Abstract
We study the application of Kasparov theory to topological insulator systems and the bulk–edge correspondence. We consider observable algebras as modelled by crossed products, where bulk and edge systems may be linked by a short exact sequence. We construct unbounded Kasparov modules encoding the dynamics of the crossed product. We then link bulk and edge Kasparov modules using the Kasparov product. Because of the anti-linear symmetries that occur in topological insulator models, real \(C^*\)-algebras and KKO-theory must be used.
Similar content being viewed by others
References
Atiyah, M.F., Bott, R., Shapiro, A.: Clifford modules. Topology 3(Suppl. 1), 3–38 (1964)
Atiyah, M.F., Singer, I.M.: Index theory for skew-adjoint Fredholm operators. Inst. Hautes Études Sci. Publ. Math. 37, 5–26 (1969)
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)
Baaj, S., Julg, P.: Théorie bivariante de Kasparov et opérateurs non bornés dans les \(C^*\)-modules hilbertiens. C. R. Acad. Sci. Paris Sér. I Math. 296(21), 875–878 (1983)
Bellissard, J.: Gap labelling theorems for Schrödinger operators. In: Waldschmidt, M., Moussa, P., Luck, J.-M., Itzkyson, C. (eds.) From Number Theory to Physics, pp. 538–630. Springer, Berlin (1992)
Bellissard, J., van Elst, A., Schulz-Baldes, H.: The noncommutative geometry of the quantum Hall effect. J. Math. Phys. 35(10), 5373–5451 (1994)
Blackadar, B.: \(K\)-Theory for Operator Algebras. Volume 5 of Mathematical Sciences Research Institute Publications. Cambridge University Press, Cambridge (1998)
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)
Bourne, C., Carey, A.L., Rennie, A.: A non-commutative framework for topological insulators. Rev. Math. Phys. 28(2), 1650004 (2016)
Brain, S., Mesland, B., van Suijlekom, W.D.: Gauge theory for spectral triples and the unbounded Kasparov product. J. Noncommut. Geom. 10(1), 135–206 (2016)
Carey, A.L., Phillips, J., Schulz-Baldes, H.: Spectral flow for real skew-adjoint Fredholm operators (2016). arXiv:1604.06994
Carey, A.L., Phillips, J., Rennie, A., Sukochev, F.: The local index formula in semifinite von Neumann algebras. I. Spectral flow. Adv. Math. 202(2), 451–516 (2006)
Carey, A.L., Phillips, J., Rennie, A., Sukochev, F.: The local index formula in semifinite von Neumann algebras. II. The even case. Adv. Math. 202(2), 517–554 (2006)
Connes, A., Moscovici, H.: The local index formula in noncommutative geometry. Geom. Funct. Anal. 5(2), 174–243 (1995)
De Nittis, G., Schulz-Baldes, H.: Spectral flows associated to flux tubes. Ann. Henri Poincaré 17(1), 1–35 (2016)
Forsyth, I., Rennie, A.: Factorisation of equivariant spectral triples in unbounded \(KK\)-theory (2015). arXiv:1505.02863
Freed, D.S., Moore, G.W.: Twisted equivariant matter. Ann. Henri Poincaré 14(8), 1927–2023 (2013)
Goffeng, M., Mesland, B., Rennie, A.: Shift tail equivalence and an unbounded representative of the Cuntz-Pimsner extension. Ergod. Theory Dyn. Syst. (2016). doi:10.1017/etds.2016.75
Gracia-Bondía, J.M., Várilly, J.C., Figueroa, H.: Elements of Noncommutative Geometry. Birkhäuser, Boston (2001)
Graf, G.M., Porta, M.: Bulk–edge correspondence for two-dimensional topological insulators. Commun. Math. Phys. 324(3), 851–895 (2013)
Grossmann, J., Schulz-Baldes, H.: Index pairings in presence of symmetries with applications to topological insulators. Commun. Math. Phys. 343(2), 477–513 (2016)
Hannabuss, K., Mathai, V., Thiang, G.C.: T-duality simplifies bulk-boundary correspondence: the general case (2016). arXiv:1603.00116
Hawkins, A.: Constructions of spectral triples on \(C^*\)-algebras. Ph.D. thesis, University of Nottingham (2013)
Kaad, J., Lesch, M.: Spectral flow and the unbounded Kasparov product. Adv. Math. 298, 495–530 (2013)
Kaad, J., Nest, R., Rennie, A.: \(KK\)-theory and spectral flow in von Neumann algebras. J. K Theory 10(2), 241–277 (2012)
Kane, C.L., Mele, E.J.: \(Z_2\) topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005)
Kasparov, G.G.: The operator \(K\)-functor and extensions of \(C^*\)-algebras. Math. USSR Izv. 16, 513–572 (1981)
Kasparov, G.G.: Equivariant \(KK\)-theory and the Novikov conjecture. Invent. Math. 91(1), 147–201 (1988)
Kellendonk, J.: On the \(C^*\)-algebraic approach to topological phases for insulators (2015). arXiv:1509.06271
