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  • Perspective
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The topology of electronic band structures

Nature Materials volume 20, pages 293–300 (2021)Cite this article

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Abstract

The study of topology as it relates to physical systems has rapidly accelerated during the past decade. Critical to the realization of new topological phases is an understanding of the materials that exhibit them and precise control of the materials chemistry. The convergence of new theoretical methods using symmetry indicators to identify topological material candidates and the synthesis of high-quality single crystals plays a key role, warranting discussion and context at an accessible level. This Perspective provides a broad introduction to topological phases, their known properties, and material realizations. We focus on recent work in topological Weyl and Dirac semimetals, with a particular emphasis on magnetic Weyl semimetals and emergent fermions in chiral crystals and their extreme responses to excitations, and we highlight areas where the field can continue to make remarkable discoveries. We further examine open questions and directions for the topological materials science community to pursue, including exploration of non-equilibrium properties of Weyl semimetals and cavity-dressed topological materials.

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Fig. 1: Dirac and Weyl physics.
Fig. 2: Large responses of topological materials.

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References

  1. Waldrop, M. M. The 1987 Nobel Prize for Physics: in one of the fastest awards on record, the prize goes to the discoverers of high-temperature superconductivity less than two years after the discovery was made. Science 238, 481–482 (1987).

    CAS  Google Scholar 

  2. Grünberg, P. A. Nobel Lecture: from spin waves to giant magnetoresistance and beyond. Rev. Mod. Phys. 80, 1531–1540 (2008).

    Google Scholar 

  3. Geim, A. K. Nobel Lecture: random walk to graphene. Rev. Mod. Phys. 83, 851–862 (2011).

    CAS  Google Scholar 

  4. Nakamura, S. Nobel Lecture: background story of the invention of efficient blue InGaN light emitting diodes. Rev. Mod. Phys. 87, 1139–1151 (2015).

    CAS  Google Scholar 

  5. Haldane, F. D. M. Nobel Lecture: Topological quantum matter. Rev. Mod. Phys. 89, 040502 (2017).

    Google Scholar 

  6. Po, H. C., Vishwanath, A. & Watanabe, H. Symmetry-based indicators of band topology in the 230 space groups. Nat. Commun. 8, 50 (2017).

    Google Scholar 

  7. Bradlyn, B. et al. Topological quantum chemistry. Nature 547, 298–305 (2017).

    CAS  Google Scholar 

  8. Watanabe, H., Po, H. C. & Vishwanath, A. Structure and topology of band structures in the 1651 magnetic space groups. Sci. Adv. 4, eaat8685 (2018).

    CAS  Google Scholar 

  9. Tang, F., Po, H. C., Vishwanath, A. & Wan, X. Efficient topological materials discovery using symmetry indicators. Nat. Phys. 15, 470–476 (2019).

    CAS  Google Scholar 

  10. Vergniory, M. G. et al. A complete catalogue of high-quality topological materials. Nature 566, 480–485 (2019).

    CAS  Google Scholar 

  11. Tang, F., Po, H. C., Vishwanath, A. & Wan, X. Comprehensive search for topological materials using symmetry indicators. Nature 566, 486–489 (2019).

    CAS  Google Scholar 

  12. Zhou, X. et al. Topological crystalline insulator states in the Ca2 As family. Phys. Rev. B 98, 241104 (2018).

    CAS  Google Scholar 

  13. Kane, C. L. & Mele, E. J. Z2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).

    CAS  Google Scholar 

  14. Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    CAS  Google Scholar 

  15. Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    CAS  Google Scholar 

  16. Maciejko, J., Hughes, T. L. & Zhang, S.-C. The quantum spin Hall effect. Annu. Rev. Condens. Matter Phys. 2, 31–53 (2011).

    CAS  Google Scholar 

  17. Hasan, M. Z. & Moore, J. E. Three-dimensional topological insulators. Annu. Rev. Condens. Matter Phys. 2, 55–78 (2011).

    CAS  Google Scholar 

  18. Fu, L. & Kane, C. L. Topological insulators with inversion symmetry. Phys. Rev. B 76, 045302 (2007).

    Google Scholar 

  19. Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).

    CAS  Google Scholar 

  20. Bernevig, B. A., Hughes, T. L. & Zhang, S.-C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).

    CAS  Google Scholar 

  21. König, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007).

    Google Scholar 

  22. Zhang, H. et al. Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface. Nat. Phys. 5, 438–442 (2009).

    CAS  Google Scholar 

  23. Xia, Y. et al. Observation of a large-gap topological-insulator class with a single Dirac cone on the surface. Nat. Phys. 5, 398–402 (2009).

    CAS  Google Scholar 

  24. Chadov, S. et al. Tunable multifunctional topological insulators in ternary Heusler compounds. Nat. Mater. 9, 541–545 (2010).

