Decoupling of Bloch Bands in the Presence of Homogeneous Fields

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Abstract
Following up an earlier communication, wave functions are constructed in Sec. 2 of this paper which are valid if a charge moves in a superposition of a periodic electric potential and a uniform magnetic field. The wave functions are not themselves solutions of the Schroedinger equation, but yield the traditional effective Hamiltonian for this problem. Contrary to the electric field case the mainfold of states linked by the "band index" does not form a Bloch band; the reason is that the cellular transforms of the Bloch-like functions are modified by the Peierls phase. At present, the derivation of these results is in closed form, but justifiable only "to all powers of the magnetic field." This was also the case for the previous electric derivation. The limitation may not be genuine. The third section of the paper does in fact prove directly the existence of closed Bloch bands in the presence of a homogeneous electric field; the case of free electrons is given as an example. One expects from this that the new results for the magnetic field are at least in part also independent of the power series method used for their justification. The fourth section extends the procedure to crossed electric and magnetic fields.