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

The Hamiltonian of a Bloch electron in a static magnetic field is $H=\frac{1}{2}{\mathrm{P}}^2+V\left(\mathrm{r}\right)$, where $V\left(\mathrm{r}\right)$ is the periodic potential, $\mathrm{P}=\mathrm{p}+\frac{\mathrm{A}}{c}$, and A is the vector potential giving rise to the magnetic field $\mathcal{H}$. We consider the case of a nondegenerate band $m$. It is then shown that, with an error vanishing with $\mathcal{H}$ like ${\mathcal{H}}^{N+1\mathrm{}}$ ($N$ arbitrary), the eigenstates of $H$ can be calculated from an equivalent Hamiltonian ${\overline{H}}_m\left(\mathrm{P}\right)$ 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 $P^{\alpha }$. (3) It has the periodicity property ${\overline{H}}_m(\mathrm{P}+\mathrm{K})={\overline{H}}_m\left(\mathrm{P}\right)$, where K is an arbitrary reciprocal lattice vector. (4) It can be written as a series ${\overline{H}}_m\left(\mathrm{P}\right)={{\Sigma }_{i=0\mathrm{}}}^N{s}^i{\overline{H}}_{m;i}\left(\mathrm{P}\right)$ where $s\equiv \frac{\mathcal{H}}{c}$ and the functions ${\overline{H}}_{m;i}\left(\mathrm{P}\right)$ are completely symmetrized in the noncommuting operators $P^{\alpha }$. Properties (3) and (4) can also be summarized in the equations ${\overline{H}}_m\left(\mathrm{P}\right)={\Sigma }_l{a}^{\mathrm{}\left(l\right)\mathrm{}}\times \exp \left[i{\mathrm{R}}^{\mathrm{}\left(l\right)\mathrm{}}·\mathrm{P}\right]$, where the ${\mathrm{R}}^{\mathrm{}\left(l\right)\mathrm{}}$ are lattice vectors and the $a^{\mathrm{}\left(l\right)\mathrm{}}$ can be expanded as $a^{\mathrm{}\left(l\right)\mathrm{}}={{\Sigma }_{i=0\mathrm{}}}^N{s}^i{a_i}^{\mathrm{}\left(l\right)\mathrm{}}$. An algorithm is given for the construction of the ${\overline{H}}_{m;i}$ 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 ${\overline{H}}_m\left(\mathrm{P}\right)$ provides a sound basis for a discussion of wave functions and energy levels of Bloch electrons in a magnetic field.