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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 $exp⁡[ie/ℏ∮A⋅dr]$ 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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That is because, in a closed loop, the states can only change by a phase factor due to Aharonov-Bohm effect. Since the magnetic flux is

$$\Phi_B = \oint\limits_{\partial S} \mathbf{A} \cdot d\boldsymbol{\ell}$$

Its clear that you aquire a zero phase in the n.n.n. hopping, and a non-zero phase $\varphi$ in the s.n.n. hopping. Ok, the issue is, you take this phase factor as a part of the hopping terms: a loop is represented by changing the hopping terms in the form

$$t_1 \rightarrow t_1 \qquad t_2 \rightarrow t_2 e^{i\varphi}$$

i.e. you are taking the effect of the magnetic flux always by shifting the hopping terms, but the $t_1$ is not changing because the flux on the unit cell is zero, so $e^0 = 1$.

You can see that introduction to topological insulators from Carpentier and Fruchart: https://arxiv.org/abs/1310.0255 .Though it has some minor errors (like the vectors in the $d$ functions of the Hamiltonian), I think this part is well explained.

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