# Berry's phase associated with monopole tunneling events of O(3) NLSM on square lattice

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I am currently reading Haldane's paper PRL. 61, 1029 on the Berry phase associated with monopole tunneling events of O(3) NLSM on square lattice. In this paper he asserts that the only non-trivial Berry phase associated with such events come from the bonds cut by the discontinuity lines connecting $-4\pi$ vortex and $+4\pi$ vortex of the field $\omega(x,y)$, and any such bond will contribute $\pi S\eta$ to the Berry phase. But I find that every bond will be counted twice since it belongs to two different plaquette, and every time I do the alternating sum $\eta(x,y) \omega(x,y)$ over a plaquette I will obtain $\pi S\eta$ from such a bond (since the field $\omega(x,y)$ will have a $4\pi$ jump across such a bond), I will in fact obtain $2\pi S\eta$ from such a bond in total, which lead to the contradiction. Can anyone explain to me what I'm doing wrong?

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