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

The Hamiltonian of a Bloch electron in a static magnetic field is H=12P²+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¯_m(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^α. (3) It has the periodicity property H¯_m(P+K)=H¯_m(P), where K is an arbitrary reciprocal lattice vector. (4) It can be written as a series H¯_m(P)=Σ_i=0^NsⁱH¯_mi(P) where s≡Hc and the functions H¯_mi(P) are completely symmetrized in the noncommuting operators P^α. Properties (3) and (4) can also be summarized in the equations H¯_m(P)=Σ_la^(l)×exp[iR^(l).P], where the R^(l) are lattice vectors and the a^(l) can be expanded as a^(l)=Σ_i=0^Nsⁱa_i^(l). An algorithm is given for the construction of the H¯_mi 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¯_m(P) provides a sound basis for a discussion of wave functions and energy levels of Bloch electrons in a magnetic field.