A rigorous Peierls-Onsager effective dynamics for semimetals in long-range magnetic fields
Abstract
We consider periodic (pseudo)differential elliptic operators of Schr\"odinger type perturbed by weak magnetic fields not vanishing at infinity, and extend our previous analysis in CIP,CHP-2,CHP-4 to the case of a semimetal having a finite family of Bloch eigenvalues whose range may overlap with the other Bloch bands but remains isolated at each fixed quasi-momentum. We do not make any assumption of triviality for the associated Bloch bundle. In this setting, we formulate a general form of the Peierls-Onsager substitution via strongly localized tight-frames and magnetic matrices. We also prove the existence of an approximate time evolution for initial states supported inside the range of the isolated Bloch family, with a precise error control.
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