approximation, we obtain $$\label{La2a}
      L\rightarrow{1\over2J}(\hat{n}\times\dot{\hat{n}})^2+\hbar\Delta S\mathbf{A}_M(\hat{n})\cdot\dot{\hat{n}},$$ where $\hat{n}=\hat{r}_A$ and $\Delta S=S_A-S_B$. The Hamiltonian of the system is $$\label{h2}
        H_M={J\over2}\left(\mathbf{\Pi}-\hbar\Delta S\mathbf{A}_M(\hat{n})\right)^2,$$ where $\mathbf{\Pi}=\dot{\hat{n}}/J$ is the canonical momentum of the rotor.

$H_M$ is the Hamiltonian of a charged particle moving on the surface of a unit sphere with a magnetic monopole of strength $|\Delta S|$ located at the center of the sphere. The eigenstates of the Hamiltonian are well known and are called the monopole spherical harmonics [@Wu76]. The most interesting feature of the monopole spherical harmonics is that they allow half-odd-integer angular momentum states (which occur when $|\Delta S|$ is a half-odd-integer). The ground state carries an angular momentum of $L=|\Delta S|$ and is $(2|\Delta S|+1)$-fold degenerate, corresponding to the degeneracy of a quantum spin of magnitude $|\Delta S|$, in agreement with the exact result for the two-spin problem.

Non-linear-$\sigma$-model
-------------------------

The two-spin problem tells us that there are two important elements that we must keep track of when a classical spin problem is replaced with the corresponding quantum spin problem: a) quantum fluctuations, originating from the (non)-commutation relation between the canonical coordinates ($\mathbf{N}$) and momenta ($\mathbf{M}$), and b) Berry’s phase, which dictates the quantization of the spins. In the following, 