HSLs/HSPLs required by CRs to be or accommodate the ENS. For P-NP and L-NP, CRs require that the ENS is simply the BN itself. For P-NL, P-NS, P-NSL, L-sNL, L-NS and PL-NS, the ENS simply consists of the neighboring HSL/HSPL(s). Note that for L-sNL and PL-NS, the HSL and HSPL where the BN is located are the ENS, respectively. The schematic diagram for each CR-BSP is also linked to Figs. A1,A2,A3,A4,A5 in the last row

Table 8 17 high-symmetric directions as expressed in Cartesian coordinates. HSL (HSPL) might be along (perpendicular to) one of these directions

4 Estimation of band gaps along unstable ENS

Other than knowing all possibilities on the ENS configurations by the complete classification [93] shown above, here we demonstrate through examples that the band gaps opened on unstable ENS can be estimated (such kind of ENS can only appear for the $k\cdot p$ model up to a smaller order than the cutoff order). In Fig. 4, we demonstrate the ENS evolution by varying the $k\cdot p$ models for $\mathcal{H}_{480}$, $\mathcal{H}_{990}$ and $\mathcal{H}_{1226}$, whose explicit forms (the constant term has been removed, not affecting the ENS solution) are provided in Eqs. (1), (3) and (2), respectively. For $\mathcal{H}_{480}$, it has to be expanded up to at least 4th order, with the corresponding ENS solution being four nodal lines (the blue lines) in Fig. 4(a) lying in the neighboring HSPLs whose normal directions are $\vec{e}_{1},\vec{e}_{2},\vec{e}_{8},\vec{e}_{9}$. These nodal lines are consistent with the CR-BSP (L-NL). For smaller expansion orders, there definitely exists additional ENS: As shown in Fig. 4(a), the ENS for the expansion order ($1,2,3$) is a nodal surface (red, green and orange surfaces, respectively). It is interesting to note that one can estimate the magnitude of gaps opened in these nodal surfaces: The gaps in the red, green surfaces and the orange surface (except the blue line) should be of order $q^{2}$, $q^{3}$ and $q^{4}$ [108, 109], respectively. Besides, the nodal lines as required by the CRs that lie in the four HSPLs can be deformed very slightly by including higher order (>4) $k\cdot p$ terms. Then for $\mathcal{H}_{1226}$ whose CR-BSP is P-NSL and which is expanded up to the 5th order, the stable ENS contains four straight nodal lines along $\vec{e}_{4},\vec{e}_{5},\vec{e}_{6},\vec{e}_{7}$ (blue lines in Fig. 4(b)) and three flat nodal surfaces whose normal directions are $\vec{e}_{1},\vec{e}_{2},\vec{e}_{3}$ (red planes in Fig. 4(b)). It can be easily found that the $k\cdot p$ model (the constant term being subtracted) only contains the 5th order terms based on Eq. (2). In this case, the gaps opened in the vicinity of the BN should be of $q^{5}$ order [108, 109] (expect the ENS, where the gaps can be viewed to be $q^{\infty}=0, q<1$).

Figure 4

The evolution of ENS by enlarging the $k\cdot p$ expansion order for $\mathcal{H}_{480}$ (a), $\mathcal{H}_{1226}$ (b) and $\mathcal{H}_{990}$ (c). The BN is at the origin (O). The cutoff $k\cdot p$ expansion order is chosen to be 4, 5 and 6, respectively, and further enlarging the expansion order would not change the number and dimensionality of the ENSs as predicted by CRs. In (a), the $k\cdot p$ model up to 1st order, 2nd order, 3rd order would all give nearly nodal surface (in red, green and orange) and thus it has to be expanded up to at least 4th order, the corresponding ENS being four nodal lines lying in four neighboring HSPLs. Note that further enlarging the $k\cdot p$ order might slightly modify the shape of these nodal lines. For (b), the stable ENSs are four straight nodal lines and three nodal surfaces whose shapes cannot be modified by further enlarging the $k\cdot p$ order. Only the expansion order takes 5 can the $k\cdot p$ model gives these nodal lines and surfaces otherwise, the ENS is actually a nodal volume for any smaller order. For (c), as the case in (a), there should exist 6 nodal lines lying in the neighboring HSPLs by CRs, which requires that the $k\cdot p$ model should be expanded up to at least 6th order. Any $k\cdot p$ model up to a smaller order would give nearly nodal surface (which is in red, green, orange, purple and gray, respectively, for the $k\cdot p$ model up to the 1st, 2nd, 3rd, 4th and 5th order). Note that there exist another 6 nodal lines threading another BN located as indicated by the red arrow, which might connect with the 6 nodal lines that thread the BN at origin or not (if not, these nodal lines are then not ENSs associated as the BN at the origin)

