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For a smooth configuration, neighboring sites lie on opposite sublattices and contribute Spin coherent-state Berry phases with opposite orientations. The terms $S\Gamma [\tilde{\mathbf{n}}]$ and $S\Gamma [-\tilde{\mathbf{n}}]$ therefore cancel pairwise modulo the quantized solid-angle ambiguity. Their smooth bulk contribution vanishes in the continuum limit.

Singular spacetime configurations can leave lattice Berry phases attached to hedgehog events, but those are outside the smooth sector used in this continuum derivation. Thus the order-parameter-only Wess–Zumino term may be dropped here.

Letwhere $dg$ is Haar measure. Left invariance gives $hK{h}^{-1}=K$ for every $h\in SU(2)$. Since the spin-$S$ representation is irreducible, Schur lemma impliesInserting this coherent-state resolution between $N$ short time steps and taking the continuum limit gives, for a noninteracting spin with zero Hamiltonian,up to the normalization generated by $c$. A Hamiltonian would add $-i\int ⟨g|H|g⟩dt$ in the exponent.

Fordirect differentiation and $S_3|\uparrow ⟩=S|\uparrow ⟩$ giveThe $\dot{\psi }$ term is a total derivative and records only the arbitrary phase used to represent a ray. It cancels against the endpoint-state phases, so the physical Spin coherent-state path integral depends only on the path on the two-sphere. For example, choosing the section $\psi =-\varphi$ gives the Spin coherent-state Berry phase proportional toChanging the surface used to fill a closed path changes its solid angle by $4\pi$. Single-valuedness of the path-integral phase requiresThus the Wess-Zumino coefficient is quantized in integer or half-integer units.

The first term is a gauge choice for the Spin coherent-state Berry phase. For a closed spin trajectory, changing the surface used to evaluate the solid angle changes the action byThe path-integral phase $e^{i{S}_B}$ must be independent of that choice, soHenceand the spin is integer or half-integer.

A spin coherent-state path integral inserts group-averaged resolutions of the identity between short time steps. Its kinetic term is the Spin coherent-state Berry phase, equal to $S$ times the oriented solid angle swept on the sphere.