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

Walter Kohn
Phys. Rev. 115, 1460 – Published 15 September 1959
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Abstract

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 HN+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 Pα. (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)=Σi=0NsiH¯m;i(P) where sHc and the functions H¯m;i(P) are completely symmetrized in the noncommuting operators Pα. Properties (3) and (4) can also be summarized in the equations H¯m(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 H¯m;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 H¯m(P) provides a sound basis for a discussion of wave functions and energy levels of Bloch electrons in a magnetic field.

  • Received 26 February 1959

DOI:https://doi.org/10.1103/PhysRev.115.1460

©1959 American Physical Society

Authors & Affiliations

Walter Kohn

  • Department of Physics, Carnegie Institute of Technology, Pittsburgh, Pennsylvania

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Issue

Vol. 115, Iss. 6 — September 1959

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