Research progress on topological nodal line semimetals

ZeYing ZHANG; BoTao FU; Zhi-Ming YU; YuGui YAO

doi:10.1360/SSPMA-2020-0149

SCIENTIA SINICA Physica, Mechanica & Astronomica

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SCIENTIA SINICA Physica, Mechanica & Astronomica, Volume 50 , Issue 9 : 090002(2020) https://doi.org/10.1360/SSPMA-2020-0149

Research progress on topological nodal line semimetals

ZeYing ZHANG ^1,2, BoTao FU ^1,3, Zhi-Ming YU ¹, YuGui YAO ^1,*

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ReceivedApr 27, 2020
AcceptedJun 17, 2020
PublishedAug 11, 2020

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Abstract

Topological quantum materials, such as topological insulators, topological semimetals, and topological superconductors, are the research focus of modern condensed matter physics and materials science. The purpose of this article is to give a brief introduction on the research progress of topological nodal line semimetal materials in recent years, and to focus on the latest research progress of our group in this field. Topological semimetals can be classified mainly by two aspects: energy band degeneracy and dispersion. In terms of energy band degeneracy, this article will introduce nodal line semimetals protected by different symmetries including nodal line semimetals protected by symmorphic space groups and non-symmorphic space groups in the system with time reversion symmetry, and nodal line semimetals coupled with magnetic and topological orders without time reversion symmetry, respectively. In terms of energy band dispersion, this article will introduce high-order nodal line semimetals and type-II nodal line semimetals.

Funded by

国家自然科学基金(11734003)

国家重点研发计划(2016YFA0300600)

中国科学院战略先导科技专项(B类)

中央高校基本科研业务费专项资金(ZY2018)

nodal line semimetal

topological semimetal

first-principles calculation

References

[1] Klitzing K V, Dorda G, Pepper M. New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance. Phys Rev Lett, 1980, 45: 494-497 CrossRef ADS Google Scholar

[2] Laughlin R B. Quantized Hall conductivity in two dimensions. Phys Rev B, 1981, 23: 5632-5633 CrossRef ADS Google Scholar

[3] Thouless D J, Kohmoto M, Nightingale M P, et al. Quantized Hall conductance in a two-dimensional periodic potential. Phys Rev Lett, 1982, 49: 405-408 CrossRef ADS Google Scholar

[4] Haldane F D M. Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the “parity anomaly”. Phys Rev Lett, 1988, 61: 2015. Google Scholar

[5] Kane C L, Mele E J. Z₂ topological order and the quantum spin hall effect. Phys Rev Lett, 2005, 95: 146802 CrossRef ADS arXiv Google Scholar

[6] Bernevig B A, Hughes T L, Zhang S C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science, 2006, 314: 1757-1761 CrossRef ADS arXiv Google Scholar

[7] Konig M, Wiedmann S, Brune C, et al. Quantum spin Hall insulator state in HgTe quantum wells. Science, 2007, 318: 766-770 CrossRef ADS arXiv Google Scholar

[8] Chang C Z, Zhang J, Feng X, et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science, 2013, 340: 167-170 CrossRef ADS arXiv Google Scholar

[9] Tokura Y, Yasuda K, Tsukazaki A. Magnetic topological insulators. Nat Rev Phys, 2019, 1: 126-143 CrossRef ADS Google Scholar

[10] Hasan M Z, Kane C L. Colloquium: Topological insulators. Rev Mod Phys, 2010, 82: 3045-3067 CrossRef ADS arXiv Google Scholar

[11] Fu L. Topological crystalline insulators. Phys Rev Lett, 2011, 106: 106802 CrossRef ADS arXiv Google Scholar

[12] Wan X, Turner A M, Vishwanath A, et al. Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys Rev B, 2011, 83: 205101 CrossRef ADS arXiv Google Scholar

[13] Nielsen H B, Ninomiya M. The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal. Phys Lett B, 1983, 130: 389-396 CrossRef Google Scholar

