Communications in Mathematical Physics

, Volume 355, Issue 2, pp 561–602 | Cite as

Differential Topology of Semimetals

  • Varghese Mathai
  • Guo Chuan Thiang


The subtle interplay between local and global charges for topological semimetals exactly parallels that for singular vector fields. Part of this story is the relationship between cohomological semimetal invariants, Euler structures, and ambiguities in the connections between Weyl points. Dually, a topological semimetal can be represented by Euler chains from which its surface Fermi arc connectivity can be deduced. These dual pictures, and the link to topological invariants of insulators, are organised using geometric exact sequences. We go beyond Dirac-type Hamiltonians and introduce new classes of semimetals whose local charges are subtle Atiyah–Dupont–Thomas invariants globally constrained by the Kervaire semicharacteristic, leading to the prediction of torsion Fermi arcs.


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  1. 1.
    Atiyah, M.F.: Vector fields on manifolds. Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen, vol. 200, pp. 7–26. VS, Cologne (1970)Google Scholar
  2. 2.
    Atiyah M.F., Dupont J.L.: Vector fields with finite singularities. Acta Math. 128(1), 1–40 (1972)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Atiyah M.F., Rees E.: Vector bundles on projective 3-space. Invent. Math. 35, 131–153 (1976)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Avila J.C., Schulz-Baldes H., Villegas-Blas C.: Topological invariants of edge states for periodic two-dimensional models. Math. Phys. Anal. Geom. 16(2), 137–170 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Avron J.E., Sadun L., Segert J., Simon B.: Topological invariants in Fermi systems with time-reversal invariance. Phys. Rev. Lett. 61, 1329 (1988)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Avron J.E., Sadun L., Segert J., Simon B.: Chern numbers, quaternions, and Berry’s phases in Fermi systems. Commun. Math. Phys. 124(4), 595–627 (1989)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Borel A., Hirzebruch F.: Characteristic classes and homogeneous spaces, I. Am. J. Math. 80(2), 458–538 (1958)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Borel A., Moore J.C.: Homology theory for locally compact spaces. Mich. Math. J. 7(2), 137–159 (1960)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bott R., Tu L.W.: Differential Forms in Algebraic Topology. Grad. Texts in Math. 82. Springer, New York (1982)CrossRefGoogle Scholar
  10. 10.
    Bradlyn, B., Cano, J., Wang, Z., Vergniory, M.G., Felser, C., Cava, R.J., Bernevig, B.A.: Beyond Dirac and Weyl fermions: unconventional quasiparticles in conventional crystals. Science 353(6299) (2016)Google Scholar
  11. 11.
    Brylinski J-L.: Loop Spaces, Characteristic Classes and Geometric Quantization. Progress in Mathematics, 107. Birkhauser Boston, Inc., Boston (1993)CrossRefGoogle Scholar
  12. 12.
    Burghelea D., Haller S.: Euler structures, the variety of representations and the Milnor–Turaev torsion. Geom. Topol. 10, 1185–1238 (2006)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Chriss N., Ginzburg V.: Representation Theory and Complex Geometry. Birkhäuser, Boston (1997)MATHGoogle Scholar
  14. 14.
    Carpentier D., Delplace P., Fruchart M., Gawędzki K.: Topological index for periodically driven time-reversal invariant 2D systems. Phys. Rev. Lett. 114(10), 106806 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    De Nittis G., Gomi K.: Classification of “Quaternionic” Bloch-bundles topological quantum Systems of type AII. Commun. Math. Phys. 339, 1–55 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Dupont J.L.: Symplectic bundles and KR-theory. Math. Scand. 24(1), 27–30 (1969)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Dwivedi V., Chua V.: Of bulk and boundaries: generalized transfer matrices for tight-binding models. Phys. Rev. B 93, 134304 (2016)ADSCrossRefGoogle Scholar
  18. 18.
    Dwivedi V., Ramamurthy S.T.: Connecting the dots: time-reversal symmetric Weyl semimetals with tunable Fermi arcs. Phys. Rev. B 94, 245143 (2016)ADSCrossRefGoogle Scholar
  19. 19.
    Freed D.S., Moore G.W.: Twisted equivariant matter. Ann. Henri Poincaré 14(8), 1927–2023 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Fu L., Kane C.L., Mele E.J.: Topological insulators in three dimensions. Phys. Rev. Lett. 98(10), 106803 (2007)ADSCrossRefGoogle Scholar
  21. 21.