    CAS  Google Scholar 

  25. Lin, H. et al. Half-Heusler ternary compounds as new multifunctional experimental platforms for topological quantum phenomena. Nat. Mater. 9, 546–549 (2010).

    CAS  Google Scholar 

  26. Shekhar, C. et al. Ultrahigh mobility and nonsaturating magnetoresistance in Heusler topological insulators. Phys. Rev. B 86, 155314 (2012).

    Google Scholar 

  27. Rasche, B. et al. Stacked topological insulator built from bismuth-based graphene sheet analogues. Nat. Mater. 12, 422–425 (2013).

    CAS  Google Scholar 

  28. Liang, A. J. et al. Observation of the topological surface state in the nonsymmorphic topological insulator KHgSb. Phys. Rev. B 96, 165143 (2017).

    Google Scholar 

  29. Ma, J. et al. Experimental evidence of hourglass fermion in the candidate nonsymmorphic topological insulator KHgSb. Sci. Adv. 3, e1602415 (2017).

    Google Scholar 

  30. Zhang, X., Zhang, H., Wang, J., Felser, C. & Zhang, S.-C. Actinide topological insulator materials with strong interaction. Science 335, 1464–1466 (2012).

    CAS  Google Scholar 

  31. Yan, B., Müchler, L., Qi, X.-L., Zhang, S.-C. & Felser, C. Topological insulators in filled skutterudites. Phys. Rev. B 85, 165125 (2012).

    Google Scholar 

  32. Yan, B., Jansen, M. & Felser, C. A large-energy-gap oxide topological insulator based on the superconductor BaBiO3. Nat. Phys. 9, 709–711 (2013).

    CAS  Google Scholar 

  33. Tokura, Y., Yasuda, K. & Tsukazaki, A. Magnetic topological insulators. Nat. Rev. Phys. 1, 126–143 (2019).

    Google Scholar 

  34. Wan, X., Turner, A. M., Vishwanath, A. & Savrasov, S. Y. Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011).

    Google Scholar 

  35. Burkov, A. A., Hook, M. D. & Balents, L. Topological nodal semimetals. Phys. Rev. B 84, 235126 (2011).

    Google Scholar 

  36. Young, S. M. et al. Dirac semimetal in three dimensions. Phys. Rev. Lett. 108, 140405 (2012).

    CAS  Google Scholar 

  37. Wang, Z. et al. Dirac semimetal and topological phase transitions in A3Bi (A = Na, K, Rb). Phys. Rev. B 85, 195320 (2012).

    Google Scholar 

  38. Weng, H., Fang, C., Fang, Z., Bernevig, B. A. & Dai, X. Weyl semimetal phase in noncentrosymmetric transition-metal monophosphides. Phys. Rev. X 5, 011029 (2015).

    Google Scholar 

  39. Xu, S.-Y. et al. Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015).

    CAS  Google Scholar 

  40. Lv, B. Q. et al. Experimental discovery of Weyl semimetal TaAs. Phys. Rev. X 5, 031013 (2015).

    Google Scholar 

  41. Yang, L. X. et al. Weyl semimetal phase in the non-centrosymmetric compound TaAs. Nat. Phys. 11, 728–732 (2015).

    CAS  Google Scholar 

  42. Jiang, J. et al. Signature of type-II Weyl semimetal phase in MoTe. Nat. Commun. 8, 13973 (2017).

    CAS  Google Scholar 

  43. Liu, Z. K. et al. Discovery of a three-dimensional topological Dirac semimetal, Na3Bi. Science 343, 864–867 (2014).

    CAS  Google Scholar 

  44. Wang, Z., Weng, H., Wu, Q., Dai, X. & Fang, Z. Three-dimensional Dirac semimetal and quantum transport in Cd3As2. Phys. Rev. B 88, 125427 (2013).

    Google Scholar 

  45. Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90, 015001 (2018).

    CAS  Google Scholar 

  46. Nielsen, H. B. & Ninomiya, M. The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal. Phys. Lett. B 130, 389–396 (1983).

    Google Scholar 

  47. Zyuzin, A. A. & Burkov, A. A. Topological response in Weyl semimetals and the chiral anomaly. Phys. Rev. B 86, 115133 (2012).

    Google Scholar 

  48. Xu, G., Weng, H., Wang, Z., Dai, X. & Fang, Z. Chern semimetal and the quantized anomalous Hall effect in HgCr2Se4. Phys. Rev. Lett. 107, 186806 (2011).

    Google Scholar 

  49. Ueda, K. et al. Magnetic-field induced multiple topological phases in pyrochlore iridates with Mott criticality. Nat. Commun. 8, 15515 (2017).

