Graduate Why is the nearest hopping kept real in Haldane model?

[lichen1983312]

I am leaning the Haldane model :
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.61.2015
Haldane imaged threading magnetic flux though a graphene sheet, and the net flux of a unit cell is zero.
He argued that since the loop integral $\exp [ie/\hbar \oint {A \cdot dr} ]$ along a path of nearest bonds vanishes, the nearest hopping is not changed.

However, I cannot see the connect between the vanishing loop integral and the unchanged nearest hopping, can anybody help?

 

[Fred Wright]

In the Haldane model of graphene the hopping strength can be complex with a phase coming from the Aharonov-Bohm effect in the presence of a magnetic field. If one moves a particle around a closed contour, then the phase difference between the final and initial states is proportional to the magnetic flux enclosed by the contour $\phi =\frac{e}{h} \iint \mathbf B \cdot \mathbf S = \frac{e}{h} \oint \mathbf A \cdot \mathbf {\mathcal l}$.
Consider three sites a, b and c. The hopping strength between these three sites is $t_{ab}$, $t_{bc}$ and $t_{ca}$. If a particle hops from a to b and then to c, then the hopping strength around the loop is: $$t_{ab}t_{bc}t_{ca}=\left | t_{ab}t_{bc}t_{ca} \right | e^{i(\phi_{ab} + \phi_{bc} + \phi_{ca})}$$
The phase picked up by the electron is: $$\phi_{ab} +\phi_{bc} + \phi_{ca} = \frac{e}{h} \iint \mathbf B \cdot \mathbf S$$
If B is nonzero inside the triangle formed by these three sites, the phase for these hoppings are nonzero. On the other hand; no net field means no phase change.