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

Abstract

The Hamiltonian of a Bloch electron in a static magnetic field is $H=\frac{1}{2}{\mathrm{P}}^{2}+V(\mathrm{r})$, where $V(\mathrm{r})$ 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 ${\mathcal{H}}^{N+1}$ ($N$ arbitrary), the eigenstates of $H$ can be calculated from an equivalent Hamiltonian ${\overline{H}}_{m}(\mathrm{P})$ 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 ${P}^{\ensuremath{\alpha}}$. (3) It has the periodicity property ${\overline{H}}_{m}(\mathrm{P}+\mathrm{K})={\overline{H}}_{m}(\mathrm{P})$, where K is an arbitrary reciprocal lattice vector. (4) It can be written as a series ${\overline{H}}_{m}(\mathrm{P})={{\ensuremath{\Sigma}}_{i=0}}^{N}{s}^{i}{\overline{H}}_{m;i}(\mathrm{P})$ where $s\ensuremath{\equiv}\frac{\mathcal{H}}{c}$ and the functions ${\overline{H}}_{m;i}(\mathrm{P})$ are completely symmetrized in the noncommuting operators ${P}^{\ensuremath{\alpha}}$. Properties (3) and (4) can also be summarized in the equations ${\overline{H}}_{m}(\mathrm{P})={\ensuremath{\Sigma}}_{l}{a}^{(l)}\ifmmode\times\else\texttimes\fi{}\mathrm{exp}[i{\mathrm{R}}^{(l)}\ifmmode\cdot\else\textperiodcentered\fi{}\mathrm{P}]$, where the ${\mathrm{R}}^{(l)}$ are lattice vectors and the ${a}^{(l)}$ can be expanded as ${a}^{(l)}={{\ensuremath{\Sigma}}_{i=0}}^{N}{s}^{i}{{a}_{i}}^{(l)}$. 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 ${\overline{H}}_{m}(\mathrm{P})$ provides a sound basis for a discussion of wave functions and energy levels of Bloch electrons in a magnetic field.