 Solution

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.