For $\mathcal{H}_{990}$ expanded up to the 6th order, whose CR-BSP is L-NL, the stable ENS should contain 6 nodal lines (blue lines in Fig. 4(c)) lying in six neighboring HSPLs (whose normal directions are $\vec{e}_{1},\vec{e}_{2}, \vec{e}_{14},\vec{e}_{15},\vec{e}_{16}$ and $\vec{e}_{17}$). All smaller orders would result in redundant ENS: The $k\cdot p$ models up to the 1st, 2nd, 3th, 4th and 5th orders all give nodal surface (red, green, orange, purple and gray surfaces in Fig. 4(c)), respectively. The gaps in these nodal surfaces can be estimated to of order, $q^{2}, q^{3}, q^{4}, q^{5}, q^{6}$, respectively [108, 109].

$$\begin{aligned} \mathcal{H}_{480}={}&\sigma _{x}[R_{10} q_{x} q_{y} \left (q_{y}^{2}-q_{x}^{2} \right )]+\sigma _{y} [R_{11} q_{x} q_{y} \left (q_{y}^{2}-q_{x}^{2} \right )] \\ &{}+\sigma _{z}[q_{z}^{2} \left (R_{18} \left (q_{x}^{2}+q_{y}^{2} \right )+R_{13}\right ) \\ &{}+q_{z} \left (R_{17} \left (q_{x}^{2}+q_{y}^{2} \right )+R_{12}\right )+R_{16} \left (q_{x}^{2}+q_{y}^{2}\right ) \\ &{}+R_{19} \left (q_{x}^{4}+q_{y}^{4}\right )+R_{20} q_{x}^{2} q_{y}^{2} \\ &{}+R_{15} q_{z}^{4}+R_{14} q_{z}^{3}]. \end{aligned}$$

(1)

$$\begin{aligned} \mathcal{H}_{1226}={}&\sigma _{x}[q_{x} q_{y} q_{z} \left (-\left ( \left (R_{4}+R_{5}\right ) q_{x}^{2}\right )+R_{5} q_{y}^{2}+R_{4} q_{z}^{2} \right )] \\ &{}+ \sigma _{y}[ (q_{x} q_{y} q_{z} \left (\left (R_{4}-R_{5}\right ) q_{x}^{2}-\left (2 R_{4}+R_{5}\right ) q_{y}^{2}\right. \\ &{}\left.+\left (R_{4}+2 R_{5}\right ) q_{z}^{2}\right ))/{\sqrt{3}}]. \end{aligned}$$

(2)

$$\begin{aligned} \mathcal{H}_{990}={}&\sigma _{x}[\frac{1}{3} R_{17} q_{x} q_{y} \left (-10 q_{x}^{2} q_{y}^{2}+3 q_{x}^{4}+3 q_{y}^{4}\right )] \\ &{}+ \sigma _{y}[ \frac{1}{3} R_{18} q_{x} q_{y} \left (-10 q_{x}^{2} q_{y}^{2}+3 q_{x}^{4}+3 q_{y}^{4}\right )] \\ &{}+\sigma _{z}[q_{z}^{4} \left (R_{29} \left (q_{x}^{2}+q_{y}^{2} \right )+R_{22}\right ) \\ &{}+q_{z}^{3} \left (R_{28} \left (q_{x}^{2}+q_{y}^{2} \right )+R_{21}\right ) \\ &{}+q_{z}^{2} \left (\left (q_{x}^{2}+q_{y}^{2} \right ) \left (R_{32} \left (q_{x}^{2}+q_{y}^{2}\right )+R_{27} \right )+R_{20}\right ) \\ &{}+q_{z} \left (\left (q_{x}^{2}+q_{y}^{2}\right ) \left (R_{31} \left (q_{x}^{2}+q_{y}^{2}\right )+R_{26}\right )+R_{19}\right ) \\ &{}+ \frac{1}{9} \left (R_{34} {\left (q_{x}^{3}-3 q_{x} q_{y}^{2}\right )}^{2}\right. \\ &{}+9 \left (q_{x}^{2}+q_{y}^{2}\right ) \left (R_{30} \left (q_{x}^{2}+q_{y}^{2} \right )+R_{25}\right ) \\ &{}\left. +3R_{33} \left (15 q_{x}^{4} q_{y}^{2}+2 q_{x}^{6}+3 q_{y}^{6}\right )\right ) \\ &{}+R_{24} q_{z}^{6}+R_{23} q_{z}^{5}]. \end{aligned}$$

