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

The Hamiltonian of a Bloch electron in a static magnetic field is H=1/2P²+V(r), where V(r) is the periodic potential, P=p+A/c, 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 HN+1 (N arbitrary), the eigenstates of H can be calculated from an equivalent Hamiltonian H̄ₘ(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 H̄ₘ(P+K)=H̄ₘ(P), where K is an arbitrary reciprocal lattice vector. (4) It can be written as a series H̄ₘ(P)=Σᵢ₌₀NsⁱH̄m;i(P) where s≡H/c and the functions H̄m;i(P) are completely symmetrized in the noncommuting operators P^α. Properties (3) and (4) can also be summarized in the equations H̄ₘ(P)=Σₗa⁽ˡ⁾×exp[iR⁽ˡ⁾·P], where the R⁽ˡ⁾ are lattice vectors and the a⁽ˡ⁾ can be expanded as a⁽ˡ⁾=Σᵢ₌₀Nsⁱaᵢ⁽ˡ⁾. An algorithm is given for the construction of the H̄m;i 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 H̄ₘ(P) provides a sound basis for a discussion of wave functions and energy levels of Bloch electrons in a magnetic field.