[14] Wang Z, Sun Y, Chen X Q, et al. Dirac semimetal and topological phase transitions in A₃Bi(A=Na, K, Rb). Phys Rev B, 2012, 85: 195320 CrossRef ADS Google Scholar

[15] Young S M, Zaheer S, Teo J C Y, et al. Dirac semimetal in three dimensions. Phys Rev Lett, 2012, 108: 140405 CrossRef ADS arXiv Google Scholar

[16] Huang H, Liu J, Vanderbilt D, et al. Topological nodal-line semimetals in alkaline-earth stannides, germanides, and silicides. Phys Rev B, 2016, 93: 201114 CrossRef ADS arXiv Google Scholar

[17] Chiu C K, Schnyder A P. Classification of reflection-symmetry-protected topological semimetals and nodal superconductors. Phys Rev B, 2014, 90: 205136 CrossRef ADS arXiv Google Scholar

[18] Armitage N P, Mele E J, Vishwanath A. Weyl and Dirac semimetals in three-dimensional solids. Rev Mod Phys, 2018, 90: 015001 CrossRef ADS arXiv Google Scholar

[19] Long Z, Wang Y, Erukhimova M, et al. Magnetopolaritons in Weyl Semimetals in a strong magnetic field. Phys Rev Lett, 2018, 120: 037403 CrossRef ADS arXiv Google Scholar

[20] Guo Z P, Lu P C, Chen T, et al. High-pressure phases of Weyl semimetals NbP, NbAs, TaP, and TaAs. Sci China-Phys Mech Astron, 2018, 61: 038211 CrossRef ADS arXiv Google Scholar

[21] Li Q Y, Lv Y Y, Wang J H, et al. Turning ZrTe₅ into a semiconductor through atom intercalation. Sci China-Phys Mech Astron, 2019, 62: 967812 CrossRef ADS Google Scholar

[22] Zhang S F, Zhang C W, Wang P J, et al. Low-energy electronic properties of a Weyl semimetal quantum dot. Sci China-Phys Mech Astron, 2018, 61: 117811 CrossRef ADS arXiv Google Scholar

[23] Zirnbauer M R. Riemannian symmetric superspaces and their origin in random‐matrix theory. J Math Phys, 1996, 37: 4986-5018 CrossRef ADS arXiv Google Scholar

[24] Altland A, Zirnbauer M R. Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys Rev B, 1997, 55: 1142-1161 CrossRef ADS arXiv Google Scholar

[25] Chiu C K, Teo J C Y, Schnyder A P, et al. Classification of topological quantum matter with symmetries. Rev Mod Phys, 2016, 88: 035005 CrossRef ADS arXiv Google Scholar

[26] Wang Z, Alexandradinata A, Cava R J, et al. Hourglass fermions. Nature, 2016, 532: 189-194 CrossRef ADS arXiv Google Scholar

[27] Bradlyn B, Elcoro L, Cano J, et al. Topological quantum chemistry. Nature, 2017, 547: 298-305 CrossRef ADS arXiv Google Scholar

[28] Alexandradinata A, Wang Z, Bernevig B A. Topological insulators from group cohomology. Phys Rev X, 2016, 6: 021008 CrossRef ADS arXiv Google Scholar

[29] Po H C, Vishwanath A, Watanabe H. Erratum: Symmetry-based indicators of band topology in the 230 space groups. Nat Commun, 2017, 8: 931 CrossRef ADS Google Scholar

[30] Watanabe H, Po H C, Vishwanath A. Structure and topology of band structures in the 1651 magnetic space groups. Sci Adv, 2018, 4: eaat8685 CrossRef ADS arXiv Google Scholar

[31] Tang F, Po H C, Vishwanath A, et al. Comprehensive search for topological materials using symmetry indicators. Nature, 2019, 566: 486-489 CrossRef ADS Google Scholar

[32] Zhang T, Jiang Y, Song Z, et al. Catalogue of topological electronic materials. Nature, 2019, 566: 475-479 CrossRef ADS arXiv Google Scholar

