Triple nodal points characterized by their nodal-line structure in all magnetic space groups

We analyze triply degenerate nodal points [or triple points (TPs) for short] in energy bands of crystalline solids. Specifically, we focus on spinless band structures, i.e., when spin-orbit coupling is negligible, and consider TPs formed along high-symmetry lines in the momentum space by a crossing of three bands transforming according to a one-dimensional (1D) and a two-dimensional (2D) irreducible corepresentation (ICR) of the little cogroup. The result is a complete classification of such TPs in all magnetic space groups, including the nonsymmorphic ones, according to several characteristics of the nodal-line structure at and near the TP. We show that the classification of the presently studied TPs is exhausted by 13 magnetic point groups (MPGs) that can arise as the little cogroup of a high-symmetry line and which support both 1D and 2D spinless ICRs. For 10 of the identified MPGs, the TP characteristics are uniquely determined without further information; in contrast, for the 3 MPGs containing sixfold rotational symmetry, two types of TPs are possible, depending on the choice of the crossing ICRs. The classification result for each of the 13 MPGs is illustrated with first-principles calculations of a concrete material candidate.

Figure 1

Triple points (TPs) and their characteristic nodal features. (a) Schematic band structure along a high-symmetry line (HSL) $k_z$ that can stabilize TPs (cf. Table 1). Note the non-degenerate crossing the doubly-degenerate band. Nodal lines (NLs) in the lower (upper) band gap of a three-band model are shown in red (blue), and the TP is indicated by a yellow disk or sphere. (b) Band structure of the material AlN along the $\mathrm{\Delta }=\mathrm{\Gamma }A$ line, which we predict to host triple points (cf. Table 5 and Figs. 16 and 17). Non-degenerate (doubly-degenerate) bands are shown as thin gray (thick black) lines and TPs are indicated by yellow arrows. (c) In the NL structure, a TP can be recognized by the change of a NL color from red to blue. (d), (e) The point where additional NLs (called NL arcs) coalesce with the central NL (along $k_z$) is called nexus. All NLs involved in the nexus are in the same gap, i.e., a red (blue) nexus involves NLs in the lower (upper) gap. The number $n_a^{\mathrm{nexus}}$ of NL arcs that coalesce with the central NL depends on the little co-group of the HSL (cf. Table 2).

Figure 2

Types and subtypes of triple points (TPs) in spinless systems. Nodal lines (NLs) are shown in red (blue) if they are in the lower (upper) band gap of the three-band model and the TP is indicated in yellow. TPs are conventionally classified as type $\mathsf{A}$ vs type $\mathsf{B}$ according to the absence vs presence of NL arcs attached to the TP besides the central NL along the high-symmetry line. Type-$\mathsf{A}$ TPs arise (a) without nexus points or (b) with nexus points that do generically (i.e., without fine tuning) not coincide with the TP. For type-$\mathsf{B}$ TPs a red and blue nexus generally coincides with the TP. We differentiate between (c) type ${\mathsf{B}}_l$ with three linearly attached NLs and (d) type ${\mathsf{B}}_q$ with six quadratically attached NLs.

Figure 3

Vertical mirror planes, i.e., the ones containing the rotation axis, of the magnetic point groups (a) $4mm$ and $4/m^{\prime }mm$, (b) ${\overline{4}}^{\prime }{2}^{\prime }m$, (c) $6mm$ and $6/m^{\prime }mm$, (d) ${\overline{6}}^{\prime }m{2}^{\prime }$, $3m$, and ${\overline{3}}^{\prime }m$. Mirror planes related by (rotation) symmetry are shown in the same color, such that in (a) and (b) the different symmetry-unrelated sets of mirror planes can be distinguished by their color. The gray boxes are guides to the eye.

Figure 4

Nodal-line (NL) structure near a type-${\mathsf{B}}_q$ triple point (TP) on a high-symmetry line (HSL) with little cogroup $6mm$, which requires a specific combination (class II) of 2D and 1D irreducible representations (cf. Table 2). NLs are shown in red (blue) if they are in the lower (upper) band gap of the three-band model and the TP is indicated in yellow. (a), (b) show the two possibilities with red (blue) NL arcs in one set of symmetry-related mirror planes and blue (red) NL arcs in the other. The insets show the view on the nodal lines from the top to clarify the arrangement of the NL arcs. An analogous discussion applies to HSLs with little cogroup $6/m^{\prime }mm$ (with a class-II combination of irreducible corepresentations).

