Mean-field Berry phase of an interacting spin-1/2 system

We study a long-range interacting spin-1/2 system in the mean-field perspective, and obtain an analytical expression for its Berry phase. The magnetic-like flux interpretation of the Berry phase shows that the source and sink of the magnetic-like field are, respectively, located at the disk-shaped level-crossing region, where the first-order quantum phase transition occurs, and its boundary, where the continuous quantum phase transition occurs. From the asymptotic distribution of the field at the infinity, we find that the source and sink as a whole can be seen as a disk-shaped monopole with a negative magnetic charge. The analogues of this monopole and its exotic field can generally exist in other interacting spin systems.