$\hat{n}(\mathbf{x})$. The first term in the action gives the kinetic energy for the rotors, which we have discussed in detail for the two-spin model. The second term represents the coupling between nearest neighboring rotors in the lattice spin model. We note that the term for the coupling between rotors has a magnitude of $\sim S^2$ and dominates over the kinetic energy in the limit of large $S$.

A more systematic derivation of the NL$\sigma$M starting from Eq.  can be achieved by writing $$\mathbf{S}_i=\eta_i S \hat{n}(x_i)\sqrt{1-\left|\frac{\mathbf{M}(x_i)}{S}\right|^2}+\mathbf{M}(x_i),$$ where $\eta_i=e^{i\pi x}$ and we still have $\mathbf{N}(\mathbf{x})\cdot\mathbf{M}(\mathbf{x})=0$. Assuming that $\mathbf{M}(\mathbf{x})$ is small, we can integrate out $\mathbf{M}(\mathbf{x})$ in a power series expansion of $\mathbf{M}(\mathbf{x})$ in the path integral. The NL$\sigma$M for $\hat{n}(\mathbf{x})$ is thus obtained to the leading (Gaussian) order [@Auerbach94].

### Topological term

We next consider the Berry’s phase contribution to the action. Following Appendix A, the total Berry’s phase contribution is $$\label{WZ2}
         S_{T}=\sum_iS_B(\hat{r}_i)\sim \hbar S\sum_i(-1)^i\Omega({\hat{n}_i}),$$ where $S\Omega(\hat{r}_i)=S\int dt\mathbf{A}_M(\hat{r}_i)\cdot \dot{\hat{r}}_i$ is the Berry’s phase for a single spin and $(-1)^i=1(-1)$ for sites on even (odd) sublattices. In the last step, we have assumed that the spins are almost anti-parallel. In the continuum limit, we obtain $$\label{WZ3}
         S_{T}\sim {\hbar S\over2^d}\int d^dx\left({\partial\over\partial_x^1}\cdots{\partial\over\partial_x^d}\right) \Omega(\hat{n}(\mathbf{x})).$$ $S_T$ is 