charge moving on the surface of the unit sphere. In this case, the Berry’s phase is simply the phase acquired by the charged particle if a magnetic monopole of strength $S$ (i.e., $\mathbf{B}(\mathbf{r})=(S/r^2)\hat{r}$) is placed at the center of the sphere. The Berry’s phase acquired is the magnetic flux enclosed by the closed path $\mathbf{C}$.

![ Berry’s phase with a magnetic monopole.[]{data-label="fig:Berry"}](Berry.eps){width="4.0cm"}

Let $S\mathbf{A}_M(\mathbf{r})$ be the vector potential associated with the monopole, i.e., $\nabla\times\mathbf{A}_M=\hat{r}/r^2$; then, in the “charge + gauge field" representation, the effect of the Berry’s phase can be described by a vector-potential term in the action: $$\label{WZ1}
         S_{B}=\hbar S\Omega(C)=\hbar S\int dt\mathbf{A}_M(\hat{r})\cdot\dot{\hat{r}}.$$ This is an example of a Wess-Zumino term for quantum particles. A more rigorous derivation of the Wess-Zumino action is given in Appendix A, where the action for a single spin in a magnetic field is derived [*via*]{} a path integral approach.

We now revisit the two-spin problem. With the Berry’s phases included, the Lagrangian of the corresponding rotor problem becomes $$\label{La2}
      L={1\over2J}(\hat{n}\times\dot{\hat{n}})^2+\hbar S_A\mathbf{A}_{M}(\hat{r}_A)\cdot\dot{\hat{r}}_A+\hbar S_B\mathbf{A}_{M}(\hat{r}_B)\cdot\dot{\hat{r}}_B,$$ where $\mathbf{N}=N\hat{n}=\mathbf{S}_A-\mathbf{S}_B$. To simplify the problem, we adopt the semi-classical approximation $\hat{r}_A=-\hat{r}_B$ in the Berry’s phase terms, which is a reasonable approximation for states close to the classical ground state. With this 