Crystallographic bulk-edge correspondence: glide reflections and twisted mod 2 indices

  • Kiyonori Gomi
  • Guo Chuan Thiang


A 2-torsion topological phase exists for Hamiltonians symmetric under the wallpaper group with glide reflection symmetry, corresponding to the unorientable cycle of the Klein bottle fundamental domain. We prove a mod 2 twisted Toeplitz index theorem, which implies a bulk-edge correspondence between this bulk phase and the exotic topological zero modes that it acquires along a boundary glide axis.


Bulk-edge correspondence Topological crystalline insulators Twisted K-theory Toeplitz index theorem 

Mathematics Subject Classification

19L50 19K56 47L80 



G.C.T. is supported by ARC grant DE170100149 and would like to thank G. De Nittis, V. Mathai and K. Hannabuss for helpful discussions. He also acknowledges H.-H. Lee for his kind hospitality at the Seoul National University, where part of this work was completed. K.G. is supported by JSPS KAKENHI Grant Number JP15K04871, and thanks I. Sasaki, M. Furuta and K. Shiozaki for useful discussions.


  1. 1.
    Atiyah, M.F.: Bott periodicity and the index of elliptic operators. Q. J. Math. Oxf. Ser. 2(19), 113–140 (1968)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Atiyah, M.F.: Algebraic topology and operators in Hilbert space. In: Taam, C.T. (ed.) Lectures in Modern Analysis and Applications I. Lecture Notes in Mathematics, vol. 103, pp. 101–121. Springer, Berlin (1969)CrossRefGoogle Scholar
  3. 3.
    Atiyah, M.F., Singer, I.M.: The index of elliptic operators I. Ann. Math. 87(3), 484–530 (1968)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bradlyn, B., Elcoro, L., Cano, J., Vergniory, M.G., Wang, Z., Felser, C., Aroyo, M.I., Bernevig, B.A.: Topological quantum chemistry. Nature 547, 298–305 (2017)ADSCrossRefGoogle Scholar
  5. 5.
    Benameur, M.-T.: Noncommutative geometry and abstract integration theory. In: Cardona, A., Paycha, S., Ocampo, H. (eds.) Geometrical and Topological Methods for Quantum Field Theory, pp. 157–227. World Scientific, River Edge (2003)CrossRefGoogle Scholar
  6. 6.
    Coburn, L.A.: The \(C^*\)-algebra generated by an isometry. II. Trans. Am. Math. Soc. 137, 211–217 (1969)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Conway, J.H., Friedrichs, O.D., Huson, D.H., Thurston, W.P.: On three-dimensional orbifolds and space groups. Beiträge Algebra Geom. 42(2), 475–507 (2001)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Cuntz, J.: \(K\)-theory and \(C^*\)-algebras. In: Bak, A. (ed.) Algebraic \(K\)-Theory, Number Theory, Geometry, and Analysis. Lecture Notes in Mathematics, vol. 1046, pp. 55–79. Springer, Berlin (1984)Google Scholar
  9. 9.
    Davidson, K.R.: \(C^*\)-algebras by example. Fields Inst. Monogr., vol. 6. Providence, RI (1996)Google Scholar
  10. 10.
    de Monvel, L.B.: On the index of Toeplitz operators of several complex variables. Invent. Math. 50, 249–272 (1979)MathSciNetCrossRefGoogle Scholar
  11. 11.
    De Nittis, G., Gomi, K.: The cohomological nature of the Fu–Kane–Mele invariant. J. Geom. Phys. 124, 124–164 (2018)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Fang, C., Fu, L.: New classes of three-dimensional topological crystalline insulators: nonsymmorphic and magnetic. Phys. Rev. B 91, 161105 (2015)ADSCrossRefGoogle Scholar
  13. 13.