Figure 5

Nodal-line (NL) structure near a type-$\mathsf{A}$ triple point (TP) on a high-symmetry line (HSL) with little cogroup $4mm$. NLs are shown in red (blue) if they are in the lower (upper) band gap of the three-band model and the TP is indicated in yellow. Type-$\mathsf{A}$ TPs can be accompanied by nearby nexus points on the central nodal line: (a) on opposite sides of the TP if $\mathrm{sign}\left(a_0\right)=-\mathrm{sign}\left(a_0^{\prime }\right)$, or (b) on the same side if $\mathrm{sign}\left(a_0\right)=\mathrm{sign}\left(a_0^{\prime }\right)$. (c) If the complex parameter $A_0$ is fine tuned to 0, the two nexus points (independently of their color) simultaneously collide with the TP. An analogous discussion applies for HSLs with little cogroup $4/m^{\prime }mm$, with the only relevant change being the reduction of $A_0$ to be a real (rather than complex) parameter.

Figure 6

Nodal-line (NL) structure near a type-$\mathsf{A}$ triple point (TP) on a high-symmetry line (HSL) with little cogroup $6mm$, which requires a specific combination (class I) of 2D and 1D irreducible representations (cf. Table 2). NLs are shown in red (blue) if they are in the lower (upper) band gap of the three-band model and the TP is indicated in yellow. As in the case of the little cogroup $4mm$ (cf. Fig. 5), type-$\mathsf{A}$ TPs in $6mm$ can be accompanied by nexus points nearby on the central nodal line (NL). However, the points of coalescence of NL arcs with the central NL in the inequivalent sets of mirror planes always coincide, such that we only distinguish (a) blue nexus points ($a_0>0$), (b) red nexus points ($a_0<0$), and (c) nexus points fine tuned to collide with the TP (if the complex parameter $A_0$ is set to zero), all with 12 NL arcs attached. An analogous discussion applies for HSLs with little cogroup $6/m^{\prime }mm$, with the only relevant change being the reduction of $A_0$ to real values.

Figure 7

Nodal-line (NL) structure near a type-${\mathsf{B}}_l$ triple point on a high-symmetry line with little cogroup ${\overline{3}}^{\prime }m$ (or $3m$). NLs are shown in red (blue) if they are in the lower (upper) band gap of the three-band model and the TP is indicated in yellow. The NL arc always changes gap (and therefore its color) when passing through the TP, but several different arrangements are possible as illustrated.

Figure 8

Nodal-line structure of ${\mathrm{Na}}_3\mathrm{N}$ near the triple point (TP) at $E_{\mathrm{TP}}=-38\mathrm{meV}$ lying on the $\mathrm{\Delta }$ line with little cogroup $6/m^{\prime }mm$ (cf. Table 5). (a) Band structure along lines of symmetry. The TP is indicated by a yellow dot and arrow, and the bands forming the TP are labeled by their irreducible corepresentations. (b) Brillouin zone (boundary in gray) with points and lines of symmetry (black dashed lines) and the two inequivalent mirror planes (orange and purple planes). Note that there are additional symmetry-related mirror planes for each of those two. (c) Size of the lower (red) and upper (blue) gaps in the two mirror planes shown in (b) encoded by the intensity of the color (higher color saturation implies smaller energy gap between the corresponding pair of bands). The gap is only plotted up to a cutoff of $0.01\mathrm{eV}$ such that white color indicates a gap larger than that. The TP (yellow) and the central nodal line are emphasized by appropriately colored overlays and the data show that the TP is type $\mathsf{A}$. Furthermore, there are no nexus points present, even though they could be stabilized.

Figure 9

Nodal-line structure of CaNaP near the triple point (TP) on the $\mathrm{\Lambda }$ line with little cogroup $4mm$. The organization of the panels is in one-to-one correspondence with Fig. 8 and the cutoff on the gap size was chosen to be $0.05\mathrm{eV}$. No nodal lines are attached to the TP, such that we conclude it to be type $\mathsf{A}$. Note the occurrence of a nexus of four (due to the fourfold rotational symmetry) red nodal lines near the TP, from which we can deduce that $n_a^{\mathrm{nexus}}=4$ and $\mu =2$. Our theoretical analysis predicts that two parameters need to be tuned to collide the nexus point with the TP.

