Topological Invariants and Corner States for Hamiltonians on a Three-Dimensional Lattice


We consider periodic Hamiltonians on a three-dimensional (3-D) lattice with a spectral gap not only on the bulk but also on two edges at the common Fermi level. By using K-theory applied for the quarter-plane Toeplitz extension, two topological invariants are defined. One is defined for the gapped bulk and edge Hamiltonians, and the non-triviality of the other means that the corner Hamiltonian is gapless. We prove a correspondence between these two invariants. Such gapped Hamiltonians can be constructed from Hamiltonians of 2-D type A and 1-D type AIII topological insulators, and its corner topological invariant is the product of topological invariants of these two phases.

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  1. 1

    Atiyah M.F., Singer I.M.: Index theory for skew-adjoint Fredholm operators. Inst. Hautes Études Sci. Publ. Math. 37, 5–26 (1969)

    MathSciNet  Article  Google Scholar 

  2. 2

    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)

    MathSciNet  Article  Google Scholar 

  3. 3

    Bellissard, J.: K-theory of C *-algebras in solid state physics. Statistical mechanics and field theory: mathematical aspects (Groningen, 1985), pp. 99–156. Springer, Berlin (1986)

  4. 4

    Bellissard J., van Elst A., Schulz-Baldes H.: The noncommutative geometry of the quantum Hall effect. J. Math. Phys. 35(10), 5373–5451 (1994)

    ADS  MathSciNet  Article  Google Scholar 

  5. 5

    Benalcazar W.A., Bernevig B.A., Hughes T.L.: Quantized electric multipole insulators. Science 357, 61–66 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  6. 6

    Blackadar, B.: K-theory for operator algebras: 2nd edn. Mathematical Sciences Research Institute Publications, vol. 5. Cambridge University Press, Cambridge (1998)

  7. 7

    Brown A., Pearcy C.: Spectra of tensor products of operators. Proc. Am. Math. Soc. 17, 162–166 (1966)

    MathSciNet  Article  Google Scholar 

  8. 8

    Böttcher A., Silbermann B.: Analysis of Toeplitz Operators, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  9. 9

    Bourne C., Carey A.L., Rennie A.: The bulk-edge correspondence for the quantum Hall effect in Kasparov theory. Lett. Math. Phys. 105(9), 1253–1273 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  10. 10

    Bourne C., Kellendonk J., Rennie A.: The K-theoretic bulk-edge correspondence for topological insulators. Ann. Henri Poincaré 18(5), 1833–1866 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  11. 11

    Coburn L.A., Douglas R.G.: C *-algebras of operators on a half-space. I. Inst. Hautes Études Sci. Publ. Math. 40, 59–67 (1971)

    MathSciNet  Article  Google Scholar 

  12. 12

    Coburn L.A., Douglas R.G., Singer I.M.: An index theorem for Wiener-Hopf operators on the discrete quarter-plane. J. Differ. Geom. 6, 587–593 (1972)

    MathSciNet  Article  Google Scholar 

  13. 13

    Douglas R.G., Howe R.: On the C *-algebra of Toeplitz operators on the quarterplane. Trans. Am. Math. Soc. 158, 203–217 (1971)

    MathSciNet  MATH  Google Scholar 

  14. 14

    Douglas, R.G.: Another look at real-valued index theory. In: Surveys of Some Recent Results in Operator Theory, Vol. II, volume 192 of Pitman Res. Notes Math. Ser., 91–120. Longman Sci. Tech., Harlow (1988)

  15. 15

    Douglas, R.G.: Banach Algebra Techniques in Operator Theory, 2nd edn. volume 179 of Graduate Texts in Mathematics. Springer, New York (1998)

    Book  Google Scholar 

  16. 16

    Elbau P., Graf G.M.: Equality of bulk and edge Hall conductance revisited. Commun. Math. Phys. 229(3), 415–432 (2002)

    ADS  MathSciNet  Article  Google Scholar 

  17. 17

    Graf G.M., Porta M.: Bulk-edge correspondence for two-dimensional topological insulators. Commun. Math. Phys. 324(3), 851–895 (2013)

    ADS  MathSciNet  Article  Google Scholar 

  18. 18

    Hashimoto K., Wu X., Kimura T.: Edge states at an intersection of edges of a topological material. Phys. Rev. B 95, 165443-1–10 (2017)

    ADS  Article  Google Scholar 

  19. 19

    Hatsugai Y.: Edge states in the integer quantum Hall effect and the Riemann surface of the Bloch function. Phys. Rev. B 48, 11851–11862 (1993)

