the general results derived in this text, consider the magnetic field $B(\bm{r})=B_0\left(\sin2\pi x/\lambda+sin 2\pi y/\lambda\right)$ with periodicity $\lambda$ in both the $x$ and $y$ directions.

It is convenient to define the magnetic length $l_m=\sqrt{\hbar/e|B_0|}$ and a dimensionless constant $\beta=e\lambda^2|B_0|/\hbar=(\lambda/l_m)^2$. According to Eq. (\[eq\_F\]), $F$ is given by $$F(\bm{r})=-\frac{\beta}{4\pi^2}\frac{B(\bm{r})}{|B_0|}.$$ This gives the clearest possible illustration that $\Psi_{\pm,0}$ are localized to regions of positive or negative $B$. The degree of localization is controlled by the ratio $\lambda/l_m$.

The renormalized Fermi velocity can be calculated explicitly as $$\tilde{v}=\frac{v}{I_0(\beta/4\pi^2)^2},\label{eq_tildev}$$ where $I_0$ is the zero’th modified Bessel function of the first kind. When $\beta\gg1$ or equivalently $l_m\ll\lambda$, the asymptotic form $\tilde{v}\simeq \beta e^{-\beta/2\pi^2}v/2\pi$ is valid. In this limit the Fermi velocity becomes very small and bands become nearly dispersionless in the region of $\bm k=0$. The result is very similar to the nanotube result.[@Lee03] This is due to the fact that the spatial profile of the magnetic field in the nanotube case is also sinusoidal. In Figure \[fig0\] the solid line shows the renormalized Fermi velocity (solid line) plotted as a function of the magnetic field strength $B_0$.

As was pointed out in Ref. and the flat bands are reminiscent of Landau levels. It should 