Theory of Bloch Electrons in a Magnetic Field: The Effective Hamiltonian

The Hamiltonian of a Bloch electron in a static magnetic field is H=12P^2+V (r), where V (r) is the periodic potential, P=p+Ac, and A is the vector potential giving rise to the magnetic field H. We consider the case of a nondegenerate band m. It is then shown that, with an error vanishing with H like H^N+1 (N arbitrary), the eigenstates of H can be calculated from an equivalent Hamiltonian H₌ (P) with the following properties: (1) It is a one-band Hamiltonian, obtained by transforming away all relevant interband matrix elements. (2) It depends only on the gauge-covariant operators P^. (3) It has the periodicity property H₌ (P+K) =H₌ (P), where K is an arbitrary reciprocal lattice vector. (4) It can be written as a series H₌ (P) ={₈=₀}^Ns^iH₌;₈ (P) where sHc and the functions H₌;₈ (P) are completely symmetrized in the noncommuting operators P^. Properties (3) and (4) can also be summarized in the equations H₌ (P) =₋a^ (l) R^ (l), where the R^ (l) are lattice vectors and the a^ (l) can be expanded as a^ (l) ={₈=₀}^Ns^i{a₈}^ (l). An algorithm is given for the construction of the H₌;₈ and carried through for i=0, 1, 2. The formalism is not restricted to the neighborhood of the bottom and top of the band. We believe that the equivalent Hamiltonian H₌ (P) provides a sound basis for a discussion of wave functions and energy levels of Bloch electrons in a magnetic field.