the magnetic field points in the negative $z$ direction are $A$ sublattice polarized while regions where the magnetic field points in the positive $z$ direction are $B$ sublattice polarized. This is consistent with the numerical results presented in Figure 3 of Ref. . There it was pointed out that the state $\Psi_{-,0}$ closely resembles a state in the zero Landau level of a graphene sample with a constant magnetic field that points in the positive $z$ direction. This would have explained the existence of a Hall effect, were it not for the state $\Psi_{+,0}$ that also belongs to the zero eigenspace. This state is similar to a zero Landau level state in a magnetic field pointing in the negative $z$ direction, and it is responsible for a Hall effect that cancels that of $\Psi_{-,0}$.

In order to have a Hall effect a perturbation that lifts the degeneracy between $\Psi_{+,0}$ and $\Psi_{-,0}$ is needed. This is the role of the electrostatic potential $U$. If the sign of $U$ is anti-correlated with that of $B$ it will raise the energy of the state $\Psi_{+,0}$ while lowering the energy of the state $\Psi_{-,0}$. This leads to a gap with associated positive mass. A 