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Reviews in Mathematical Physics Vol. 28, No. 10 (2016) 1630003 (57 pages) ⃝c World Scientific Publishing Company DOI: 10.1142/S0129055X1630003X Notes on topological insulators Ralph M. Kaufmann∗ and Dan Li† Department of Mathematics, Purdue University, 150 N University St, West Lafayette, IN 47907, USA ∗rkaufman@math.purdue.edu †li1863@math.purdue.edu Birgit Wehefritz-Kaufmann Department of Mathematics and Department of Physics and Astronomy, Purdue University, 150 N University St, West Lafayette, IN 47907, USA ebkaufma@math.purdue.edu Received 23 June 2016 Accepted 18 October 2016 Published 8 November 2016 This paper is a survey of the Z2-valued invariant of topological insulators used in con- densed matter physics. The Z-valued topological invariant, which was originally called the TKNN invariant in physics, has now been fully understood as the first Chern num- ber. The Z2 invariant is more mysterious; we will explain its equivalent descriptions from different points of view and provide the relations between them. These invariants provide the classification of topological insulators with different symmetries in which K-theory plays an important role. Moreover, we establish that both invariants are realizations of index theorems which can also be understood in terms of condensed matter physics. Keywords: Topological insulators; index theory; topological K-theory. Mathematics Subject Classification 2010: 82-02, 19K56, 81V70 Contents 1. Introduction 2 2. First Chern Number as the Z Invariant 4 2.1. Landaulevels............................... 5 2.2. Blochbundle ............................... 6 2.3. DiracHamiltonian ............................ 9 2.4. Bulk-edgecorrespondence ........................ 10 3. Z2 Topological Insulators 1630003-1 12 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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