states. They satisfy $H(\bm{k})\psi_{n,\bm{k}}=\varepsilon_{n,\bm{k}}\psi_{n,\bm{k}}$ where $\varepsilon_{n,\bm{k}}$ are the energies of the eigenstates. The effective Hamiltonian $H(\bm{k})$ is given by $$H(\bm{k})=v\bm{\sigma}\cdot\left(-i\hbar\bm{\partial}_{\bm{r}}+\hbar\bm{k}+e\bm{A}\right)+U=H+\hbar v\bm{\sigma}\cdot\bm{k}.$$ Normalizing $\psi_{n,\bm{k}}(\bm{r})$ to unity over $UC$ ensures that $\Psi_{n,\bm{k}}$ is normalized to unity over the whole sample. The zero-eigenstates of Eq. (\[eq\_sols2\]) are $UC$-periodic, and therefore correspond to $\bm{k}=0$ solutions in this labeling scheme. The corresponding normalized zero-energy eigenstates of $H(\bm{k}=0)$ (with $U=0$) are $\psi_{\pm,0}=\Omega\Psi_{\pm,0}$.

Near Zero Eigenstates and a non-zero but small scalar potential {#near_zero}
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The near-zero energy eigenstates in the vicinity of $\bm{k}=0$ can be studied by treating the term $\hbar v\bm{\sigma}\cdot\bm{k}$ in $H(\bm{k})$ perturbatively. The effect of a weak periodic scalar potential $U$ can be treated perturbatively at the same time.

To leading order in $\bm{k}$ and $U$, the problem is solved by Bloch states of the form $$u_{\eta,\bm{k}}(\bm{r})=\chi_{\eta,\bm{k}}^+\psi_{+,0}(\bm{r})+\chi_{\eta,\bm{k}}^-\psi_{-,0}(\bm{r}),$$ where $\chi_{\eta,\bm{k}}=(\chi_{\eta,\bm{k}}^+,\chi_{\eta,\bm{k}}^-)$ satisfies $h(\bm{k})\chi_{\eta,\bm{k}}=\varepsilon_{\eta,\bm{k}}\chi_{\eta,\bm{k}}$. Here $\eta=\pm$ labels the bands ($-$ for the valence band and $+$ for the conduction band), $\varepsilon_{\eta,\bm{k}}$ is the energy of the state with label $\eta,\bm{k}$ and $h(\bm{k})$ is the Hamiltonian $H(\bm{k})$ projected onto the zero eigenspace. It has the form of a $2\times2$ Dirac Hamiltonian in k-space: $$\begin{aligned}
&h(\bm{k})_{\eta\eta'}=\int_{UC}dr^2\,\psi_{\eta,0}(\bm{r})^\dagger
\left[\hbar v\bm{\sigma}\cdot\bm{k}+U(\bm{r})\right]\psi_{\eta',0}(\bm{r}),\nonumber\\
&\implies h(\bm{k})=\hbar \tilde{v}\,\bm{\sigma}\cdot\bm{k}+\mu\,\sigma_z\end{aligned}$$ where a term proportional to identity is 