(3)

5 Additional emanating nodal line by $k\cdot p$ models

As discussed in the above sections, CRs can dictate the form of ENS requiring bands near a BN to split or not to split from the purely symmetry point of view. The corresponding CR-BSP or CR-required ENS need to be fulfilled by the ENS solution by the explicitly constructed $k\cdot p$ model up to a reasonable expansion order. Interestingly, through systematically comparing the CR-BSP/CR-required ENS and the ENS by $k\cdot p$ model for all BNs that can be realized in the 1651 MSGs in the spinless and spinful settings, it was found that there might exist additional emanating nodal lines which cannot be predicted by a pure symmetry analysis using CRs [93] (the corresponding CR-BSP might be P-NP, L-NP and P-NL). In fact, these nodal lines are protected either by spinless PT symmetry or mirror/glide symmetry. It is worth pointing out that such nodal line, themselves are not constrained to thread one BN pinning at HSP/HSL. However, threading a BN at HSP/HSL is allowed, and such nodal line can be predicted by computing the representation of bands at the corresponding HSP/HSL. Spinless PT symmetry protects two-fold degenerate nodal line that can be located in GP, already well-known and a large number of materials (with negligible SOC) have been proposed to have such kind of nodal line near the Fermi level. The identification of such kind of nodal line need a dense k mesh and Berry phase encircling the nodal line usually need to be computed. However, though not required to be subjectto other symmetry except the spinless PT symmetry, additional symmetry could pin one point in such kind of nodal line at HSP/HSL: Then this point is the BN and such kind of nodal line is the corresponding ENS. Similarly, mirror/glide could protect nodal line within an HSPL corresponding to the mirror/glide plane that is formed by the crossing of two bands carrying opposite mirror/glide eigenvalues, but not definitely pinned at HSP/HSL.

For some $k\cdot p$ models, though the corresponding CR-BSP is P-NP/L-NP (whose ENS should be a nodal point), there exists additional emanating nodal line(s): Most of these $k\cdot p$ models are $2\times 2$ (see Table 10 for the case of P-NP and Table 9 for the case of L-NP) and the rest are $4\times 4$ only for P-NP (Table 11). For these $4\times 4$ $k\cdot p$ models, the additional nodal line should be protected by one mirror/gilde operation, since spinless PT symmetry can only protect two-fold degenerate nodal line. For the rest $2\times 2$ $k\cdot p$ models, the additional nodal line can be protected by spinless PT symmetry, mirror/glide, or both, as provided in Tables 10 and 9. An example of the additional nodal lines is provided in Fig. 5 using $\mathcal{H}_{1234}$, for which the CR-BSP is P-NP (also, the possible setting for this $k\cdot p$ model is the spinless setting), requiring that the BN should be a nodal point by CRs. See Eq. (4) for the $k\cdot p$ model, and one can easily find that to find the ENS, we only need to solve two equations (as required by the spinless PT symmetry), which guarantees that there is a chance for the ENS solution to be nodal line: The explicit $k\cdot p$ model indeed predicts three nodal lines as shown in Fig. 5.