[33] Vergniory M G, Elcoro L, Felser C, et al. A complete catalogue of high-quality topological materials. Nature, 2019, 566: 480-485 CrossRef ADS arXiv Google Scholar

[34] Bradlyn B, Cano J, Wang Z, et al. Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals. Science, 2016, 353: aaf5037 CrossRef Google Scholar

[35] Burkov A A, Hook M D, Balents L. Topological nodal semimetals. Phys Rev B, 2011, 84: 235126 CrossRef ADS arXiv Google Scholar

[36] Fang C, Weng H, Dai X, et al. Topological nodal line semimetals. Chin Phys B, 2016, 25: 117106 CrossRef ADS arXiv Google Scholar

[37] Bradlyn B, Elcoro L, Vergniory M G, et al. Band connectivity for topological quantum chemistry: Band structures as a graph theory problem. Phys Rev B, 2018, 97: 035138 CrossRef ADS arXiv Google Scholar

[38] Bzdušek T, Wu Q S, Rüegg A, et al. Nodal-chain metals. Nature, 2016, 538: 75-78 CrossRef ADS arXiv Google Scholar

[39] Fu B, Fan X, Ma D, et al. Hourglasslike nodal net semimetal in Ag₂BiO₃. Phys Rev B, 2018, 98: 075146 CrossRef ADS arXiv Google Scholar

[40] Bradley C, Cracknell A. Mathematical Theory of Symmetry in Solids: Representation Theory for Point Groups and Space Groups. Oxford: Oxford University Press, 2009. Google Scholar

[41] Soluyanov A A, Gresch D, Wang Z, et al. Type-II Weyl semimetals. Nature, 2015, 527: 495-498 CrossRef ADS arXiv Google Scholar

[42] Li S, Yu Z M, Liu Y, et al. Type-II nodal loops: Theory and material realization. Phys Rev B, 2017, 96: 081106 CrossRef ADS arXiv Google Scholar

[43] Yu Z M, Wu W, Sheng X L, et al. Quadratic and cubic nodal lines stabilized by crystalline symmetry. Phys Rev B, 2019, 99: 121106 CrossRef ADS arXiv Google Scholar

[44] Ma D S, Zhou J, Fu B, et al. Mirror protected multiple nodal line semimetals and material realization. Phys Rev B, 2018, 98: 201104 CrossRef ADS arXiv Google Scholar

[45] Li X P, Deng K, Fu B, et al. Type-III Weyl semimetals and its materialization, arXiv: 1909.12178. arXiv Google Scholar

[46] Wang S S, Liu Y, Yu Z M, et al. Hourglass Dirac chain metal in rhenium dioxide. Nat Commun, 2017, 8: 1844 CrossRef ADS arXiv Google Scholar

[47] Yu R, Wu Q, Fang Z, et al. From nodal chain semimetal to Weyl semimetal in HfC. Phys Rev Lett, 2017, 119: 036401 CrossRef ADS arXiv Google Scholar

[48] Weng H, Liang Y, Xu Q, et al. Topological node-line semimetal in three-dimensional graphene networks. Phys Rev B, 2015, 92: 045108 CrossRef ADS arXiv Google Scholar

[49] Chen W, Lu H Z, Hou J M. Topological semimetals with a double-helix nodal link. Phys Rev B, 2017, 96: 041102 CrossRef ADS arXiv Google Scholar

[50] Yan Z, Bi R, Shen H, et al. Nodal-link semimetals. Phys Rev B, 2017, 96: 041103 CrossRef ADS arXiv Google Scholar

[51] Chang P Y, Yee C H. Weyl-link semimetals. Phys Rev B, 2017, 96: 081114 CrossRef ADS arXiv Google Scholar

[52] Zhou Y, Xiong F, Wan X, et al. Hopf-link topological nodal-loop semimetals. Phys Rev B, 2018, 97: 155140 CrossRef ADS Google Scholar