    Freed, D.S., Moore, G.: Twisted equivariant matter. Ann. Henri Poincaré 14(8), 1927–2023 (2013)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Gohberg, I.C., Krein, M.G.: The basic propositions on defect numbers, root numbers and indices of linear operators. Am. Math. Soc. Transl. 2(13), 185–264 (1960)MathSciNetGoogle Scholar
  15. 15.
    Gomi, K.: Twists on the torus equivariant under the 2-dimensional crystallographic point groups. SIGMA Symmetry Integr. Geom. Methods Appl. 13, 014 (2017)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Gomi, K.: Freed–Moore \(K\)-theory. arXiv:1705.09134
  17. 17.
    Gomi, K.: A variant of \(K\)-theory and topological T-duality for real circle bundles. Commun. Math. Phys. 334(2), 923–975 (2015)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Gomi, K., Thiang, G.C.: Crystallographic T-duality. arXiv:1806.11385
  19. 19.
    Haldane, F.D.M.: Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett. 61(18), 2015–2018 (1988)ADSCrossRefGoogle Scholar
  20. 20.
    Halperin, B.I.: Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential. Phys. Rev. B 25, 2185 (1982)ADSCrossRefGoogle Scholar
  21. 21.
    Hannabuss, K., Mathai, V., Thiang, G.C.: T-duality simplifies bulk-boundary correspondence: the noncommutative case. Lett. Math. Phys. 108(5), 1163–1201 (2018)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Hatsugai, Y.: Chern number and edge states in the integer quantum Hall effect. Phys. Rev. Lett. 71, 3697 (1993)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Hatsugai, Y.: Bulk-edge correspondence in graphene with/without magnetic field: Chiral symmetry, Dirac fermions and Edge states. Solid State Commun. 149, 1061 (2009)ADSCrossRefGoogle Scholar
  24. 24.
    Hsieh, D., et al.: A tunable topological insulator in the spin helical Dirac transport regime. Nature 460(7259), 1101–1105 (2009)ADSCrossRefGoogle Scholar
  25. 25.
    Karoubi, M.: \(K\)-theory: an introduction. In: Grundlehren math. Wiss., vol 226. Springer, Berlin (1978)Google Scholar
  26. 26.
    Karoubi, M.: Twisted bundles and twisted \(K\)-theory. In: Cortinãs, G. (ed.) Topics in noncommutative geometry, Clay Math. Proc., vol. 16, pp. 223–257. Providence, RI (2012)Google Scholar
  27. 27.
    Kellendonk, J., Richter, T., Schulz-Baldes, H.: Edge current channels and Chern numbers in the integer quantum Hall effect. Rev. Math. Phys. 14(01), 87–119 (2002)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Kitaev, A.: Periodic table for topological insulators and superconductors. AIP Conf. Ser. 1134, 22–30 (2009)ADSCrossRefGoogle Scholar
  29. 29.
    Kopsky, V., Litvin, D.B., eds.: International Tables for Crystallography, Volume E: Subperiodic groups, E (5th ed.), Berlin, New York (2002)Google Scholar
  30. 30.
    Kruthoff, J., de Boer, J., van Wezel, J., Kane, C.L., Slager, R.-J.: Topological classification of crystalline insulators through band structure combinatorics. Phys. Rev. X 7, 041069 (2017)Google Scholar
  31. 31.
    Kubota, Y.: Notes on twisted equivariant K-theory for \(C^*\)-algebras. Int. J. Math. 27(6), 1650058 (2016)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Kubota, Y.: Controlled topological phases and bulk-edge correspondence. Commun. Math. Phys. 349(2), 493–525 (2017)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Lück, W., Stamm, R.: Computations of \(K\)- and \(L\)-theory of cocompact planar groups. \(K\)-theory 21, 249–292 (2000)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Mendez-Diez, S., Rosenberg, J.: \(K\)-theoretic matching of brane charges in S- and U-duality. Adv. Theor. Math. Phys. 16(6), 1591–1618 (2012)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Michel, L., Zak, J.: Connectivity of energy bands in crystals. Phys. Rev. B 59, 5998 (1999)ADSCrossRefGoogle Scholar
