Why is the nearest hopping kept real in Haldane model?

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SUMMARY

The discussion focuses on the Haldane model, specifically the behavior of nearest hopping in a graphene sheet subjected to magnetic flux. Haldane demonstrated that the loop integral along nearest bonds vanishes, which leads to the conclusion that the nearest hopping remains unchanged. The relationship between the Aharonov-Bohm effect and the complex hopping strength is emphasized, where the phase difference is directly proportional to the magnetic flux enclosed. The analysis of hopping strengths between three sites (a, b, and c) illustrates that a non-zero magnetic field results in a phase change, while a net zero field maintains the original hopping characteristics.

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  • Understanding of the Haldane model in condensed matter physics
  • Familiarity with the Aharonov-Bohm effect
  • Knowledge of complex hopping terms in quantum mechanics
  • Basic concepts of magnetic flux and its implications in quantum systems
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Physicists, materials scientists, and researchers interested in quantum mechanics and the electronic properties of graphene, particularly those studying the effects of magnetic fields on hopping phenomena.

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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?
 
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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.