Figure 5

Additional nodal lines (pinned at one BN) predictable by constructing $k\cdot p$ model rather than by CRs in $\mathcal{H}_{1234}$

Table 9 All $k\cdot p$ models $\mathcal{H}_{i}$’s for L-NP allowing additional nodal line. All these models are $2\times 2$, and the values of i are given in the 1st, 5th and 9th columns. These additional nodal lines cannot be protected by mirror/glide symmetry, otherwise the CR-BSP would become L-NL

Table 10 All $2\times 2$ $k\cdot p$ models $\mathcal{H}_{i}$’s for P-NP allowing additional nodal line. The values of i are given in the 1st, 5th and 9th columns. The additional nodal line can be protected by spinless PT symmetry or mirror/glide symmetry. Once the additional nodal line can/cannot be protected by the spinless PT symmetry, we print ✓/× in the column (with the header being “$(PT)^{2}=1$”) following that of $\mathcal{H}_{i}$. Once the additional nodal line can/cannot be protected by mirror/glide symmetry, we print ✓/× in the subsequent column (with the header being “Mirror/glide”). The number of mirror/glide planes is given in the parentheses. In some case, we print “Partial” in the parentheses which means that not all realizations are protected by the corresponding symmetry: For example, $\mathcal{H}_{24}$ gives additional nodal lines other that CRs required ones, and these additional nodal lines can be protected by spinless PT symmetry and one mirror/glide symmetry. However, for the realization in the spinless setting, not all the corresponding MSGs contains PT symmetry

Table 11 All $4\times 4$ $k\cdot p$ models $\mathcal{H}_{i}$’s for P-NP allowing additional nodal line. The values of i are given in the 1st, 5th and 9th columns. All the additional nodal lines here are protected by one mirror/glide symmetry

Besides, it was also found that for some $k\cdot p$ models (the corresponding CR-BSP being P-NL), there also exist additional emanating nodal line(s), as listed in Table 12. These $k\cdot p$ models are all $2\times 2$, and the ENS for the corresponding BNs consists of not only straight nodal line(s) along neighboring HSL(s) as required by CRs and also additional nodal line(s) that can only be predicted through the explicitly constructed $k\cdot p$ model.

$$\begin{aligned} \mathcal{H}_{1234}={}&\sigma _{x}[q_{x}^{2} \left (R_{6} q_{y}+R_{5} q_{z} \right ) \\ &{}+q_{x} \left (R_{7} q_{y} q_{z}+R_{5} q_{y}^{2}+R_{6} q_{z}^{2}+R_{3} \right ) \\ &{}+R_{4} q_{x}^{3}+q_{y} \left (R_{5} q_{z}^{2}+R_{3}\right )+R_{6} q_{y}^{2} q_{z} \\ &{}+R_{4} q_{y}^{3}+q_{z} \left (R_{4} q_{z}^{2}+R_{3}\right )] \\ &{}+\sigma _{y} [q_{x}^{2} \left (R_{11} q_{y}+R_{10} q_{z}\right ) \\ &{}+q_{x} \left (R_{12} q_{y} q_{z}+R_{10} q_{y}^{2}+R_{11} q_{z}^{2}+R_{8} \right ) \\ &{}+R_{9} q_{x}^{3}+q_{y} \left (R_{10} q_{z}^{2}+R_{8}\right )+R_{11} q_{y}^{2} q_{z} \\ &{}+R_{9} q_{y}^{3}+q_{z} \left (R_{9} q_{z}^{2}+R_{8}\right )]. \end{aligned}$$

(4)

Table 12 All $k\cdot p$ models $\mathcal{H}_{i}$’s for P-NL allowing additional nodal line. All these models are $2\times 2$, and the values of i are given in the 1st, 5th and 9th columns. The additional nodal line can be protected by spinless PT symmetry or mirror/glide symmetry. Once the additional nodal line can/cannot be protected by the spinless PT symmetry, we print ✓/× in the column (with the header being “$(PT)^{2}=1$”) following that of $\mathcal{H}_{i}$. Once the additional nodal line can/cannot be protected by mirror/glide symmetry, we print ✓/× in the subsequent column (with the header being “Mirror/glide”). The number of mirror/glide planes is given in the parentheses. In some case, we print “Partial” in the parentheses which means that not all realizations are protected by the corresponding symmetry: For example, $\mathcal{H}_{860}$ gives additional nodal lines other that CRs required ones, and these additional nodal lines can be protected by spinless PT symmetry and 3 mirror/glide symmetries. However, for the realization in the spinless setting, not all the corresponding MSGs contains PT symmetry

