Fig. 1. A schematic figure for the topological boundary state in a topological insulator.关于拓扑绝缘体中拓扑边缘态的简单图像

Fig. 2. Schematic figure of a nodal loop: (a) Nodal loop formed by two crossing bands; (b) the Berry phase of a closed path circling the nodal loop (green circle) is π^[61]. 节线的示意图　(a)由两条能带交叉所形成的节线; (b)绿色的环代表节线, 是环绕节线一个路径, 沿着 走一圈的贝利相为π^[61]

Fig. 3. Chiral symmetry protected nodal line in a Dirac superconductor: (a) A Dirac node can evolve into a nodal ring or two Weyl nodes under different symmetry breaking; (b)−(d) illustrate the different topological protection for the degeneracies in (a). Here, the nodal ring is protected by the winding number^[51]. 在狄拉克超导体中出现的由手征对称性保护的节线　(a)当空间反演或者时间反演破坏时, 一个狄拉克点会变为一个节环或两个外尔点; (b)−(d)刻画了(a)中几种简并点的拓扑保护机制, 其中节环是由拓扑绕数所保护^[51]

Fig. 4. Nodal lines found in three carbon allotropes: (a) 3D carbon with Mackay-Terrones crystal structur^[52]; (b) 3D hyperhoneycomb carbon^[53]; (c) 3D graphene network structure^[56]. 在三种碳材料中发现的节线　(a) Mackay-Terrones结构的三维碳和节线在动量空间的表示^[52]; (b) hyperhoneycomb结构的三维碳和节线在动量空间的表示^[53]; (c)三维的石墨烯网络结构和节线在动量空间的表示^[56]

Fig. 5. Nodal line protected by the glide mirror symmetry: (a) Shows the glide-mirror-invariant plane in Brillouin zone, O and X are two TRIM points with different glide mirror eigenvalues; (b) shows the band structure along a path L connecting O and X (as in (a)); it displays an hourglass shaped spectrum. The degeneracy point P in the hourglass traces out a nodal loop in the glide mirror plane. 滑移镜面所保护的节线　(a) O和X是滑移镜面上对应两个不同配对类型的TRIM点; (b)展示了沿着连接O和X的一条路径L上的能带特征, 这里每四条能带都会形成一种沙漏形的结构; 沙漏脖子处的交叉点P在滑移镜面上会形成一条节线

Fig. 6. Material examples with glide-mirror-protected nodal rings: (a) ReO₂^[73]; (b) X₃SiTe₆(X = Ta, Nb)^[74], the hourglass dispersions can be observed in their band structures. 具有滑移镜面所保护的节线的例子　(a) ReO₂的晶体结构和能带结构, 可以看到高对称线上的沙漏型色散^[73]; (b) X₃SiTe₆(X = Ta, Nb)的晶体结构和能带结构, 以及在高对称线上的沙漏型能量色散^[74]

Fig. 7. Three types of nodal lines classified by the energy dispersion: (a) Type-I nodal line; (b) type-II nodal lines; (c) hybrid nodal lines; (d)−(f) show the typical shapes of the constant energy surface for the three types^[64]. 三种不同色散类型的节线　(a) type-I节线; (b) type-II节线; (c) hybird节线; (d)−(f)三种节线的等能面^[64]

Fig. 8. Unique properties of type-II and hybrid nodal lines: (a) Comparison between type-I and type-II nodal lines in terms of JDOS and optical absorption rate^[61]; (b) the magnetic breakdown and its feature in anisotropic magnetic oscillation for a hybrid nodal loop^[64]. Type-II节线和hybrid节线的特殊物理性质　(a) Type-II节线和type-I节线的光学性质的比较^[61]; (b) hybrid节线导致的磁坍塌效应和磁振荡中的各向异性^[64]

Fig. 9. (a) Schematic figure for the higher order nodal lines; (b)−(d) show the quadratic nodal line in ZrPtGa: (c) the band structure of ZrPtGa, the blue solid curve indicates the quadratic nodal line; (d) shows the band dispersion in the plane perpendicular toΓ-A, which clearly demonstrates a quadratic dispersion^[83]. (a)按照节线的色散次数进行分类的示意图; (b)−(d)展示了一个具有二次节线的材料ZrPtGa, (c)是ZrPtGa的能带结构, 蓝色实线标记了二次节线, (d)是这个节线在垂直于Γ-A的平面上的色散, 可以清楚地看到是二次色散^[83]

