In this article we study the Bistritzer-MacDonald (BM) model with external magnetic field. We study the spectral properties of the Hamiltonian in an external magnetic field with a particular emphasis on the flat band of the chiral model at magic angles. Our analysis includes different types of interlayer tunneling potentials, the so-called chiral and anti-chiral limits. One novelty of our article is that we show that using a magnetic field one can discriminate between flat bands of different multiplicities, as they lead to different Chern numbers in the presence of magnetic fields, while for zero magnetic field their Chern numbers always coincide.
Abstract. We consider quenched random perturbations of skew products of rotations on the unit circle over uniformly expanding maps on the unit circle. It is known that if the skew product satisfies a certain condition (shown to be generic in the case of linear expanding maps), then the transfer operator of the skew product has a spectral gap. Using semiclassical analysis we show that the spectral gap is preserved under small random perturbations. This implies exponential decay of quenched random correlation functions for smooth observables at small noise levels.
We obtain microlocal analogues of results by L. Hörmander about inclusion relations between the ranges of first order differential operators with coefficients in C ∞ that fail to be locally solvable. Using similar techniques, we study the properties of the range of classical pseudodifferential operators of principal type that fail to satisfy condition (Ψ ).
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