6 Summary and outlook

In this work, we first review the recent works on complete classification on all BNs in the 1651 MSGs using CRs and $k\cdot p$ models and by the exhaustive classification of all BNs, we provide several new results including all the 25 ENS configurations by CRs near BN, all realizations around BNs given any of the 2574 $k\cdot p$ models and all protection mechanisms on nodal lines that cannot be captured simply by CRs. Other than the BNs appearing at GP or within HSPL carrying two identical irreps/co-irreps of the little group, which can be identified through wave-functions-overlap-based method, such as computing Chern number and Berry phase, very fruitful BNs appear at HSPs/HSLs/HSPLs by one (co-)irrep, or two different (co-)irreps of HSLs/HSPLs, which can be identified by computing representation of the BN, and there are 73411 such BNs in total considering both spinless and spinful settings in the 1651 MSGs. Note that some BNs can definitely appear without any realistic calculations: For example, Weyl point definitely appears at time-reversal invariant k points for SG 2 (with an additional TRS) in the spinful setting, since only one two-fold degenerate co-irrep is allowed for these k⃗ points [110]. Constructing $k\cdot p$ models is thought to be a useful method of studying BNs while the expansion order was usually chosen to be truncated at a leading order. Owing to transforming the construction of $k\cdot p$ model to any expansion order to simply a linear combination problem [91], expanding the $k\cdot p$ model to seemingly very large order becomes feasible, and could reveal something new to the community as the hexagonal warping of energy contour of the surface states in the Bi₂Se₃ family of materials. For the BNs here, the expansion order need to take a least value to reproduce the ENS as predicted from CRs, which can be found as large as 6. Accordingly, hierarchical structure of ENS by varying the $k\cdot p$ order can be expected, among which some nearly nodal lines/surfaces can be gapped by higher order $k\cdot p$ terms.

On future directions, one might expand the classification scheme on BNs in the normal states for the 1651 MSGs to other symmetry settings, such as the recent revived spin space groups [111–120], symmetries for non-Hermitian Hamiltonians [121–123] and superconductors [124–128]. To relate the gaps opened along the nearly ENS, which might of magnitude $\sim q^{6}$, with concrete physical observable, such as spin texture, is also an interesting topic. Note that the symmetry group is kept unchanged during the analysis of BNs, while symmetry breaking can lead to a subgroup, the relation of BNs in the subgroup and those in the parent group can be utilized to study the evolution of the BNs: e.g. the ENS can change from a nodal line to a nodal point. Besides, according to the mapping from $k\cdot p$ model to its realization in MSGs, one can anticipate that one $k\cdot p$ model can correspond to different little groups, thus approximate symmetry (not belonging to the MSG) in the low-energy zone can be designed easily by this mapping. The low-energy approximate symmetry can be utilized to impose additional constraints on quantities such as Berry curvature and orbital magnetic moment. It is also worth pointing out that the parameters allowed in the generic $k\cdot p$ models for BNs should be of physical origin related with detailed crystal structure (belonging to the corresponding MSG), e.g. electronic hopping amplitude, strength of SOC, magnetic moments and so on, which can be expected to study analogous properties among different freedoms (e.g. charge/spin/orbital). Finally, the generic $k\cdot p$ models for BNs can be compared with a target effective model (that have been thought to be endowed with fascinating properties but lacking materials realization) to unveil whether the model can be completely determined by little group or partial parameters in the generic $k\cdot p$ model should be vanishing, thus aiding in narrowing down materials candidates for detailed experimental or theoretical studies.

Appendix

We provide all possible ENS configurations for all the ten CR-BSPs in Figs. A1, A2, A3, A4 and A5. In total, 25 different configurations can be concluded. Besides, we showcase the usage of “BN_2574kp.mx” in Figs. A6, A7, A8 and A9. In the following, a toy model is first introduced to produce one charge-4 Weyl point.

