In the case of larger bubbles, the two paths preserve the number of domain walls (top). Their respective amplitudes only depend on the change in the number of spins, making them opposite in sign and cancelling each other out, meaning that larger bubbles cannot directly hop. However, when next to each other, n -bubbles can exchange spins. The interface between them is a single spin and one of the two paths changes the number of domain walls, leading to a different amplitude (bottom). This type of spin exchange is not possible for 1-bubbles since no bubble can get smaller. For n > 1, these interactions lead to bubbles of size other than n . Through multiple consecutive exchanges, even 1-bubbles can emerge, which are then able to hop.