    CAS  Google Scholar 

  50. Ueda, K. et al. Spontaneous Hall effect in the Weyl semimetal candidate of all-in all-out pyrochlore iridate. Nat. Commun. 9, 3032 (2018).

    Google Scholar 

  51. LaBarre, P. G., Dong, L., Trinh, J., Siegrist, T. & Ramirez, A. P. Evidence for undoped Weyl semimetal charge transport in Y2Ir2O7. J. Phys. Condens. Matter 32, 02LT01 (2019).

    Google Scholar 

  52. Hirschberger, M. et al. The chiral anomaly and thermopower of Weyl fermions in the half-Heusler GdPtBi. Nat. Mater. 15, 1161–1165 (2016).

    CAS  Google Scholar 

  53. Shekhar, C. et al. Anomalous Hall effect in Weyl semimetal half-Heusler compounds RPtBi (R = Gd and Nd). Proc. Natl Acad. Sci. USA 115, 9140–9144 (2018).

    CAS  Google Scholar 

  54. Wang, Z. et al. Time-reversal-breaking Weyl fermions in magnetic heusler alloys. Phys. Rev. Lett. 117, 236401 (2016).

    Google Scholar 

  55. Chang, G. et al. Room-temperature magnetic topological Weyl fermion and nodal line semimetal states in half-metallic Heusler CoTiX (X=Si, Ge, or Sn). Sci. Rep. 6, 38839 (2016).

    CAS  Google Scholar 

  56. Kübler, J. & Felser, C. Weyl points in the ferromagnetic Heusler compound Co2MnAl. Europhys. Lett. 114, 47005 (2016).

    Google Scholar 

  57. Burkov, A. A. Anomalous Hall effect in Weyl metals. Phys. Rev. Lett. 113, 187202 (2014).

    CAS  Google Scholar 

  58. Kübler, J. & Felser, C. Berry curvature and the anomalous Hall effect in Heusler compounds. Phys. Rev. B 85, 012405 (2012).

    Google Scholar 

  59. Manna, K. et al. From colossal to zero: controlling the anomalous Hall effect in magnetic Heusler compounds via Berry curvature design. Phys. Rev. X 8, 041045 (2018).

    CAS  Google Scholar 

  60. Sakai, A. et al. Giant anomalous Nernst effect and quantum-critical scaling in a ferromagnetic semimetal. Nat. Phys. 14, 1119–1124 (2018).

    CAS  Google Scholar 

  61. Guin, S. N. et al. Anomalous Nernst effect beyond the magnetization scaling relation in the ferromagnetic Heusler compound Co2MnGa. npg Asia Mater. 11, 16 (2019).

    Google Scholar 

  62. Liu, E. et al. Giant anomalous Hall effect in a ferromagnetic kagome-lattice semimetal. Nat. Phys. 14, 1125–1131 (2018).

    CAS  Google Scholar 

  63. Belopolski, I. et al. Discovery of topological Weyl fermion lines and drumhead surface states in a room temperature magnet. Science 365, 1278–1281 (2019).

    CAS  Google Scholar 

  64. Liu, D. F. et al. Magnetic Weyl semimetal phase in a kagomé crystal. Science 365, 1282–1285 (2019).

    CAS  Google Scholar 

  65. Morali, N. et al. Fermi-arc diversity on surface terminations of the magnetic Weyl semimetal Co3Sn2S2. Science 365, 1286–1291 (2019).

    CAS  Google Scholar 

  66. Kübler, J. & Felser, C. Non-collinear antiferromagnets and the anomalous Hall effect. Europhys. Lett. 108, 67001 (2014).

    Google Scholar 

  67. Chen, H., Niu, Q. & MacDonald, A. H. Anomalous Hall effect arising from noncollinear antiferromagnetism. Phys. Rev. Lett. 112, 017205 (2014).

    Google Scholar 

  68. Nakatsuji, S., Kiyohara, N. & Higo, T. Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature. Nature 527, 212–215 (2015).

    CAS  Google Scholar 

  69. Nayak, A. K. et al. Large anomalous Hall effect driven by a nonvanishing Berry curvature in the noncolinear antiferromagnet Mn3Ge. Sci. Adv. 2, e1501870 (2016).

    Google Scholar 

  70. Sanchez, D. S. et al. Topological chiral crystals with helicoid-arc quantum states. Nature 567, 500–505 (2019).

    CAS  Google Scholar 

  71. Schröter, N. B. M. et al. Chiral topological semimetal with multifold band crossings and long Fermi arcs. Nat. Phys. 15, 759–765 (2019).

    Google Scholar 

  72. Bradlyn, B. et al. Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals. Science 353, aaf5037 (2016).

    Google Scholar 

  73. Schröter, N. B. M. et al. Observation and control of maximal Chern numbers in a chiral topological semimetal. Science 369, 179–183 (2020).