[53] Bzdušek T, Sigrist M. Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems. Phys Rev B, 2017, 96: 155105 CrossRef ADS arXiv Google Scholar

[54] Wu W, Liu Y, Li S, et al. Nodal surface semimetals: Theory and material realization. Phys Rev B, 2018, 97: 115125 CrossRef ADS arXiv Google Scholar

[55] Türker O, Moroz S. Weyl nodal surfaces. Phys Rev B, 2018, 97: 075120 CrossRef ADS arXiv Google Scholar

[56] Bian G, Chang T R, Zheng H, et al. Drumhead surface states and topological nodal-line fermions in TlTaSe₂. Phys Rev B, 2016, 93: 121113 CrossRef ADS arXiv Google Scholar

[57] Bian G, Chang T R, Sankar R, et al. Topological nodal-line fermions in spin-orbit metal PbTaSe₂. Nat Commun, 2016, 7: 10556 CrossRef ADS Google Scholar

[58] Feng B, Fu B, Kasamatsu S, et al. Experimental realization of two-dimensional Dirac nodal line fermions in monolayer Cu₂Si. Nat Commun, 2017, 8: 1007 CrossRef ADS arXiv Google Scholar

[59] Feng B, Zhang R W, Feng Y, et al. Discovery of Weyl nodal lines in a single-layer ferromagnet. Phys Rev Lett, 2019, 123: 116401 CrossRef ADS arXiv Google Scholar

[60] Hu J, Tang Z, Liu J, et al. Evidence of topological nodal-line Fermions in ZrSiSe and ZrSiTe. Phys Rev Lett, 2016, 117: 016602 CrossRef ADS arXiv Google Scholar

[61] Li C, Wang C M, Wan B, et al. Rules for phase shifts of quantum oscillations in topological nodal-line semimetals. Phys Rev Lett, 2018, 120: 146602 CrossRef ADS arXiv Google Scholar

[62] Joynt R. Discrete scale invariance and ln(B) periodic quantum oscillations in topological semimetals. Sci China-Phys Mech Astron, 2019, 62: 37431 CrossRef ADS Google Scholar

[63] Li R, Li J, Wang L, et al. Underlying topological Dirac nodal line mechanism of the anomalously large electron-phonon coupling strength on a Be (0001) surface. Phys Rev Lett, 2019, 123: 136802 CrossRef ADS arXiv Google Scholar

[64] Li J, Xie Q, Liu J, et al. Phononic Weyl nodal straight lines in MgB₂. Phys Rev B, 2020, 101: 024301 CrossRef ADS Google Scholar

[65] Li J, Wang L, Liu J, et al. Topological phonons in graphene. Phys Rev B, 2020, 101: 081403 CrossRef ADS Google Scholar

[66] Chan Y H, Chiu C K, Chou M Y, et al. Ca₃P₂ and other topological semimetals with line nodes and drumhead surface states. Phys Rev B, 2016, 93: 205132 CrossRef ADS arXiv Google Scholar

[67] Young S M, Kane C L. Dirac semimetals in two dimensions. Phys Rev Lett, 2015, 115: 126803 CrossRef ADS arXiv Google Scholar

[68] Fang C, Chen Y, Kee H Y, et al. Topological nodal line semimetals with and without spin-orbital coupling. Phys Rev B, 2015, 92: 081201 CrossRef ADS arXiv Google Scholar

[69] Li S, Liu Y, Fu B, et al. Almost ideal nodal-loop semimetal in monoclinic CuTeO₃ material. Phys Rev B, 2018, 97: 245148 CrossRef ADS arXiv Google Scholar

[70] Li S, Liu Y, Wang S S, et al. Nonsymmorphic-symmetry-protected hourglass Dirac loop, nodal line, and Dirac point in bulk and monolayer X₃SiTe₆(X = Ta, Nb). Phys Rev B, 2018, 97: 045131 CrossRef ADS arXiv Google Scholar