Figure A1

Configurations of ENSs: The ENS might be a nodal point (a) or one straight nodal line along an HSL (b,c) and one nodal surface along an HSPL (d,e,f). The red sphere represents the band node while the blue line/plane represents the nodal line/surface, respectively. In each plot, the CR-BSP (P-NP, L-NP, P-NL, L-sNL, P-NS, L-NS and PL-NS) is provided

Figure A2

Configurations of ENSs: The ENS might be two nodal surfaces (a,c) or three nodal surfaces (b) or existing straight nodal line(s) and nodal surface(s) (d,e,f). The red sphere represents the band node while the blue line/plane represents the nodal line/surface, respectively. In each plot, the CR-BSP (P-NS, L-NS, P-NSL) is provided

Figure A3

Configurations of ENSs: The ENS might be one or multiple nodal lines that lie within one or multiple neighboring HSPLs. In (a) and (b), only one nodal line (blue line) is anticipated penetrating the band node (red sphere). In (c-f), 2, 3, 4 and 6 nodal lines are anticipated lying in 2, 3, 4 and 6 neighboring HSPLs, respectively. The HSPLs accommodating the emanating nodal lines are shown by orange planes. The CR-BSPs (PL-NL or L-NL) are given in each plot

Figure A4

Configurations of ENSs corresponding to P-NL: In (a,b,c,d), the emanating (2,3,4,6) nodal lines all lie with one plane, respectively. In (e,f,g,h), there exists one more nodal line perpendicular to this plane compared with (a,b,c,d), respectively

Figure A5

Configurations of ENSs corresponding to P-NL with 4, 7, 9 and 13 emanating nodal lines in (a,b,c,d), respectively. In (a), the 4 nodal lines are along $\vec{e}_{4}, \vec{e}_{5}, \vec{e}_{6}, \vec{e}_{7}$. In (b), the 7 nodal lines are along $\vec{e}_{1}, \vec{e}_{2}, \vec{e}_{3}, \vec{e}_{4}, \vec{e}_{5}, \vec{e}_{6}, \vec{e}_{7}$. In (c), the 9 nodal lines are along $\vec{e}_{1}, \vec{e}_{2}, \vec{e}_{3}, \vec{e}_{8}, \vec{e}_{9}, \vec{e}_{10}, \vec{e}_{11}, \vec{e}_{12},\vec{e}_{13}$. In (d), the 13 nodal lines are along the 13 direction as listed in Table 8

Figure A6

Demonstration of usage of “BN_2574kp.mx” by $\mathcal{H}_{1201}$: BN is imported by the Mathematica command: BN=Import$\left [\text{``BN\_2574kp.mx''}\right ]$. Then the command as in the blue box, gives the explicit $k\cdot p$ model $\mathcal{H}_{1201}$, the corresponding CR-BSP(s) and the locations of the ENS, respectively. Here since the CR-BSP is P-NP, then there exists no neighboring HSL/HSPL for the ENS

Figure A7

Demonstration of usage of “BN_2574kp.mx” by $\mathcal{H}_{1201}$: BN is imported by the Mathematica command: BN=Import$\left [\text{``BN\_2574kp.mx''}\right ]$. Here the command as in the blue box, gives realization in the spinless and spinful settings, respectively. Here there is no realization in the spinful setting

Figure A8

Demonstration of usage of “BN_2574kp.mx” by $\mathcal{H}_{2421}$: BN is imported by the Mathematica command: BN=Import$\left [\text{``BN\_2574kp.mx''}\right ]$. Then the commands as in the blue box, gives the explicit $k\cdot p$ model $\mathcal{H}_{2421}$, the corresponding CR-BSP(s) and the locations of the ENS. Here since the CR-BSP is P-NSL, then there exist neighboring HSL and HSPL being the ENSs. The direction of the HSL is along $(0,1,0)$ while the normal direction of neighboring HSPL is $(0,1,0)$, thus the ENS configuration is that as in Fig. A2(d)

