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The Non-Commutative Geometry of the Complex Classes of Topological Insulators

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Prodan E.
Topological Quantum Matter
|
2014
|
vol. 1
|
issue 1
EN
Alain Connes’ Non-Commutative Geometry program [1] has been recently carried out [2, 3] for the entire A- and AIII-symmetry classes of topological insulators, in the regime of strong disorder where the insulating gap is completely filled with dense localized spectrum. This is a short overview of these results, whose goal is to highlight the methods of Non-Commutative Geometry involved in these studies. The exposition proceeds gradually through the cyclic cohomology, quantized calculus with Fredholm-modules, local formulas for the odd and even Chern characters and index theorems for the odd and even Chern numbers. The characterization of the A- and AIIIsymmetry classes in the presence of strong disorder and magnetic fields emerges as a natural application of these tools.
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