[71] Zhang R W, Liu C C, Ma D S, et al. Nodal-line semimetal states in the positive-electrode material of a lead-acid battery: Lead dioxide family and its derivatives. Phys Rev B, 2018, 98: 035144 CrossRef ADS arXiv Google Scholar

[72] Zhang R W, Liu C C, Ma D S, et al. From node-line semimetals to large-gap quantum spin Hall states in a family of pentagonal group-IVA chalcogenide. Phys Rev B, 2018, 97: 125312 CrossRef ADS arXiv Google Scholar

[73] Zhang T T, Yu Z M, Guo W, et al. From type-II triply degenerate nodal points and three-band nodal rings to type-II Dirac points in centrosymmetric zirconium oxide. J Phys Chem Lett, 2017, 8: 5792-5797 CrossRef Google Scholar

[74] Li R, Ma H, Cheng X, et al. Dirac node lines in pure alkali earth metals. Phys Rev Lett, 2016, 117: 096401 CrossRef ADS arXiv Google Scholar

[75] Topp A, Queiroz R, Grüneis A, et al. Surface floating 2D bands in layered nonsymmorphic semimetals: ZrSiS and related compounds. Phys Rev X, 2017, 7: 041073 CrossRef ADS arXiv Google Scholar

[76] Fu B B, Yi C J, Zhang T T, et al. Dirac nodal surfaces and nodal lines in ZrSiS. Sci Adv, 2019, 5: eaau6459 CrossRef ADS Google Scholar

[77] Yang L M, Bačić V, Popov I A, et al. Two-dimensional Cu₂Si monolayer with planar hexacoordinate copper and silicon bonding. J Am Chem Soc, 2015, 137: 2757-2762 CrossRef Google Scholar

[78] Curson N J, Bullman H G, Buckland J R, et al. Interaction of silane with Cu(111): Surface alloy and molecular chemisorbed phases. Phys Rev B, 1997, 55: 10819-10829 CrossRef ADS Google Scholar

[79] Ménard H, Horn A B, Tear S P. Methylsilane on Cu(111), a STM study of the R30°-Cu₂Si surface silicide. Surf Sci, 2005, 585: 47-52 CrossRef ADS Google Scholar

[80] Fan X, Ma D, Fu B, et al. Cat’s-cradle-like Dirac semimetals in layer groups with multiple screw axes: Application to two-dimensional borophene and borophane. Phys Rev B, 2018, 98: 195437 CrossRef ADS arXiv Google Scholar

[81] Mostofi A A, Yates J R, Lee Y S, et al. Wannier90: A tool for obtaining maximally-localised wannier functions. Comput Phys Commun, 2008, 178: 685-699 CrossRef ADS arXiv Google Scholar

[82] Lopez Sancho M P, Lopez Sancho J M, Sancho J M L, et al. Highly convergent schemes for the calculation of bulk and surface Green functions. J Phys F-Met Phys, 1985, 15: 851-858 CrossRef ADS Google Scholar

[83] Deibele S, Jansen M. Bismuth in Ag₂BiO₃: Tetravalent or internally disproportionated?. J Solid State Chem, 1999, 147: 117-121 CrossRef ADS Google Scholar

[84] Muhlbauer S, Binz B, Jonietz F, et al. Skyrmion lattice in a chiral magnet. Science, 2009, 323: 915-919 CrossRef ADS arXiv Google Scholar

[85] Wang Z, Vergniory M G, Kushwaha S, et al. Time-reversal-breaking Weyl Fermions in magnetic heusler alloys. Phys Rev Lett, 2016, 117: 236401 CrossRef ADS arXiv Google Scholar

[86] Huang X, Zhao L, Long Y, et al. Observation of the chiral-anomaly-induced negative magnetoresistance in 3D Weyl semimetal TaAs. Phys Rev X, 2015, 5: 031023 CrossRef ADS arXiv Google Scholar