Figure A9

Demonstration of usage of “BN_2574kp.mx” by $\mathcal{H}_{2421}$: BN is imported by the Mathematica command: BN=Import$\left [\text{``BN\_2574kp.mx''}\right ]$. Here the command as in the blue box, gives realization in the spinless and spinful settings, respectively. Here there is no realization in the spinful setting

1.1 A1 A toy model

We construct a toy model with a type II MSG symmetry (214.68), with a body-centered cubic lattice. The primitive lattice basis vectors are chosen as: $\vec{a}_{1}=(-\frac{a}{2},\frac{a}{2},\frac{a}{2}),\vec{a}_{2}=( \frac{a}{2},-\frac{a}{2},\frac{a}{2}), \vec{a}_{3}=(\frac{a}{2}, \frac{a}{2},-\frac{a}{2})$. There are six atoms in each primitive unit cell, as listed below (on the basis of $\vec{a}_{1},\vec{a}_{2},\vec{a}_{3}$):

$$\begin{aligned} &\vec{\tau }_{1}=(\frac{1}{4},\frac{3}{8},\frac{1}{8}) \\ &\vec{\tau }_{2}=(\frac{3}{4},\frac{1}{8},\frac{3}{8}) \\ &\begin{aligned} &\vec{\tau }_{3}=(\frac{3}{8},\frac{3}{4},\frac{1}{8}) \\ &\vec{\tau }_{4}=(\frac{1}{8},\frac{1}{4},\frac{3}{8}) \end{aligned} \\ &\vec{\tau }_{5}=(\frac{3}{8},\frac{1}{8},\frac{1}{4}) \\ &\vec{\tau }_{6}=(\frac{1}{8},\frac{3}{8},\frac{3}{8}). \end{aligned}$$

(A1)

We consider the 12 nearest neighbors and 12 next nearest neighbors, as listed below (and the hopping amplitudes are $t_{1}$ and $t_{2}$, respectively). The 12 nearest neighbors hopping vectors are represented by:

$$\begin{aligned} &\vec{\tau }_{1}\rightarrow \vec{\tau }_{3} \\ &\vec{\tau }_{1}\rightarrow \vec{\tau }_{4} \\ &\vec{\tau }_{2}\rightarrow \vec{\tau }_{4}+\vec{a}_{1} \\ &\vec{\tau }_{2}\rightarrow \vec{\tau }_{3}-\vec{a}_{2} \\ &\vec{\tau }_{2}\rightarrow \vec{\tau }_{5} \\ &\begin{aligned} &\vec{\tau }_{4}\rightarrow \vec{\tau }_{6} \\ &\vec{\tau }_{2}\rightarrow \vec{\tau }_{6}+\vec{a}_{1} \end{aligned} \\ &\vec{\tau }_{4}\rightarrow \vec{\tau }_{5} \\ &\vec{\tau }_{1}\rightarrow \vec{\tau }_{6}-\vec{a}_{3} \\ &\vec{\tau }_{1}\rightarrow \vec{\tau }_{5} \\ &\vec{\tau }_{3}\rightarrow \vec{\tau }_{6}-\vec{a}_{3} \\ &\vec{\tau }_{3}\rightarrow \vec{\tau }_{5}+\vec{a}_{2}, \end{aligned}$$

(A2)

and the 12 next nearest neighbors hopping vectors are represented by:

$$\begin{aligned} &\vec{\tau }_{1}\rightarrow \vec{\tau }_{2}-\vec{a}_{1}-\vec{a}_{3} \\ &\vec{\tau }_{1}\rightarrow \vec{\tau }_{2} \\ &\vec{\tau }_{1}\rightarrow \vec{\tau }_{2}+\vec{a}_{2} \\ &\vec{\tau }_{3}\rightarrow \vec{\tau }_{4}+\vec{a}_{2} \\ &\vec{\tau }_{5}\rightarrow \vec{\tau }_{6}+\vec{a}_{1} \\ &\begin{aligned} &\vec{\tau }_{1}\rightarrow \vec{\tau }_{2}-\vec{a}_{1} \\ &\vec{\tau }_{5}\rightarrow \vec{\tau }_{6}-\vec{a}_{3} \end{aligned} \\ &\vec{\tau }_{3}\rightarrow \vec{\tau }_{4}-\vec{a}_{3} \\ &\vec{\tau }_{5}\rightarrow \vec{\tau }_{6}-\vec{a}_{2}-\vec{a}_{3} \\ &\vec{\tau }_{3}\rightarrow \vec{\tau }_{4}+\vec{a}_{1}+\vec{a}_{2} \\ &\vec{\tau }_{5}\rightarrow \vec{\tau }_{6} \\ &\vec{\tau }_{3}\rightarrow \vec{\tau }_{4}. \end{aligned}$$

