scalar potential $U$ whose spatial profile is correlated with that of $B$ will do the opposite and leads to gap associated with a negative mass.

Turning to the expression \[Eq. (\[eq\_u\]\] for the $U\not=0$ valence band Bloch states $u_{-,\bm{k}}$, one sees that the dominant term in the linear combination is proportional to $e^{-{\rm sign}(\mu) F}$. As a result, valence band states are confined to regions of positive $B$ when $U$ is anti-correlated with $B$, and confined to regions of negative $B$ when $U$ is correlated with $B$. This is consistent with the response of $\Psi_{\pm,0}$ to $U$ discussed above.

One thus arrives at the fundamental result that there is a finite positive (negative) magnetic flux through the region of space occupied valence band electrons when the mass is positive (negative). The valence band will then experience a ‘usual’ Hall effect. An electric field $\bm{E}=E\hat{\bm{y}}$ will, through the Lorentz force, lead to a current in the direction $-{\rm sign}(\mu)\, \hat{\bm{x}}$. Electrons in the conduction band would have seen an opposite magnetic field and produced an opposite Hall effect. However, the Fermi energy is in the gap between the two bands so that no conduction band states are occupied. The sign obtained 