Triviality of Bloch and Bloch-Dirac bundles

Details Triviality of Bloch and Bloch-Dirac bundles Triviality of Bloch and Bloch-Dirac bundles

2006-01-18

arxiv.org/abs/math-ph/0601034v1

Ann. Henri Poincare' 8 (2007), 995-1011.

Physics

Mathematical Physics

20 pages, no figures

Scientific paper

10.1007/s00023-007-0326-8

In the framework of the theory of an electron in a periodic potential, we reconsider the longstanding problem of the existence of smooth and periodic quasi-Bloch functions, which is shown to be equivalent to the triviality of the Bloch bundle. By exploiting the time-reversal symmetry of the Hamiltonian and some bundle-theoretic methods, we show that the problem has a positive answer for any d < 4, thus generalizing a previous result by G. Nenciu. We provide a general formulation of the result, aiming at the application to the Dirac equation with a periodic potential and to piezoelectricity.

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