Figure 10

Nodal-line structure of ${\mathrm{TiB}}_2$ near the triple point (TP) at $E_{\mathrm{TP}}=1.3\mathrm{eV}$ lying on the $\mathrm{\Delta }$ line with little cogroup $6/m^{\prime }mm$. The organization of the panels is in one-to-one correspondence with Fig. 8 with a cutoff of $0.05\mathrm{eV}$ on the gap size that is shown in (c). Note the NL arcs attaching quadratically ($k_z\propto k_x^2+k_y^2$) to the TP, i.e., the two nexus points coincide with the TP, implying that the TP is type ${\mathsf{B}}_q$. Due to the rotational symmetry, there are six nodal-line arcs in each of the two gaps (shown in red and blue, respectively), implying $n_a^{\mathrm{nexus}}=12$.

Figure 11

Nodal-line structure of ${\mathrm{B}}_2\mathrm{CN}$ near the triple point (TP) on the $\mathrm{\Delta }$ line with little cogroup $3m$. The organization of the panels is in one-to-one correspondence with Fig. 8 (except for the fact that only one inequivalent high-symmetry plane is now present) and the cutoff on the gap size was chosen to be $0.02\mathrm{eV}$. Note the NL arcs attaching to the TP, i.e., the two nexus points coincide with the TP and that the NLs scale linearly, $k_z\propto \sqrt{k_x^2+k_y^2}$, implying a type-${\mathsf{B}}_l$ TP. Due to the threefold rotational symmetry, there are $n_a^{\mathrm{nexus}}=6$ nodal-line arcs in total.

Figure 12

Nodal-line structure of ZrO near the triple point (TP) at $E_{\mathrm{TP}}=2.0\mathrm{eV}$ lying on the $\mathrm{\Delta }$ line with little cogroup $4/m^{\prime }mm$. The organization of the panels is in one-to-one correspondence with Fig. 8 and the cutoff on the gap size was chosen to be $0.02\mathrm{eV}$. The inset of (c) shows the nodal-line (NL) structure of the minimal $\mathbf{k}·\mathbf{p}$ model given in Appendix pp4-s2e with parameters tuned to qualitatively reproduce the situation in ZrO. We observe that there are two red nexus points, each with four NL arcs, coinciding with the TP, such that $\mathrm{sign}\left(a_3{a}_4\right)=+1$ in the $\mathbf{k}·\mathbf{p}$ model in Eq. (D20) (in contrast to another TP in this compound, illustrated in Fig. 24). Therefore, there are in total eight red NLs attached to the TP. This is accidental (see text and Fig. 25) and we therefore classify the TP despite the NL arcs as type $\mathsf{A}$ and consider the two nexus points (separately) with $n_a^{\mathrm{nexus}}=4$ and $\mu =2$.

Figure 13

Construction of the leading terms contributing to the discriminant of $6/m^{\prime }mm$ class I given the Hamiltonian in Eq. (D12) (red line). The black dots denote elements of the set $M$ as defined in Eq. (26), and the red dots are elements of $L\left(M\right)$ as defined in Eq. (28).

Figure 14

Nodal-line structure of ${\mathrm{Na}}_2\mathrm{LiN}$ near the type-$\mathsf{A}$ triple point on the $\mathrm{\Lambda }$ line with little cogroup $4/m^{\prime }mm$. (a) Band structure along lines of symmetry. The triple point is indicated by a yellow dot and arrow and the bands forming the triple point are labeled by their irreducible corepresentations. (b) Brillouin zone (boundary in gray) with points and lines of symmetry (black dashed lines) and the two inequivalent mirror planes (orange and purple planes). (c) Size of the lower (red) and upper (blue) gaps in the two mirror planes shown in (b) encoded by the intensity of the color (with a cutoff at a gap size of $0.01\mathrm{eV}$, i.e., gaps larger than that are shown in white). The triple point (yellow) and the central nodal line are emphasized by appropriately colored overlays.

Figure 15

Nodal-line structure of P near the type-${\mathsf{B}}_l$ triple point on the $\mathrm{\Lambda }$ line with little cogroup ${\overline{3}}^{\prime }m$. The organization of the panels is in one-to-one correspondence with Fig. 14 with the difference that there is only one mirror plane. The cutoff on the gap size is $0.1\mathrm{eV}$. Due to the threefold rotational symmetry, we conclude that $n_a^{\mathrm{nexus}}=6$.

