# Berry phase for instanton events in 2d $O(3)$ nonlinear sigma model

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I am reading the paper "http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.61.1029" by Haldane where he calculates the Berry phase associated with instanton or hedgehog events in the O(3) nonlinear sigma model (described by the unit vectors Ω(x,y,t)). On page 2 last paragraph he says that a hedgehog event will create ±4π vortex of w(x,y) at the spatial center of the hedghog, where w(x,y) is supposed to be the Berry phase obtained by the closed loop defined by the time evolution of eΩ(x,y,t). I am confused by this statement, because after the hedgehog event, the initial field configuration has changed to a configuration with a different skyrmion number and Ω(x,y,t) does not define a closed loop. So, I don't understand what he means by w(x,y) in this case. If somebody can clarify this, it will be very helpful.

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