    Gawędzki K.: 2d Fu–Kane–Mele invariant as Wess–Zumino action of the sewing matrix. Lett. Math. Phys. 107(4), 733–755 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Hatsugai Y.: Edge states in the integer quantum Hall effect and the Riemann surface of the Bloch function. Phys. Rev. B 48(16), 11851–11862 (1993)ADSCrossRefGoogle Scholar
  23. 23.
    Hatsugai Y.: Symmetry-protected \({{\mathbb{Z}}_2}\) -quantization and quaternionic Berry connection with Kramers degeneracy. New J. Phys. 12, 065004 (2010)ADSCrossRefGoogle Scholar
  24. 24.
    Herring C.: Accidental degeneracy in the energy bands of crystals. Phys. Rev. 52, 365–373 (1937)ADSCrossRefGoogle Scholar
  25. 25.
    Huang S.-M. et al.: New type of Weyl semimetal with quadratic double Weyl fermions. Proc. Natl. Acad. Sci. USA 113(5), 1180–1185 (2016)ADSCrossRefGoogle Scholar
  26. 26.
    Hutchings M.: Reidemeister torsion in generalized Morse theory. Forum Math. 14, 209–244 (2002)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Iversen B.: Cohomology of Sheaves. Springer, Berlin (1986)CrossRefMATHGoogle Scholar
  28. 28.
    Kato T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995)CrossRefGoogle Scholar
  29. 29.
    Kaufmann R.M., Li D., Wehefritz-Kaufmann B.: Notes on topological insulators. Rev. Math. Phys. 28(10), 1630003 (2016)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Korbaš J.: Distributions, vector distributions, and immersions of manifolds in Euclidean spaces. In: Krupka, D., Saunders, D. (eds) Handbook of Global Analysis, pp. 665–724. Elsevier Science, Amsterdam (2008)CrossRefGoogle Scholar
  31. 31.
    Kraus Y.E., Lahini Y., Ringel Z., Verbin M., Zilberberg O.: Topological states and adiabatic pumping in quasicrystals. Phys. Rev. Lett. 109, 106402 (2012)ADSCrossRefGoogle Scholar
  32. 32.
    Kuchment P.: An overview of periodic elliptic operators. Bull. Am. Math. Soc. 53, 343–414 (2016)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Lawson H.B., Michelsohn M.-L.: Spin Geometry. Princeton University Press, Princeton (1989)MATHGoogle Scholar
  34. 34.
    Lin H., Yau S.-T.: On exotic sphere fibrations, topological phases, and edge states in physical systems. Int. J. Modern Phys. B 27(19), 1350107 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Lindner N.H., Refael G., Galitski V.: Floquet topological insulator in semiconductor quantum wells. Nat. Phys. 7, 490–495 (2011)CrossRefGoogle Scholar
  36. 36.
    Liu, J., Fang, C., Fu, L.: Tunable Weyl fermions and Fermi arcs in magnetized topological crystalline insulators. arXiv:1604.03947
  37. 37.
    Lv B.Q. et al.: Experimental discovery of Weyl semimetal TaAs. Phys. Rev. X 5, 031013 (2015)Google Scholar
  38. 38.
    Mathai V., Thiang G.C.: T-duality of topological insulators. J. Phys. A: Math. Theor. 48(42), 42FT02. (2015) arXiv:1503.01206 MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Mathai V., Thiang G.C.: T-duality simplifies bulk-boundary correspondence. Commun. Math. Phys. 345(2), 675–701. (2016) arXiv:1505.05250 ADSMathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Mathai V., Thiang G.C.: T-duality simplifies bulk-boundary correspondence: some higher dimensional cases. Ann. Henri Poincaré 17(12), 3399–3424. (2016) arXiv:1506.04492 ADSMathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Mathai V., Thiang G.C.: Global topology of Weyl semimetals and Fermi arcs. J. Phys. A: Math. Theor. (Letter) 50(11), 11LT01 (2017) arXiv:1607.02242 MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Milnor J.: On manifolds homeomorphic to the 7-sphere. Ann. Math. (2) 64(2), 399–405 (1956)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Milnor J.: Topology from the Differentiable Viewpoint. Based on Notes by David W. Weaver. The University Press of Virginia, Charlottesville (1965)MATHGoogle Scholar
  44. 44.
    Molina O.M.: Co-Euler structures on bordisms. Topol. Appl. 193, 51–76 (2015)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Murray M., Stevenson D.: The basic bundle gerbe on unitary groups. J. Geom. Phys. 58(11), 1571–1590 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Murray M., Stevenson D.: Bundle gerbes: stable isomorphism and local theory. J. Lond. Math. Soc. (2) 62(3), 925–937 (2000)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Nielsen H.B., Ninomiya M.: Absence of neutrinos on a lattice: (II). Intuitive topological proof. Nucl. Phys. B 193, 173–194 (1981)ADSMathSciNetCrossRefGoogle Scholar
  48. 48.
