however be pointed out that flat bands do not guarantee a Hall effect: The non-relativistic Schrödinger equation in the presence of the magnetic field of this example was studied numerically in Ref. . Flat bands were found in the limit $\lambda/l_m\gg1$. For that system the Chern numbers of flat bands were found to be zero, leading to a zero Hall conductivity.

Neither are the flat bands a necessary condition for the Hall effect to occur. The effect also occurs for $l_m >\lambda$ when the renormalized Fermi velocity is of the same order of magnitude as the unrenormalized Fermi velocity. Counter-intuitively, it is not the flat bands, but the Dirac cones at the touching points between bands, that lead to the Hall effect.

Returning to the analysis of the example I now introduce a scalar potential $U(\bm{r})=-U_0 B(\bm{r})/B_0$ with the same spatial profile as the magnetic field. The induced mass is $$\mu={\rm sign}(B_0)U_0\frac{I_1(\beta/4\pi^2)}{I_0(\beta/4\pi^2)}\simeq{\rm sign}(B_0)U_0,\label{eq_muii}$$ where $I_1$ is the first modified Bessel function of the first kind. The asymptotic form is valid in the same limit as before, namely when $l_m\ll\lambda$. This result is straight forward to interpret. In the strong magnetic field limit the eigenstates become exponentially well-confined to the maxima 