Fig. 10. Nodal lines with different kinds of distribution in Brillouin zone: (a) Nodal lines in a carbon allotrope, which traverse the Brillouin zone^[56]; (b) nodal line in CuTeO₃, which is located around a point in Brillouin zone^[69]. 具有不同形态的节线　(a)穿越布里渊区的一对节线^[56]; (b)局域在布里渊区某个点周围的节线^[69]

Fig. 11. Different structures formed by nodal lines: (a) Crossed nodal rings^[38]; (b) nodal box^[89]; (c) inter-connected nodal loops^[90]; (d) nodal Hopf link^[91]; (e) weyl chain; (f) dirac chain^[73]. 节环可以形成的一些复杂结构　(a)笼子状的结构^[38]; (b)骨架状的结构^[89];(c)三能带形成的结状节线^[90]; (d) Hopf链环^[91]; (e)外尔链; (f)狄拉克链^[73]

Fig. 12. Stable nodal lines under SOC in 2D: (a)−(c) GaTeI family materials^[94]; (d)−(f) MnN monolayer, here MnN is a half metal, so the nodal loops are fully spin^[85]. 二维材料中在SOC作用下仍然稳定的节线　(a)−(c)二维GaTeI中的节线^[94]; (d)−(f)单层MnN中的节线, 单层MnN是一个铁磁材料, 在费米面处只存在一个自旋通道, 因此这里的节线是完全自旋极化的^[85]

Fig. 13. Surface states of nodal line metals: (a) Drumhead surface states for nodal rings in superconductors^[51]; (b) drumhead surface states in a 3D carbon allotrope^[52]; (c), (d) show the double drumhead surface states in ReO₂^[73]and Ta₃SiTe₆^[74]; (e) surface states of cubic nodal line, which spreads over the whole BZ^[83]. 节线对应的拓扑表面态　(a)狄拉克超导体中节线导致的鼓膜态^[51]; (b)碳的同素异形体中的鼓膜态^[52]; (c), (d) ReO₂^[73]和Ta₃SiTe₆^[74]中的双鼓膜态; (e)对应着三次节线的遍布布里渊区的环面表面态^[83]

Fig. 14. Two kinds of nodal surfaces: (a) Nodal surfaces in a 3D carbon allotrope^[63]; (b) nodal surface in BaMX₃^[60]. 两种不同的节面　(a)三维碳材料中的节面^[63]; (b) BaMX₃中的节面^[60]

Fig. 15. Materials with Class-II nodal surfaces: (a) K₆YO₄; (b) TlMo₃Te₃^[63]. 具有第二类节面的材料　(a) K₆YO₄; (b) TlMo₃Te₃^[63]

Fig. 16. Nodal surface robust against SOC: (a) Crystal structure of Ta₃TeI₇; (b) is the band structure of Ta₃TeI₇ in the presence of SOC with no gap opening^[63]. SOC作用下稳定的节面　(a)展示了Ta₃TeI₇晶体结构; (b)是Ta₃TeI₇在考虑SOC时的能带结构; 能带在考虑SOC时没有打开能隙^[63]

Fig. 17. Nodal surface in magnetic materials: (a) The crystal structure of CsCrI₃; (b) the band structure of CsCrI₃ without SOC; (c) and (d) band structures with magnetic moment along x and z directions respectively^[63]. 磁性材料中的节面　(a) CsCrI₃晶体结构; (b)不考虑SOC时的能带结构; (c), (d)考虑SOC时, 磁矩分别沿面内和面外时的能带结构^[63]

Fig. 18. Materials with multiple nodal surfaces: (a) Cu₃Se₂; (b) Rb₂Se₅. The location of the nodal surfaces is indicated by the orange color^[63]. 存在多个节面的材料　(a) Cu₃Se₂; (b) Rb₂Se₅; 布里渊区中的节面分布用橙色标记^[63]

Fig. 19. A method to circumvent the Nielson-Ninomiya no-go theorem: (a) Schematic figure showing the single Weyl point; (b) Berry curvature distribution; (c), (d) show that there is no surface Fermi arc emitted from the Weyl point, the white dot labels the surface projection of the Weyl point^[117]. 绕过Nielson-Ninomiya不可行定理的方法　(a)一个单独外尔点的示意图; (b)贝利曲率分布; (c), (d)显示了在表面上不存在连接单外尔点的费米弧表面, 白色点标记了体内外尔点在表面的投影^[117]