(A3)

We can thus get $H(\vec{k})$ in momentum space and for Γ point, there exists one charge-4 Weyl point, whose $k\cdot p$ model is $\mathcal{H}_{1257}$:

$$\begin{aligned} \mathcal{H}_{1257}={}&R_{1}(q_{x}^{2}+q_{y}^{2}+q_{z}^{2})\sigma _{0}+R_{2}(q_{y}^{2}-q_{x}^{2}) \sigma _{x} \\ &{}+R_{3}(q_{x}q_{y}q_{z})\sigma _{y}- R_{2}( \frac{q_{x}^{2}+q_{y}^{2}-2q_{z}^{2}}{\sqrt {3}})\sigma _{z}. \end{aligned}$$

(A4)

Projecting into the subspace spanned by the two Bloch states for the charge-4 Weyl point by $H_{eff}=P^{\dagger }H(\vec{k})P$ and writing the low-energy model in q⃗ ($\vec{q}=\vec{k}-\vec{K}=\vec{k}$ and K⃗ is the central wave vector and here it’s simply the Γ point) where $P=\left [ \begin{array}{c@{\quad}c} -\frac{1}{2\sqrt {3}}&\frac{1}{2} \\ -\frac{1}{2\sqrt {3}}&\frac{1}{2} \\ -\frac{1}{2\sqrt {3}}&-\frac{1}{2} \\ -\frac{1}{2\sqrt {3}}&-\frac{1}{2} \\ \frac{1}{\sqrt {3}}&0 \\ \frac{1}{\sqrt {3}}&0\end{array} \right ]$ denotes two eigenstates of the charge-4 Weyl point at Γ point, we have:

$$\begin{aligned} H_{eff}(\vec{q})={}&\left [ \textstyle\begin{array}{l} -2t_{1}+4t_{2}+\frac{a^{2}(9t_{1}-44t_{2})(q_{x}^{2}+q_{y}^{2})}{192}- \frac{a^{2}t_{2}q_{z}^{2}}{6}\\ -\frac{\mathrm{i}\sqrt {3}a^{3}t_{1}}{128}q_{x}q_{y}q_{z}+ \frac{a^{2}(3t_{1}-4t_{2})(q_{x}^{2}-q_{y}^{2})}{64\sqrt {3}} \end{array}\displaystyle \right. \\ &\left. \textstyle\begin{array}{l} \frac{\mathrm{i}\sqrt {3}a^{3}t_{1}}{128}q_{x}q_{y}q_{z}+ \frac{a^{2}(3t_{1}-4t_{2})(q_{x}^{2}-q_{y}^{2})}{64\sqrt {3}} \\ -2t_{1}+4t_{2}+ \frac{a^{2}(t_{1}-12t_{2})(q_{x}^{2}+q_{y}^{2})}{64}+ \frac{a^{2}(t_{1}-4t_{2})q_{z}^{2}}{16}\end{array}\displaystyle \right ] \\ &{}+O(q^{4}), \end{aligned}$$

(A5)

(noting that $-2t_{1}+4t_{2}$ is the energy of the charge-4 Weyl point) by which we know that the $k\cdot p$ model parameters in Eq. (A4) are:

$$\begin{aligned} &R_{1}=\frac{(3t_{1}-20t_{2})a^{2}}{96} \\ &R_{2}=\frac{(4t_{2}-3t_{1})a^{2}}{64\sqrt {3}} \\ &R_{3}=-\frac{\sqrt {3} t_{1} a^{3}}{128}, \end{aligned}$$

(A6)

showing that the 3rd $k\cdot p$ term in Eq. (A4) is proportional to the nearest neighbor hopping amplitude.