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the level of Kasparov’s analytic bivariant K-theory, the bulk-edge correspondence is the Kasparov product with a carefully selected Kasparov module according to the interpretation of index maps in KK-theory. For quantum Hall effect, such a bulk- edge correspondence is explicitly constructed in [20], where the authors represent the bulk conductance as a Kasparov product of the boundary K-homology with a specific extension class (i.e. KK1-class) as the correspondence. A similar model for time reversal invariant topological insulators is expected to exist, and the bulk-edge correspondence is expected to be realized by Kasparov product in real KK-theory. 6.2. Noncommutative geometry There are different sources of noncommutativity in condensed matter physics, which call for generalizations of classical models in the noncommutative world. For instance, the introduction of magnetic fields into a lattice model breaks the translational symmetry, that is, the effective Hamiltonian does not commute with the translation operators any longer, which is the key assumption for the classical Bloch theory of crystal wavefunctions. Another source of noncommutativity is cre- ated by nontrivial quasi-particle statistics such as non-abelian anyons in fractional quantum Hall systems. In addition, systems with impurities or defects such as dis- ordered topological insulators are of realistic importance, which is the main source of noncommutative geometry models up to date. Noncommutative geometry [23] has already achieved some success in modeling the integer quantum Hall effect and topological insulators with disorders. The work by Bellissard, van Elst and Schulz-Baldes [16] set up the framework for studying topological invariants of a disordered system by noncommutative geometry. The dis- ordered integer quantum Hall effect can be characterized by the noncommutative first Chern number, which was obtained by combining the Connes–Chern character with the Kubo–Chern formula [16]. The noncommutative second Chern number is a natural generalization for time reversal invariant models, and the general non- commutative nth Chern number was also studied in [58]. A survey on the noncommutative approach to disordered topological insulators was written by Prodan [57]. The noncommutative odd Chern character was dis- cussed in a recent paper [60] by Prodan and Schulz-Baldes in the same framework as in [16]. A noncommutative formula for the isotropic magneto-electric response of disordered insulators under magnetic fields was also proposed in [53]. It would be interesting to see more noncommutative generalizations of both the analytical and topological Z2 index of topological insulators in the future. Acknowledgments The authors would like to thank the Max-Planck Institute for Mathematics in Bonn for its hospitality. Major parts of this paper were written there. We would like thank Charles Kane and Jonathan Rosenberg for discussions and inspiration. We would also like to thank Barry Simon, Joel Moore, Duncan Haldane and Masoud 1630003-53 Notes on topological insulators Rev. Math. Phys. 2016.28. Downloaded from www.worldscientific.com by PURDUE UNIVERSITY on 08/11/17. For personal use only.PDF Image | Notes on topological insulators
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