    Google Scholar 

  74. Cook, A. M., Fregoso, B. M., de Juan, F., Coh, S. & Moore, J. E. Design principles for shift current photovoltaics. Nat. Commun. 8, 14176 (2017).

    CAS  Google Scholar 

  75. Wu, L. et al. Giant anisotropic nonlinear optical response in transition metal monopnictide Weyl semimetals. Nat. Phys. 13, 350–355 (2016).

    Google Scholar 

  76. Osterhoudt, G. B. et al. Colossal mid-infrared bulk photovoltaic effect in a type-I Weyl semimetal. Nat. Mater. 18, 471–475 (2019).

    CAS  Google Scholar 

  77. Garcia, C. A. C., Coulter, J. & Narang, P. Optoelectronic response of the type-I Weyl semimetals TaAs and NbAs from first principles. Phys. Rev. Res. 2, 013073 (2020).

    CAS  Google Scholar 

  78. Lindenberg, A. M. et al. Time-resolved X-Ray diffraction from coherent phonons during a laser-induced phase transition. Phys. Rev. Lett. 84, 111–114 (2000).

    CAS  Google Scholar 

  79. Matsunaga, R. et al. Light-induced collective pseudospin precession resonating with Higgs mode in a superconductor. Science 345, 1145–1149 (2014).

    CAS  Google Scholar 

  80. Juraschek, D. M., Meier, Q. N. & Narang, P. Parametric excitation of an optically silent goldstone-like phonon mode. Phys. Rev. Lett. 124, 117401 (2020).

    CAS  Google Scholar 

  81. Mitrano, M. et al. Possible light-induced superconductivity in K3C60 at high temperature. Nature 530, 461–464 (2016).

    CAS  Google Scholar 

  82. Scruby, C. B., Williams, P. M. & Parry, G. S. The role of charge density waves in structural transformations of 1T TaS2. Philos. Mag. 31, 255–274 (1975).

    CAS  Google Scholar 

  83. Shin, D. et al. Phonon-driven spin-Floquet magneto-valleytronics in MoS2. Nat. Commun. 9, 638 (2018).

    Google Scholar 

  84. He, R.-H. et al. From a single-band metal to a high-temperature superconductor via two thermal phase transitions. Science 331, 1579–1583 (2011).

    CAS  Google Scholar 

  85. Basov, D. N., Averitt, R. D. & Hsieh, D. Towards properties on demand in quantum materials. Nat. Mater. 16, 1077–1088 (2017).

    CAS  Google Scholar 

  86. Rivera, N., Flick, J. & Narang, P. Variational theory of nonrelativistic quantum electrodynamics. Phys. Rev. Lett. 122, 193603 (2019).

    CAS  Google Scholar 

  87. Patankar, S. et al. Resonance-enhanced optical nonlinearity in the Weyl semimetal TaAs. Phys. Rev. B 98, 359 (2018).

    Google Scholar 

  88. Gooth, J. et al. Axionic charge-density wave in the Weyl semimetal (TaSe)I. Nature 575, 315–319 (2019).

    CAS  Google Scholar 

  89. Rajamathi, C. R. et al. Weyl semimetals as hydrogen evolution catalysts. Adv. Mater. 29, (2017).

  90. Yan, B. & Felser, C. Topological materials: Weyl semimetals. Annu. Rev. Condens. Matter Phys. 8, 337–354 (2017).

    Google Scholar 

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Acknowledgements

We thank M. Hopkins (Harvard University), D. Nenno (Harvard University), Y. Wang (Harvard University) and J. Gooth (Max Planck) for fruitful discussions and feedback. P.N. acknowledges helpful discussions with D. Basov (Columbia University) and S. Parkin (Max Planck) on various aspects of this work. This work was supported by the DOE Photonics at Thermodynamic Limits Energy Frontier Research Center under grant no. DE-SC0019140. C.A.C.G. is supported by the NSF Graduate Research Fellowship Program under grant no. DGE-1745303. P.N. is a Moore Inventor Fellow and gratefully acknowledges support through grant GBMF8048 from the Gordon and Betty Moore Foundation, and support as a CIFAR Azrieli Global Scholar.

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Authors and Affiliations

  1. John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA

    Prineha Narang, Christina A. C. Garcia & Claudia Felser

  2. Max-Planck-Institut für Chemische Physik fester Stoffe, Dresden, Germany

    Claudia Felser

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  1. Prineha Narang
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P.N., C.A.C.G. and C.F. have all contributed equally to this Perspective, the writing and the ideas presented here.

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Correspondence to Prineha Narang.

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Narang, P., Garcia, C.A.C. & Felser, C. The topology of electronic band structures. Nat. Mater. 20, 293–300 (2021). https://doi.org/10.1038/s41563-020-00820-4

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