[87] Hanke J P, Freimuth F, Niu C, et al. Mixed Weyl semimetals and low-dissipation magnetization control in insulators by spin-orbit torques. Nat Commun, 2017, 8: 1479 CrossRef ADS Google Scholar

[88] Zhang R W, Zhang Z, Liu C C, et al. Nodal line spin-gapless semimetals and high-quality candidate materials. Phys Rev Lett, 2020, 124: 016402 CrossRef ADS arXiv Google Scholar

[89] Wu H, Ma D S, Fu B, et al. Weyl nodal point-line Fermion in ferromagnetic Eu₅Bi₃. J Phys Chem Lett, 2019, 10: 2508-2514 CrossRef Google Scholar

[90] Zhang Z, Gao Q, Liu C C, et al. Magnetization-direction tunable nodal-line and Weyl phases. Phys Rev B, 2018, 98: 121103 CrossRef ADS Google Scholar

[91] Zhou X, Zhang R W, Zhang Z, et al. Fully spin-polarized nodal loop semimetals in alkaline metal monochalcogenide monolayers. J Phys Chem Lett, 2019, 10: 3101-3108 CrossRef Google Scholar

[92] Hepworth M A, Jack K H, Peacock R D, et al. The crystal structures of the trifluorides of iron, cobalt, ruthenium, rhodium, palladium and iridium. Acta Cryst, 1957, 10: 63-69 CrossRef Google Scholar

[93] Müller B G, Serafin M. The crystal structure of manganese tetrafluoride. Z Kristallogr, 1987, 42: 1102. Google Scholar

[94] Liu G B, Shan W Y, Yao Y, et al. Three-band tight-binding model for monolayers of group-VIB transition metal dichalcogenides. Phys Rev B, 2013, 88: 085433 CrossRef ADS arXiv Google Scholar

[95] Effenberger H, Mereiter Κ, Zemann J. Crystal structure refinements of magnesite, calcite, rhodochrosite, siderite, smithonite, and dolomite, with discussion of some aspects of the stereochemistry of calcite type carbonates. Z Kristallogr, 1981, 156: 233. Google Scholar

[96] Moreno L C, Valencia J S, Landínez Téllez D A, et al. Preparation and structural study of LaMnO₃ magnetic material. J Magn Magn Mater, 2008, 320: e19-e21 CrossRef ADS Google Scholar

[97] Huber M, Deiseroth H J. Crystal structure of titanium(III) borate, TiBO₃. Z Kristallogr, 1995, 210: 685. Google Scholar

[98] Kopnin N B, Heikkilä T T, Volovik G E. High-temperature surface superconductivity in topological flat-band systems. Phys Rev B, 2011, 83: 220503 CrossRef ADS arXiv Google Scholar

[99] Chen H, Zhu W, Xiao D, et al. CO oxidation facilitated by robust surface states on Au-covered topological insulators. Phys Rev Lett, 2011, 107: 056804 CrossRef ADS arXiv Google Scholar

[100] Takayama H, Bohnen K P, Fulde P. Magnetic surface anisotropy of transition metals. Phys Rev B, 1976, 14: 2287-2295 CrossRef ADS Google Scholar

[101] Kim Y S, Chung Y C. Magnetic and half-metallic properties of Cr-doped /spl beta/-SiC. IEEE Trans Magn, 2005, 41: 2733-2735 CrossRef ADS Google Scholar

[102] Shoenberg D. Magnetic Oscillations in Metals. Cambridge: Cambridge University Press, 1984. Google Scholar

[103] Mikitik G P, Sharlai Y V. Manifestation of Berry’s phase in metal physics. Phys Rev Lett, 1999, 82: 2147-2150 CrossRef ADS Google Scholar

[104] Zumdick M F, Hoffmann R D, Pöttgen R. The intermetallic zirconium compounds ZrNiAl, ZrRhSn, and ZrPtGa-structural distortions and metal-metal bonding in Fe₂P related compounds. Zeitschrift für Naturforschung B, 2014, 54: 45. Google Scholar

