only the former is relevant for present purposes. As a consequence of the smoothness of $\bm A$, $UC$ is much larger than the unit cell of the graphene crystal lattice. Because $B$ averages to zero, the magnetic flux through $UC$ is zero.

The graphene sample is assumed to be much larger than $UC$ and only bulk effects will be considered. It is therefore convenient to impose periodic boundary conditions $$\Psi(\bm{r})=\Psi(\bm{r}+\Omega\bm{a}_k),~~k=1,2,\label{eq_bc}$$ where $\Omega\gg1$ is an integer. All functions with this property will be called sample-periodic. Note that because $\Omega$ is an integer, any $UC$-periodic function is also sample-periodic.

A reciprocal lattice is associated with the real space lattice. The basis vectors $\bm{b}_j$, $j=1,\,2$ of the reciprocal lattice are defined through the equations $\bm{b}_j\cdot\bm{a}_k=2\pi\delta_{j,k}$. The Brillouin zone is a primitive unit cell of the reciprocal lattice, and will be denoted $BZ$. It is convenient to define the Fourier components $$B_{mn}=l^{-2}\int_{UC} dr^2\,e^{-i\bm{k}_{mn}\cdot\bm{r}}B(\bm{r}),\label{eq_bmn}$$ of the magnetic field. Here $\bm{k}_{mn}=m\bm{b}_1+n\bm{b}_2$ with $m$ and $n$ integers and $l^2=|\bm{a}_1\times\bm{a}_2|$ is the area of $UC$. Note that $B_{00}=0$ due to the zero-flux condition imposed on $B$ and $B_{m,n}=B_{-m,-n}^*$ because $B(\bm{r})$ is real.

Zero energy modes {#two}
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The task is now to analyze the low-energy (i.e. small $|\varepsilon|$) 