    Paechter G.F.: The groups πr(V n,m). Q. J. Math. 7(1), 249–268 (1956)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Polyakov A.M.: Particle spectrum in quantum field theory. Pisma. Zh. Eksp. Theor. Fiz. 20, 430 (1974) [JETP Lett. 20 194 (1974)]Google Scholar
  50. 50.
    Prodan E.: Virtual topological insulators with real quantized physics. Phys. Rev. B 91, 245104 (2015)ADSCrossRefGoogle Scholar
  51. 51.
    Read N.: Compactly-supported Wannier functions and algebraic K-theory. Phys. Rev. B 95, 115309 (2017)ADSCrossRefGoogle Scholar
  52. 52.
    Rechtsman M.C.: Photonic Floquet topological insulators. Nature 496, 196–200 (2013)ADSCrossRefGoogle Scholar
  53. 53.
    Reed M., Simon B.: Methods of Modern Mathematica Physics. Vol. IV: Analysis of Operators. Elsevier, Amsterdam (1978)MATHGoogle Scholar
  54. 54.
    Simon B.: Holonomy, the quantum adiabatic theorem, and Berry’s phase. Phys. Rev. Lett. 51(24), 2167 (1983)ADSMathSciNetCrossRefGoogle Scholar
  55. 55.
    Simons, J., Sullivan, D.: The Mayer–Vietoris property in differential cohomology. arXiv:1010.5269
  56. 56.
    Soluyanov A.A., Gresch D., Wang Z., Wu Q., Troyer M., Dai X., Bernevig B.A.: Type-II Weyl semimetals. Nature 527, 495–498 (2015)ADSCrossRefGoogle Scholar
  57. 57.
    Shaw R., Lever J.: Irreducible multiplier corepresentations of the extended Poincaré group. Commun. Math. Phys. 38(4), 279–297 (1974)ADSCrossRefMATHGoogle Scholar
  58. 58.
    t’ Hooft G.: Magnetic monopoles in unified gauge theories. Nucl Phys. B 79(2), 276–284 (1974)ADSMathSciNetCrossRefGoogle Scholar
  59. 59.
    Tang Z., Zhang W.: A generalization of the Atiyah–Dupont vector fields theory. Commun. Contemp. Math. 4(4), 777–796 (2002)MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Thiang G.C.: On the K-theoretic classification of topological phases of matter. Ann. Henri Poincaré. 17(4), 757–794 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Thomas E.: The index of a tangent 2-field. Comment. Math. Helv. 42(1), 86–110 (1967)MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Thomas E.: Vector fields on manifolds. Bull. Am. Math. Soc. 75(4), 643–683 (1969)MathSciNetCrossRefMATHGoogle Scholar
  63. 63.
    Turaev V.: Euler structures, nonsingular vector fields, and torsions of Reidemeister type. Izv. Math. 34(3), 627–662 (1990)MathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    Turaev V.: Torsion invariants of Spin c-structures on 3-manifolds. Math. Res. Lett. 4, 679–695 (1997)MathSciNetCrossRefMATHGoogle Scholar
  65. 65.
    Turner A.M., Vishwanath A.: Beyond band insulators: topology of semimetals and interacting phases. In: Franz, M., Molenkamp, L. (eds) Contemp. Concepts Cond. Mat. Sci. 6, Topological Insulators, pp. 293–324. Elsevier, Amsterdam (2013)Google Scholar
  66. 66.
    Varadarajan, V.S.: Supersymmetry for Mathematicians: An Introduction, vol. 11. Courant Institute of Mathematical Sciences at New York University, New York (2004)Google Scholar
  67. 67.
    von Neumann J., Wigner E.P.: Über merkwürdige diskrete Eigenwerte. Physik. Zeits. 30, 467–470 (1929)MATHGoogle Scholar
  68. 68.
    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)ADSCrossRefGoogle Scholar
  69. 69.
    Wigner E.P.: Unitary representations of the inhomogeneous Lorentz group including reflections. In: Gürsey, F. (eds) Group Theoretical Concepts in Elementary Particle Physics, vol. 1., pp. 37–80. Gordon and Breach, New York (1964)Google Scholar
  70. 70.
    Witten E.: Three lectures on topological phases of matter. La Rivista del Nuovo Cimento 39(7), 313–370 (2016)ADSGoogle Scholar
  71. 71.
    Xu S.-Y. et al.: Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015)ADSCrossRefGoogle Scholar
  72. 72.
    Xu S.-Y. et al.: Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide. Nat. Phys. 11, 748–754 (2015)CrossRefGoogle Scholar
  73. 73.
    Xu Y., Zhang F., Zhang C.: Structured Weyl points in Spin-orbit coupled fermionic superfluids. Phys. Rev. Lett. 115, 265304 (2015)ADSCrossRefGoogle Scholar
  74. 74.
    Zhang C. et al.: Signatures of the Adler–Bell–Jackiw chiral anomaly in a Weyl fermion semimetal. Nat. Commun. 7, 10735 (2016)ADSCrossRefGoogle Scholar
  75. 75.
    Zhang S.-C., Lian B.: Five-dimensional generalization of the topological Weyl semimetal. Phys. Rev. B 94, 041105(R) (2016)ADSCrossRefGoogle Scholar
  76. 76.
    Zhao Y.X., Wang Z.D.: Topological classification and stability of Fermi surfaces. Phys. Rev. Lett 110, 240404 (2013)ADSCrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of AdelaideAdelaideAustralia

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