Figure 16

Nodal-line structure of AlN near the type-$\mathsf{A}$ triple point at $E_{\mathrm{TP}}=-0.28\mathrm{eV}$ on the $\mathrm{\Delta }$ line with little cogroup $6mm$. The organization of the panels is in one-to-one correspondence with Fig. 14 and the cutoff on the gap size is $0.05\mathrm{eV}$. Note the occurrence of the nearby nexus of $n_a^{\mathrm{nexus}}=12$ (due to the sixfold rotational symmetry) blue nodal-line (NL) arcs with $\mu =2$ in (c). The inset of (c) shows the NL structure of the minimal $\mathbf{k}·\mathbf{p}$ model given in Appendix pp4-s2e with parameters tuned to qualitatively reproduce the situation in AlN. As expected, the $\mathbf{k}·\mathbf{p}$ model does not reproduce the different curvature of the NL arcs in the two inequivalent sets of mirror planes which is clearly visible in the data. As remarked upon in Sec. 3b2 this would be reflected in terms of higher order in $k_x,k_y$ in the $\mathbf{k}·\mathbf{p}$ expansion. Our theoretical arguments suggest that two parameters need be tuned to collide the nexus with the type-$\mathsf{A}$ TP.

Figure 17

Nodal-line structure of AlN near the type-${\mathsf{B}}_q$ triple point at $E_{\mathrm{TP}}=-0.85\mathrm{eV}$ on the $\mathrm{\Delta }$ line with little cogroup $6mm$. The organization of the panels is in one-to-one correspondence with Fig. 14 and the cutoff on the gap size is $0.05\mathrm{eV}$. Due to the sixfold rotational symmetry, we conclude that $n_a^{\mathrm{nexus}}=12$.

Figure 18

Nodal-line structure of ${\mathrm{Li}}_4$N near the type-$\mathsf{A}$ triple point on the $\mathrm{\Delta }$ line with little cogroup ${\overline{6}}^{\prime }m{2}^{\prime }$. The organization of the panels is in one-to-one correspondence with Fig. 14 with the difference that there is only one mirror plane. The cutoff on the gap size is $0.01\mathrm{eV}$. Note the occurrence of a nearby nexus of $n_a^{\mathrm{nexus}}=6$ (due to the threefold rotational symmetry) blue nodal-line arcs with $\mu =2$ in (c). Our theoretical arguments suggest that only a single parameter needs be tuned to collide the nexus with the type-$\mathsf{A}$ TP.

Figure 19

Nodal-line structure ${\mathrm{Na}}_2\mathrm{O}$ near the type-$\mathsf{A}$ triple point on the $\mathrm{\Delta }$ line with little cogroup ${\overline{6}}^{\prime }m{2}^{\prime }$. The organization of the panels is in one-to-one correspondence with Fig. 14 with the difference that there is only one mirror plane. The cutoff on the gap size is $0.01\mathrm{eV}$. Note again the occurrence of a nearby nexus of $n_a^{\mathrm{nexus}}=6$ (due to the threefold rotational symmetry) blue nodal-line arcs with $\mu =2$ in (c). Our theoretical arguments suggest that only a single parameter needs be tuned to collide the nexus with the type-$\mathsf{A}$ TP.

Figure 20

Nodal-line structure of ${\mathrm{Li}}_2\mathrm{NaN}$ near the type-$\mathsf{A}$ triple point on the $\mathrm{\Delta }$ line with little cogroup $6/m^{\prime }mm$. The organization of the panels is in one-to-one correspondence with Fig. 14 and the cutoff on the gap size is $0.02\mathrm{eV}$. Our recent work [45] used this material to illustrate a relation between type-$\mathsf{A}$ TPs of spinless $\mathcal{P}\mathcal{T}$-symmetric crystals to multiband nodal links.

Figure 21

Nodal-line structure of ${\mathrm{TiB}}_2$ near the type-$\mathsf{A}$ triple point at $E_{\mathrm{TP}}=0.57\mathrm{eV}$ on the $\mathrm{\Delta }$ line with little cogroup $6/m^{\prime }mm$. The organization of the panels is in one-to-one correspondence with Fig. 14 and the cutoff on the gap size is $0.05\mathrm{eV}$. We do not identify a nexus point in this case; the nodal lines discernible in the right part of (c) seem to connect to the high-symmetry point $K$ such that they cannot be considered to be near the triple point.