Kellendonk, J.: Cyclic cohomology for graded \(C^{\ast , r}\)-algebras and its pairings with van Daele \(K\)-theory (2016). arXiv:1607.08465
Kellendonk, J., Richard, S.: Topological boundary maps in physics. In: Boca, F., Purice, R., Strătilă, Ş. (eds.) Perspectives in Operator Algebras and Mathematical Physics. Theta Series in Advanced Mathematics, vol. 8. Theta, Bucharest, pp. 105–121 (2008). arXiv:math-ph/0605048
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)
Kellendonk, J., Schulz-Baldes, H.: Quantization of edge currents for continuous magnetic operators. J. Funct. Anal. 209(2), 388–413 (2004)
Kellendonk, J., Schulz-Baldes, H.: Boundary maps for \(C^*\)-crossed products with an application to the quantum Hall effect. Commun. Math. Phys. 249(3), 611–637 (2004)
Kitaev, A.: Periodic table for topological insulators and superconductors. In: Lebedev, V., Feigel’man, M., (eds.) American Institute of Physics Conference Series, vol. 1134, pp. 22–30 (2009)
Kubota, Y.: Notes on twisted equivariant \(K\)-theory for \(C^*\)-algebras. Int. J. Math. 27, 1650058 (2016)
Kubota, Y.: Controlled topological phases and bulk–edge correspondence. Commun. Math. Phys. (2016). doi:10.1007/s00220-016-2699-3
Kucerovsky, D.: The \(KK\)-product of unbounded modules. K Theory 11, 17–34 (1997)
Laca, M., Neshveyev, S.: KMS states of quasi-free dynamics on Pimsner algebras. J. Funct. Anal. 211(2), 457–482 (2004)
Li, D., Kaufmann, R.M., Wehefritz-Kaufmann, B.: Topological insulators and \(K\)-theory (2015). arXiv:1510.08001
Lord, S., Rennie, A., Várilly, J.C.: Riemannian manifolds in noncommutative geometry. J. Geom. Phys. 62(7), 1611–1638 (2012)
Loring, T.A.: \(K\)-theory and pseudospectra for topological insulators. Ann. Phys. 356, 383–416 (2015)
Mathai, V., Thiang, G.C.: T-duality simplifies bulk-boundary correspondence. Commun. Math. Phys. 345(2), 675–701 (2016)
Mathai, V., Thiang, G.C.: T-duality simplifies bulk-boundary correspondence: some higher dimensional cases. Ann. Henri Poincaré 17(12), 3399–3424 (2016)
Mesland, B.: Unbounded bivariant \(K\)-theory and correspondences in noncommutative geometry. J. Reine Angew. Math. 691, 101–172 (2014)
Mesland, B.: Spectral triples and \(KK\)-theory: a survey. In: Clay Mathematics Proceedings. Volume 16: Topics in Noncommutative Geometry, pp. 197–212 (2012)
Mesland, B., Rennie, A.: Nonunital spectral triples and metric completeness in unbounded \(KK\)-theory. J. Funct. Anal. 271(9), 2460–2538 (2016)
Packer, J.A., Raeburn, I.: Twisted crossed products of \(C^*\)-algebras. Math. Proc. Camb. Philos. Soc. 106, 293–311 (1989)
Pask, D., Rennie, A.: The noncommutative geometry of graph \(C^*\)-algebras. I: The index theorem. J. Funct. Anal. 233(1), 92–134 (2006)
Pimsner, M., Voiculescu, D.: Exact sequences for \(K\)-groups and Ext-groups of certain cross-product \(C^*\)-algebras. J. Oper. Theory 4(1), 93–118 (1980)
Prodan, E.: Intrinsic Chern–Connes characters for crossed products by \({\mathbb{Z}}^d\) (2015). arXiv:1501.03479
Prodan, E., Schulz-Baldes, H.: Bulk and Boundary Invariants for Complex Topological Insulators: From \(K\)-theory to Physics. Springer, Berlin (2016). arXiv:1510.08744
Rennie, A., Robertson, D., Sims, A.: The extension class and KMS states for Cuntz-Pimsner algebras of some bi-Hilbertian bimodules. J. Topol. Anal. (2016). doi:10.1142/S1793525317500108
Schröder, H.: \(K\)-theory for Real \(C^*\)-algebras and Applications. Taylor & Francis, New York (1993)
Schulz-Baldes, H.: Persistence of spin edge currents in disordered quantum spin Hall systems. Commun. Math. Phys. 324(2), 589–600 (2013)
Schulz-Baldes, H.: \(\mathbb{Z}_2\) indices of odd symmetric Fredholm operators. Doc. Math. 20, 1481–1500 (2015)
Schulz-Baldes, H., Kellendonk, J., Richter, T.: Simultaneous quantization of edge and bulk Hall conductivity. J. Phys. A 33(2), L27–L32 (2000)
Thiang, G.C.: On the K-theoretic classification of topological phases of matter. Ann. Henri Poincaré 17(4), 757–794 (2016)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jean Bellissard.
Rights and permissions
About this article
Cite this article
Bourne, C., Kellendonk, J. & Rennie, A. The K-Theoretic Bulk–Edge Correspondence for Topological Insulators. Ann. Henri Poincaré 18, 1833–1866 (2017). https://doi.org/10.1007/s00023-016-0541-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-016-0541-2