Phasenfestlegung in den Wannier-Funktionen und einfache Herleitung des Ersatzoperators für Gitterelektronen im Magnetfeld

Abstract

The motion of cristal electrons in a magnetic field can be described by the well known Hamiltonian, first given byPeierls to zeroth order inB and derived in full generality byKohn. A derivation of this most general result is given here in the most elementary and direct way, which seems possible (as the author believes). The results summed up in section 1 are derived using the gaugeA=1/2B×r which proves to be the most advatageous one for this purpose and using a special choice of phases in constructing Wannier functions from Bloch functions. This choice of phases given and discussed in section 2 seems to be of general interest for any use of Wannier-functions. In section 3 one gets the new Hamiltonian in thek-representation in a closed form by a straightforward calculation after acting with the initial Hamiltonian on certain unorthogonal modified Bloch functions, whose Fourier coefficients with respect tok are the Wannier-Luttinger functions. It consists of an one-band-Hamiltonian for each band and a band-coupling-Hamiltonian for any two bands as well as like metric operators because of the nonorthogonality of the original basis.

In section 4 it is shown that the expansion of the operators in a power series ofB is always divergent but asymptotically right for smallB. In the same sense any two nonintersecting bands can be decoupled; the decoupling to first order inB and the zeroth and first order of the one-band-Hamiltonians are given and can in principle be calculated to any order. Section 5 contains some remarks about the range of validity of the asymptotic expansion in powers ofB.