  36. 36.
    Mislin, G.: Equivariant \(K\)-homology of the classifying space for proper actions. In: Proper group actions and the Baum–Connes conjecture, pp. 1–78. Birkhäuser, Basel (2003)CrossRefGoogle Scholar
  37. 37.
    Moutuou, E.M.: Twisted groupoid \(KR\)-Theory. Ph.D. thesis, Université de Lorraine (2012).
  38. 38.
    Murphy, G.J.: \(C\)*-Algebras and Operator Theory. Academic Press, Boston (1990)zbMATHGoogle Scholar
  39. 39.
    Phillips, N.C.: The Toeplitz operator proof of noncommutative Bott periodicity. J. Austral. Math. Soc. (Ser. A) 53, 229–251 (1992)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Pimsner, M., Voiculescu, D.: Exact sequences for \(K\)-groups and \(EXT\)-groups of certain cross-product \(C^*\)-algebras. J. Oper. Theory 4, 93–118 (1980)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Prodan, E., Schulz-Baldes, H.: Bulk and Boundary Invariants for Complex Topological Insulators: From \(K\)-Theory to Physics. Springer, Basel (2016)CrossRefGoogle Scholar
  42. 42.
    Prodan, E., Schulz-Baldes, H.: Generalized Connes–Chern characters in \(KK\)-theory with an application to weak invariants of topological insulators. Rev. Math. Phys. 28(10), 1650024 (2016)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Ryu, S., Hatsugai, Y.: Topological origin of zero-energy edge states in particle-hole symmetric systems. Phys. Rev. Lett. 89, 077002 (2002)ADSCrossRefGoogle Scholar
  44. 44.
    Schneider, A.: Equivariant T-duality of Locally Compact Abelian Groups. arXiv:0906.3734
  45. 45.
    Shiozaki, K., Sato, M., Gomi, K.: \(\mathbb{Z}_2\)-topology in nonsymmorphic crystalline insulators: Möbius twist in surface states. Phys. Rev. B 91, 155120 (2015)ADSCrossRefGoogle Scholar
  46. 46.
    Shiozaki, K., Sato, M., Gomi, K.: Topological crystalline materials: general formulation, module structure, and wallpaper groups. Phys. Rev. B 95(23), 235425 (2017)ADSCrossRefGoogle Scholar
  47. 47.
    Stolz, S.: Concordance classes of positive scalar curtavure metrics. Preprint Accessed 9 Aug 2018
  48. 48.
    Taylor, K.F.: \(C^*\)-algebras of crystal groups. Oper. Theory Adv. Appl. 41, 511–518 (1989)MathSciNetGoogle Scholar
  49. 49.
    Thiang, G.C.: On the \(K\)-theoretic classification of topological phases of matter. Ann. Henri Poincaré 17(4), 757–794 (2016)ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    Thiang, G.C.: Topological phases: isomorphism, homotopy and K-theory. Int. J. Geom. Methods Mod. Phys. 12, 1550098 (2015)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Valette, A.: Introduction to the Baum–Connes conjecture. Lectures Math. ETH Zürich. Birkhäuser Verlag, Basel (2002)CrossRefGoogle Scholar
  52. 52.
    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
  53. 53.
    Wieder, B.J.: Wallpaper fermions and the nonsymmorphic dirac insulator. Science 361(6399), 246–251 (2018)ADSCrossRefGoogle Scholar
  54. 54.
    Witten, E.: D-branes and \(K\)-theory. J. High Energy Phys. 12, 019 (1998)ADSMathSciNetCrossRefGoogle Scholar
  55. 55.
    Xu, S.-Y.: Discovery of a Weyl Fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015)ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesShinshu UniversityMatsumotoJapan
  2. 2.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

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