spectrum of $H$.

For $U=0$, zero energy eigenstates $\Psi=(\phi_+,\phi_-)$ of $H$ solve $$\left[\partial_x\pm i\partial_y+i(\alpha_x\pm i\alpha_y\right)]\phi_\pm=0,\label{eq_zero_eigs}$$ with $\bm{\alpha}=e\bm{A}/\hbar$. Following Jackiw,[@Jac84] I work in the Coulomb gauge where $\bm{\partial}_{\bm{r}}\cdot\bm{A}=0$. Then Eq. (\[eq\_zero\_eigs\]), subject to the sample-periodic boundary conditions of Eq. (\[eq\_bc\]), is solved by $$\phi_\pm(\bm{r})=c_\pm\exp[\pm F(\bm{r})],\label{eq_phipm}$$ where $c_\pm$ are arbitrary constants. The real function $F$ is $UC$-periodic and satisfies $$\bm{\partial}_{\bm{r}}^2F=eB/\hbar.\label{eq_d2F}$$ In terms of the Fourier components $B_{mn}$ of the magnetic field, $F$ is given by $$F(\bm{r})=-\frac{e}{\hbar}\sum_{mn\not=0}\frac{B_{m,n}}{|\bm{k}_{mn}|^2}e^{i\bm{k}_{mn}\cdot\bm{r}}.
\label{eq_F}$$

One demonstrates uniqueness of the solutions in Eq. (\[eq\_phipm\]) as follows. Consider the equation for $\phi_-$. Suppose that $\tilde{\phi}_-(\bm{r})$ solves Eq. (\[eq\_zero\_eigs\]) and is sample-periodic as required by the boundary conditions. Define a function $g(\bm{r})=\tilde{\phi}(\bm{r})e^{F(\bm{r})}$. Since $F$ is $UC$-periodic, $g$ is sample-periodic. The function $g$ satisfies the Cauchy-Riemann equation $(\partial_x-i\partial_y)g=0$. Thus $g$ is an analytical function of the complex variable $z=x+iy$. As a consequence $g$ is either constant or unbounded in some directions at large $\bm{r}$. Unboundedness is incompatible with $g$ being sample-periodic and therefore $g$ must be constant so that $\tilde{\phi}_-(\bm{r})=ge^{-F(\bm{r})}$ is proportional to $\phi_-(\bm{r})=c_-e^{-F(\bm{r})}$. The uniqueness of $\phi_+$ is demonstrated in the same manner.

The Dirac equation $H\Psi=E\Psi$ with $U=0$ therefore has exactly two zero energy solutions which I will take to be 