Figure 22

Nodal-line structure of ${\mathrm{Na}}_3\mathrm{N}$ near the type-${\mathsf{B}}_q$ triple point at $E_{\mathrm{TP}}=-93\mathrm{meV}$ on the $\mathrm{\Delta }$ line with little cogroup $6/m^{\prime }mm$. The organization of the panels is in one-to-one correspondence with Fig. 14 and the cutoff on the gap size is $0.01\mathrm{eV}$. Due to the sixfold rotational symmetry, we conclude that $n_a^{\mathrm{nexus}}=12$.

Figure 23

Nodal-line structure of ${\mathrm{C}}_3{\mathrm{N}}_4$ (SG 215) near the type-$\mathsf{A}$ triple point on the $\mathrm{\Delta }$ line with little cogroup ${\overline{4}}^{\prime }{2}^{\prime }m$. The organization of the panels is in one-to-one correspondence with Fig. 14 with the difference that there is only one mirror plane. The cutoff on the gap size is $0.02\mathrm{eV}$. We do not identify a nexus point in this case; the nodal lines discernible in (c) connect to the high-symmetry point $\mathrm{\Gamma }$ such that they cannot be considered to be near the triple point.

Figure 24

Nodal-line structure of ZrO near the type-$\mathsf{A}$ triple point at $E_{\mathrm{TP}}=0.12\mathrm{eV}$ on the $\mathrm{\Delta }$ line with little cogroup $4/m^{\prime }mm$. The organization of the panels is in one-to-one correspondence with Fig. 14 and the cutoff on the gap size is $0.02\mathrm{eV}$. Note the extremely close adjacency of the triple point to two nexus points of $n_a^{\mathrm{nexus}}=4$ (due to the fourfold rotational symmetry) nodal-line arcs each. According to our theoretical arguments, this is achieved by tuning a single model parameter. Furthermore, observe that the two nexus points are of different colors (in contrast to Fig. 12), suggesting $\mathrm{sign}\left(a_3{a}_4\right)=-1$ in the $\mathbf{k}·\mathbf{p}$ model in Eq. (D20).

Figure 25

Nodal-line structure of ZrO with $5\%$ uniaxial tensile strain in $z$ direction near the type-$\mathsf{A}$ triple point at $E_{\mathrm{TP}}=2.0\mathrm{eV}$ on the $\mathrm{\Delta }$ line with little cogroup $4/m^{\prime }mm$ (cf. Fig. 12 for the same data in ZrO without strain). Size of the lower (red) and upper (blue) gaps in the two mirror planes shown in Fig. 12 encoded by the intensity of the color (higher color saturation implies smaller energy gap between the corresponding pair of bands). The gap is only plotted up to a cutoff of $0.02\mathrm{eV}$ such that white color indicates a gap larger than that. The triple point (yellow) and the central nodal line are emphasized by appropriately colored overlays. In contrast to Fig. 12, we recognize two separate nexus points (indicated by black disks) not coinciding with the triple point. Thus, we conclude that the triple point is indeed type $\mathsf{A}$ and (due to the fourfold rotational symmetry) that $n_a^{\mathrm{nexus}}=4$.

Figure 26

Nodal-line structure of ${\mathrm{SiO}}_2$ near the type-$\mathsf{A}$ triple point on the $\mathrm{\Lambda }$ line with little cogroup ${\overline{4}}^{\prime }$. The organization of the panels is in one-to-one correspondence with Fig. 14 with the difference that the shown planes are not mirror planes. The cutoff on the gap size is $0.005\mathrm{eV}$. The little cogroup ${\overline{4}}^{\prime }$ contains neither a vertical mirror symmetry nor $\mathcal{P}\mathcal{T}$ symmetry, such that nodal lines away from the rotation axis cannot be stabilized. We have performed DFT calculations in planes perpendicular to the central line (see [63]) and verified that.

Figure 27

Nodal-line structure of ${\mathrm{Li}}_4\mathrm{HN}$ near the type-$\mathsf{A}$ triple point on the $\mathrm{\Lambda }$ line with little cogroup $4/m^{\prime }$. (a) Band structure along lines of symmetry. The triple point is indicated by a yellow dot and arrow and the bands forming the triple point are labeled by their irreducible corepresentations. (b) Brillouin zone (boundary in gray) with points and lines of symmetry (black dashed lines). There are no mirror planes in this space group. (c) Size of the lower (red) and upper (blue) gaps, encoded by the intensity of the color (with a cutoff in gap size at $0.02\mathrm{eV}$), in two horizontal planes (at fixed $k_z$) one slightly below (left) and one slightly above (right) the triple point. We find that no nodal-line arcs attach to the triple point, but we show the attachment of nodal-line arcs to a nexus point in Fig. 28.

