Fig. 4: Schematic of bubble dynamics.

a,b, Second-order processes for bubble hopping and interactions. The amplitude of each contributing path is shown in units of \(\kappa ={h}_{x}^{2}/(2{h}_{z})=-n{h}_{x}^{2}/(4J)\). The rate of the process is given by the sum of all the paths. a, 1-bubbles can always hop to neighbouring sites via a second-order process. For n = 1, the lower path cannot be used (crossed out) as it is resonant and is, therefore, already accounted for by first-order processes. For n > 1, the two paths do not cancel out since one changes the number of domain walls and the other does not. b, 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. c, Bubble dynamics at the n = 1 resonance. 1-bubbles are created, which then hop around the system; furthermore, no larger bubbles can be produced. d, Bubble dynamics at the n = 2 resonance, which is representative of all n > 1. 2-bubbles are created and cannot move, after which neighbouring 2-bubbles create 1- and 3-bubbles through interaction effects. Larger bubbles cannot move, whereas 1-bubbles can hop around the system. The colours of the spins in c and d correspond to the size of the bubble as in Fig. 2.