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R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann 3.1. Backgroundandexamples........................ 13 3.1.1. Timereversalsymmetry..................... 13 3.1.2. EffectiveBrillouinzone ..................... 14 3.1.3. QuantumspinHalleffect .................... 15 3.1.4. 3dtopologicalinsulators..................... 16 3.2. Kane–Meleinvariant........................... 17 3.2.1. Kramersdegeneracy ....................... 17 3.2.2. Geometricsetup ......................... 19 3.2.3. Definition............................. 21 3.2.4. Spectralflow ........................... 22 3.2.5. StrongZ2invariant........................ 23 3.2.6. Holonomy............................. 24 3.3. Topologicalbandtheory......................... 26 3.3.1. Quaternionicstructure...................... 26 3.3.2. QuaternionicK-theory...................... 27 3.3.3. Determinantlinebundle..................... 28 3.4. Homotopytheory............................. 29 4. Z2 Invariant as an Index Theorem 32 4.1. AnalyticalZ2index............................ 32 4.1.1. Maslovindex ........................... 32 4.2. TopologicalZ2index........................... 35 4.2.1. CS/WZWcorrespondence.................... 36 4.2.2. OddCherncharacter....................... 38 4.2.3. Green’sfunction ......................... 40 4.2.4. Parityanomaly.......................... 41 4.3. Summary ................................. 43 5. K-Theoretic Classification 44 5.1. Tenfoldway................................ 45 5.2. Nonlinearσ-model ............................ 45 5.3. Cliffordmodules ............................. 47 5.4. TwistedequivariantK-theory...................... 50 6. Further Discussions 51 6.1. Bulk-edgecorrespondencerevisited................... 51 6.2. Noncommutativegeometry ....................... 53 1. Introduction Topological insulators [41] are new materials observed in nature which behave like insulators in the bulk but have conducting edge or surface states on the boundary. In physical terms, topological insulators are gapped electronic systems which show topologically protected non-trivial phases in the presence of the time reversal Z2- symmetry. Because of the (odd) time reversal symmetry, topological insulators are characterized by a Z2-valued invariant. We will use Z2 to denote the group Z/2Z, 1630003-2 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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