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Progress on topological nodal line and nodal surface

Wang Shan-Shan ,  Wu Wei-Kang ,  Yang Sheng-Yuan

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Progress on topological nodal line and nodal surface

Wang Shan-Shan, Wu Wei-Kang, Yang Sheng-Yuan
Acta Phys. Sin., 2019, 68(22): 227101.
doi: 10.7498/aps.68.20191538
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  • Abstract

    Electronic band crossing can not only form zero-dimensional nodal points, but also one dimensional nodal lines and two dimensional nodal surfaces. These topological band features have been attracting significant research interest, as they may lead to many special physical properties. In this article, we review the progress in this field, including the conceptual development, the character and classification of these nodal structures, and the material realization.

    Authors and contacts

    • 1.

      School of Physics, Southeast University, Nanjing 211189, China

    • 2.

      Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore

      • Corresponding author: Wang Shan-Shan, 101012564@seu.edu.cn
    • Funds: Project supported by the Excellent Scholar Project of Southeast University, Class A, China and the Singapore Ministry of Education Academic Research Fund Tier 2 (Grant No. MOE-2017-T2-2-108)

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  • 图 1  关于拓扑绝缘体中拓扑边缘态的简单图像

    Figure 1.  A schematic figure for the topological boundary state in a topological insulator.

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    图 2  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, $\ell $是环绕节线一个路径, 沿着$\ell $走一圈的贝利相为π[61]

    Figure 2.  Schematic figure of a nodal loop: (a) Nodal loop formed by two crossing bands; (b) the Berry phase of a closed path $\ell $ circling the nodal loop (green circle) is π[61].

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    图 3  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]

    Figure 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].

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    图 4  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]

    Figure 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].

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    图 5  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

    Figure 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.

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    图 6  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]

    Figure 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.

    DownLoad: Full-Size Img PowerPoint

    图 7  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]

    Figure 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].

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    图 8  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]

    Figure 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].

    DownLoad: Full-Size Img PowerPoint

    图 9  (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]

    Figure 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 to Γ-A, which clearly demonstrates a quadratic dispersion[83].

    DownLoad: Full-Size Img PowerPoint

    图 10  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]

    Figure 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].

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    图 11  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]

    Figure 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].

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    图 12  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]

    Figure 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].

    DownLoad: Full-Size Img PowerPoint

    图 13  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]

    Figure 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].

    DownLoad: Full-Size Img PowerPoint

    图 14  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]

    Figure 14.  Two kinds of nodal surfaces: (a) Nodal surfaces in a 3D carbon allotrope[63]; (b) nodal surface in BaMX3[60].

    DownLoad: Full-Size Img PowerPoint

    图 15  具有第二类节面的材料　(a) K6YO4; (b) TlMo3Te3[63]

    Figure 15.  Materials with Class-II nodal surfaces: (a) K6YO4; (b) TlMo3Te3[63].

    DownLoad: Full-Size Img PowerPoint

    图 16  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]

    Figure 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].

    DownLoad: Full-Size Img PowerPoint

    图 17  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]

    Figure 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].

    DownLoad: Full-Size Img PowerPoint

    图 18  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]

    Figure 18.  Materials with multiple nodal surfaces: (a) Cu3Se2; (b) Rb2Se5. The location of the nodal surfaces is indicated by the orange color[63].

    DownLoad: Full-Size Img PowerPoint

    图 19  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]

    Figure 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].

    DownLoad: Full-Size Img PowerPoint
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    [20] Guo Zeng-Bao. . Acta Physica Sinica, 2002, 51(10): 2344-2348. doi: 10.7498/aps.51.2344
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  • Abstract views:  25474
  • PDF Downloads:  1381
  • Cited By: 0
Publishing process
  • Received Date:  09 October 2019
  • Accepted Date:  04 November 2019
  • Available Online:  19 November 2019
  • Published Online:  20 November 2019

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