Figure 28

Nodal-line structure of ${\mathrm{Li}}_4\mathrm{HN}$ near the nexus point on the $\mathrm{\Lambda }$ line with little cogroup $4/m^{\prime }$. Size of the lower (red) and upper (blue) gaps, encoded by the intensity of the color (with a cutoff in gap size at $0.02\mathrm{eV}$), in horizontal planes at $k_z\in \{0.48,0.5,0.52,0.54,0.56\}$. We observe that four red nodal-line arcs attach to the nexus point lying on the red central nodal line, such that $n_a^{\mathrm{nexus}}=4$. Furthermore, the nodal lines' symmetric arrangement indicates a quadratic attachment to the nexus point and therefore $\mu =2$.

Figure 29

Nodal-line structure of ${\mathrm{MgH}}_2{\mathrm{O}}_2$ near the type-${\mathsf{B}}_l$ triple point on the $P$ line with little cogroup ${\overline{3}}^{\prime }$. The organization of the panels is in one-to-one correspondence with Fig. 14 with the difference that the shown plane is not a mirror plane. The cutoff on the gap size is $0.05\mathrm{eV}$. Instead, we have performed DFT calculations in planes perpendicular to the central line and determined the approximate planes in which nodal lines attaching to the triple point appear (see [63]). Here that plane is the $\mathrm{\Gamma }\mathrm{KH}$ plane and its three mirror-symmetry-related copies, implying $n_a^{\mathrm{nexus}}=6$.

Figure 30

Nodal-line structure of ${\mathrm{Li}}_2{\mathrm{Co}}_{12}{\mathrm{P}}_7$ near the type-$\mathsf{A}$ triple point on the $\mathrm{\Delta }$ line with little cogroup ${\overline{6}}^{\prime }$. The organization of the panels is in one-to-one correspondence with Fig. 14 with the difference that the shown planes are not mirror planes. The cutoff on the gap size is $0.01\mathrm{eV}$. The little cogroup ${\overline{6}}^{\prime }$ contains neither a vertical mirror symmetry nor $\mathcal{P}\mathcal{T}$ symmetry, such that nodal lines away from the rotation axis cannot be stabilized.

Figure 31

Nodal-line structure of ${\mathrm{C}}_3{\mathrm{N}}_4$ (SG 176) near the type-${\mathsf{B}}_q$ triple point at $E_{\mathrm{TP}}=-9.3\mathrm{eV}$ on the $\mathrm{\Delta }$ line with little cogroup $6/m^{\prime }$ The organization of the panels is in one-to-one correspondence with Fig. 14 with the difference that the shown plane is not a mirror plane. The cutoff on the gap size is $0.05\mathrm{eV}$. Using additional DFT calculations in planes perpendicular to the central nodal line (see [63]), we have determined that six nodal lines approximately lie in the planes indicated in (b), each implying $n_a^{\mathrm{nexus}}=12$. The green plane is spanned by ${\mathbf{k}}_{\mathrm{\Gamma }\mathrm{A}}$ and $\mathbf{k}=0.2{\mathbf{k}}_{\mathrm{\Gamma }\mathrm{M}}+0.11{\mathbf{k}}_{\mathrm{\Gamma }{\mathrm{M}}^{\prime }}$, where $M^{\prime }$ is the $M$ point rotated by $\frac{\pi }{3}$, and the red plane by ${\mathbf{k}}_{\mathrm{\Gamma }\mathrm{A}}$ and $\mathbf{k}=0.02{\mathbf{k}}_{\mathrm{\Gamma }\mathrm{M}}+0.2{\mathbf{k}}_{\mathrm{\Gamma }{\mathrm{M}}^{\prime }}$.

Figure 32

Nodal-line structure of ${\mathrm{C}}_3{\mathrm{N}}_4$ (SG 176) near the type-$\mathsf{A}$ triple point at $E_{\mathrm{TP}}=-2.6\mathrm{eV}$ on the $\mathrm{\Delta }$ line with little cogroup $6/m^{\prime }$. The organization of the panels is in one-to-one correspondence with Fig. 27 (but with a cutoff of $0.02\mathrm{eV}$) and again there are no nodal-line arcs connecting to the triple point.