I consider the Hofstadter and the Harper operators, regarded as e ective models for a Bloch electron in a uniform magnetic eld, in the limit of weak and strong eld respectively. For each value of the Fermi energy in a spectral gap, I prove that the corresponding Fermi projectors exhibit a geometric duality, expressed in terms of some vector bundles canonically associated to the projectors. As a corollary, I get a rigorous geometric derivation of the TKNN equations. More generally, I prove that analogous equations hold true for any orthogonal projector in the rational rotation C -algebra, alias the algebra of the (rational) noncommutative torus.
| Identifer | oai:union.ndltd.org:CCSD/oai:tel.archives-ouvertes.fr:tel-00674271 |
| Date | 29 October 2010 |
| Creators | De Nittis, Giuseppe |
| Source Sets | CCSD theses-EN-ligne, France |
| Language | English |
| Detected Language | English |
| Type | PhD thesis |
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