thought indicates that our mapping of the spin problem to the rotor problem cannot be totally correct. What happens if $\mathbf{S}_A$ is an integer spin and $\mathbf{S}_B$ is a half-odd-integer spin? Elementary quantum mechanics tells us that the ground state should carry half-odd-integer angular momentum. The possibility of such a scenario is missing in our rotor mapping, in which the spin magnitudes $S_{A(B)}$ do not appear.

Berry’s phase
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The missing piece in our mapping of the two-spin problem to the rotor model is the Berry’s phase [@Berry84], which is carried by spins but is absent in rotors. The correct spin-quantization rule is recovered only after this piece of physics is properly added into the rotor problem. First, let us review the Berry’s phase carried by a single spin.

We recall that for a spin tracing out a closed path $\mathbf{C}$ on the surface of the unit sphere, the spin wavefunction acquires a Berry’s phase $\gamma(\mathbf{C})=S\Omega(\mathbf{C})$, where $S$ is the spin magnitude and $\Omega(\mathbf{C})$ is the surface area under the closed path $\mathbf{C}$ on the unit sphere (see Fig. \[fig:Berry\]). $S\Omega(\mathbf{C})$ can be represented more conveniently by imagining the spin trajectory as the trajectory of a particle carrying a unit 