    ADS  Article  Google Scholar 

  20. 20

    Hatsugai Y.: Chern number and edge states in the integer quantum Hall effect. Phys. Rev. Lett. 71, 3697–3700 (1993)

    ADS  MathSciNet  Article  Google Scholar 

  21. 21

    Hayashi S.: Bulk-edge correspondence and the cobordism invariance of the index. Rev. Math. Phys. 29, 1750033 (2017)

    MathSciNet  Article  Google Scholar 

  22. 22

    Higson, N., Roe, J.: Analytic K-homology. Oxford Mathematical Monographs. Oxford University Press, Oxford Science Publications, Oxford (2000)

  23. 23

    Jiang X.: On Fredholm operators in quarter-plane Toeplitz algebras. Proc. Am. Math. Soc. 123(9), 2823–2830 (1995)

    MathSciNet  Article  Google Scholar 

  24. 24

    Kellendonk J., Richter T., Schulz-Baldes H.: Edge current channels and Chern numbers in the integer quantum Hall effect. Rev. Math. Phys. 14(1), 87–119 (2002)

    MathSciNet  Article  Google Scholar 

  25. 25

    Kitaev A.: Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22–30 (2009)

    ADS  Article  Google Scholar 

  26. 26

    Kohmoto M.: Topological invariant and the quantization of the hall conductance. Ann. Phys. 160, 343–354 (1985)

    ADS  MathSciNet  Article  Google Scholar 

  27. 27

    Kubota Y.: Controlled topological phases and bulk-edge correspondence. Commun. Math. Phys. 349(2), 493–525 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  28. 28

    Mathai V., Thiang G.C.: T-duality simplifies bulk-boundary correspondence. Commun. Math. Phys. 345, 675–701 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  29. 29

    Murphy G.J.: C *-Algebras and Operator Theory. Academic Press, Inc., Boston, MA (1990)

    MATH  Google Scholar 

  30. 30

    Park E.: Index theory and Toeplitz algebras on certain cones in \({\mathbb{Z}^2}\) . J. Oper. Theory 23(1), 125–146 (1990)

    Google Scholar 

  31. 31

    Park E., Schochet C.: On the K-theory of quarter-plane Toeplitz algebras. Int. J. Math. 2(2), 195–204 (1991)

    MathSciNet  Article  Google Scholar 

  32. 32

    Phillips J.: Self-adjoint Fredholm operators and spectral flow. Can. Math. Bull. 39(4), 460–467 (1996)

    MathSciNet  Article  Google Scholar 

  33. 33

    Prodan, E., Schulz-Baldes, H.: Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics. Springer, Berlin (2016)

    Book  Google Scholar 

  34. 34

    Rørdam, M., Larsen, F., Laustsen, N.: An Introduction to K-Theory for C *-Algebras. Volume 49 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge (2000)

  35. 35

    Schnyder A.P., Ryu S., Furusaki A., Ludwig W.W.: Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125-1–22 (2008)

    ADS  Google Scholar 

  36. 36

    Schulz-Baldes H., Kellendonk J., Richter T.: Simultaneous quantization of edge and bulk Hall conductivity. J. Phys. A 33(2), L27–L32 (2000)

    ADS  MathSciNet  Article  Google Scholar 

  37. 37

    Simonenko I.B.: Operators of convolution type in cones. Mat. Sb. (N.S.) 74(116), 298–313 (1967)

    MathSciNet  Google Scholar 

  38. 38

    Thouless D.J., Kohmoto M., Nightingale M.P., den Nijs M.: Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982)

    ADS  Article  Google Scholar 

  39. 39

    Wegge-Olsen, N.E.: K-Theory and C *-Algebras. A Friendly Approach. Oxford Science Publications, New York (1993)

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This work is part of a Ph.D. thesis, defended at the University of Tokyo in 2017. The author would like to expresses his gratitude for the support and encouragement of his supervisor, Mikio Furuta. This work was inspired by the author’s collaborative research with Mikio Furuta, Motoko Kotani, Yosuke Kubota, Shinichiroh Matsuo, and Koji Sato. He would like to thank them for many stimulating conversations and much encouragement. He also would like to thank Christopher Bourne, Ken-Ichiro Imura, Takeshi Nakanishi, and Yukinori Yoshimura for their many discussions, and thank Emil Prodan for sharing the information regarding [5]. This work was supported by JSPS Grant-in-Aid for Scientific Research on Innovative Areas, “Discrete Geometric Analysis for Materials Design”: Grant Number JP17H06461.

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Correspondence to Shin Hayashi.

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Communicated by Y. Kawahigashi

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Hayashi, S. Topological Invariants and Corner States for Hamiltonians on a Three-Dimensional Lattice. Commun. Math. Phys. 364, 343–356 (2018).

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