have canceled.

There are two ways to change the sign of the mass term \[cf. Eq. (\[eq\_mass\])\]. One may either change the sign of the external potential $U$, or one may change the sign of the magnetic field and hence of $F$. According to the theory, both methods give the same change in the Hall conductivity, $\Delta\sigma_{xy}=\Delta\sigma_{xy}(U\to-U)=\Delta\sigma_{xy}(B\to-B)$. The transformation $B\to-B$ is equivalent to time-reversal, which changes the sign of the Hall conductivity: $\sigma_{xy}=-\sigma_{xy}$. Therefore $\Delta\sigma_{xy}=2\sigma_{xy}$. Thus, knowing the change in $\sigma_{xy}$ when the mass changes sign, one also knows the actual value of $\sigma_{xy}$.

If $\sigma_{xy}^\pm$ is the Hall conductivity of a sample in which the mass term has sign $\pm$ then $\sigma_{xy}^+-\sigma_{xy}^-=-2e^2/h$. Therefore, according to the above argument $$\sigma_{xy}^{\pm}=\mp \frac{e^2}{h}.\label{eq_hall}$$

Origin of non-zero Hall conductivity {#five}
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Suppose that the topological theory discussed in the previous section was not known. Would it have been possible to show that the graphene system studied here displays a Hall effect, without at the same time essentially deriving the topological theory? As I show in this section, the answer is ‘yes’. In the process, the underlying physical mechanism that here produces the anomalous Hall effect is identified.

The starting point of the argument 