[105] Mullen K, Uchoa B, Glatzhofer D T. Line of Dirac nodes in hyperhoneycomb lattices. Phys Rev Lett, 2015, 115: 026403 CrossRef ADS arXiv Google Scholar

[106] O’Brien T E, Diez M, Beenakker C W J. Magnetic breakdown and Klein tunneling in a type-II Weyl semimetal. Phys Rev Lett, 2016, 116: 236401 CrossRef ADS arXiv Google Scholar

[107] Chang T R, Pletikosic I, Kong T, et al. Realization of a type-II nodal-line semimetal in Mg₃Bi₂. Adv Sci, 2019, 6: 1800897 CrossRef Google Scholar

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Figure 1
(Color online) (a) Crystal structure of Cu₂Si; (b) band structure when spin-orbit coupling is not considered; (c) Fermi surface of Cu₂Si; (d) derivative constant energy contours measured using 30-eV p-polarized photons. Pictures are taken from ref. [58] with permission.
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Figure 2
(Color online) (a) Crystal structure of Ag₂BiO₃; (b) the band structure at the XURS plane. The eigenvalues of ${M_{x} | 0 \frac{1}{2} \frac{1}{2}}$ are shown.(c) The band structure of Ag₂BiO₃ along high symmetry line; (d) the Fermi surface of Ag₂BiO₃; (e) the calculated surface states of Ag₂BiO₃ on the (001) surface plane; (f) display of the gapless points in the extended Brillouin zone. The color represents the energy dispersion of each point with respect to the Fermi level. Pictures are taken from ref. [39] with permission.
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Figure 3
(Color online) (a) Crystal structure of XF₃ (X=Pd, Mn); (b) band structure of PdF₃, the red line stand for the first-principles calculation results, the black line is from the tight-binding mode. (c), (d) The profile of the gapless points of PdF₃ in Brillouin zone, including the top views and the side views. Different colors correspond to different orientations of the nodal lines. (e) The surface states of PdF₃ for surface (111). Pictures are taken from ref. [88] with permission.
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Figure 4
(Color online) (a) Schematic diagram of the D_2d lattice model, including three C₂ rotation axes and two mirror operations; (b) band structure with (the magnetization direction is parallel to the z-axis) and without spin-orbit coupling; (c), (d) the gapless point of the system in the Brillouin zone with respect to θ, where (c) is side view and (d) is top view. Pictures are taken from ref. [90] with permission.
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Figure 5
(Color online) (a)-(c) are the top view and side views of the crystal structure and the Brillouin zone of ZrPtGa, respectively; (d) calculated band structure along high symmetry lines for ZrPtGa with spin-orbit coupling; (e) the band dispersion around a generic point, where P and Q are the midpoints of Г-A and K-H, respectively. Pictures are taken from ref. [43] with permission.
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Figure 6
(Color online) (a) Schematic figure of a nodal loop, where q₁ and q₂ label the two mutually perpendicular transverse directions; (b) and (c) illustrate the type-I and type-II nodal loops dispersions along the q₁ direction, respectively. Pictures are taken from ref. [42] with permission.

Table 1 All possible higher-order nodal line protected by crystalline symmetry. The zero-term in the effective Hamiltonian is omitted because it does not affect the classification of the nodal line. The form is taken from ref. [43]

空间群号	对应点群	节线的阶	$k$ 点路径	不可约表示	有效哈密顿量
174	C_3h	二次型	Г-A	{Г₄, Г₅}	$α q_{-}^{2} σ_{+} + H . c .$
187-190	D_3h	二次型	Г-A	Г₄	$α q_{-}^{2} σ_{+} + H . c .$
183-186	C_6v	三次型	Г-A	Г₉	$i (α q_{-}^{3} + β q_{-}^{3}) σ_{+} + H . c .$

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03.65.Vf	Phases: geometric; dynamic or topological
71.20.Gj	Other metals and alloys
75.50.-y	Studies of specific magnetic materials

Research progress on topological nodal line semimetals

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