Spin coherent-state Berry phase

The spin coherent-state path integral contains a Berry phase equal, modulo gauge choices, to the spin $S$ times the solid angle swept on the Bloch sphere. Single-valued path-integral amplitudes require $2S\in \mathbb{Z}$.

For a spin coherent-state history, a Wess--Zumino term records the oriented target-sphere area bounded by the spin path. Its variation is local even though a single gauge-patch expression for the area need not be global.

In a bipartite antiferromagnet, the smooth staggered contributions to the spin Wess--Zumino terms cancel between sublattices. Their term linear in uniform canting is proportional to $\mathbf{m}\cdot (\mathbf{n}\times {\partial }_t\mathbf{n})$.

The CP1 spinor representation writes a unit vector as $n_i=z^{†}{\sigma }_{i}z$ with $z^{†}z=1$. The phase redundancy $z\mapsto e^{i\alpha }z$ identifies normalized spinors that represent the same point of the two-sphere.

The ferromagnetic Wess–Zumino term is the first-order kinetic term $i\kappa z^{†}\dot{z}$ in the CP1 description. A local phase change of $z$ changes it only by the total derivative $-\kappa \dot{\alpha }$.