Nonperturbative Matrix Mechanics Approach to Spin-Split Landau Levels and the $g$ Factor in Spin-Orbit Coupled Solids

We propose a fully quantum approach to nonperturbatively calculate the spin-split Landau levels and $g$ factor of various spin-orbit coupled solids based on the $\mathit{k}·\mathit{p}$ theory in the matrix mechanics representation. The new method considers the detailed band structure and the multiband effect of spin-orbit coupling irrespective of the magnetic-field strength. We show an application of this method to PbTe, a typical Dirac electron system. Contrary to popular belief, we show that the spin-splitting parameter $M$, which is the ratio of the Zeeman to cyclotron energy, exhibits a remarkable magnetic-field dependence. This field dependence can rectify the existing discrepancy between experimental and theoretical results. We also show that $M$ evaluated from the fan diagram plot is different from that determined as the ratio of the Zeeman to cyclotron energy, which also overturns common belief.

Figure 1

Schematic image of the $\pi$ matrix method. The kinematical momentum operator $\mathit{\pi }$ in the $\mathit{k}·\mathit{p}$ Hamiltonian is expressed in the matrix mechanics representation.

Figure 2

Magnetic-field dependence of (a) spin-split Landau levels, (b) Zeeman $E_Z$ and cyclotron $E_c$ energy, and (c) spin-splitting parameter $M_{\mathrm{ZC}}$ ($E_{Z,c}$ and $M_{\mathrm{ZC}}$ are obtained for $n=2$). The solid and dashed lines represent the results obtained by the $\pi$ matrix method and the extended Dirac model, respectively.

Figure 3

Spin-splitting parameter $M_{\mathrm{ZC}}$ for ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Te}$. The band inversion (therefore, the topological transition) occurs at $x=0.38$.

Figure 4

Plot of the reciprocal fields $1/B_n$ for PbTe, where $\mu$ touches the bottom of the $n$th Landau level. Here, $M_{\mathrm{fan}}=x_{-}-x_{+}=0.63$. The inset shows the $\mu$ dependence of $M_{\mathrm{fan}}$.