is the lattice spacing and $z=2d$ is the coordination number. We have assumed a square (cubic)-type lattice and have adopted the slowly varying approximation $$\mathbf{M}(x_{i+1})+\mathbf{M}(x_{i-1})\sim2\mathbf{M}(x_i),$$ $$\mathbf{N}(x_{i+1}))+\mathbf{N}(x_{i-1})\sim2\mathbf{N}(x_i)+a^2\partial^2_x\mathbf{N}(x_i),$$ etc. in deriving the above result. We have also assumed $\mathbf{M}(\mathbf{x})$ to be small and have neglected all non-linear terms in $\mathbf{M}(\mathbf{x})$ in deriving Eq. .

To proceed further, we consider the situation in which all spins have the same magnitude $S$. Then, it is easy to see from Eq.  that $\mathbf{N}(\mathbf{x})^2+\mathbf{M}(\mathbf{x})^2=S^2$ and $\mathbf{N}(\mathbf{x})\cdot\mathbf{M}(\mathbf{x})=0$. Assuming that $M=|\mathbf{M}(\mathbf{x})|\ll N=|\mathbf{N}(\mathbf{x})|\sim S$, we find from Eq.  that $M\sim\omega/(zJ)$ and $\omega\sim\sqrt{z}JaS|\mathbf{k}|$, where $\omega$ and $\mathbf{k}$ are the frequency and wavevector, respectively, of the fluctuations in $\mathbf{N}$. In particular, $M\ll N$ when $ak\ll \sqrt{z}$, i.e., when $\mathbf{N}(\mathbf{x})$ is slowly varying in space.

In the following, we adopt the approximation $N\sim S$ and write $\mathbf{N}(\mathbf{x})=S\hat{n}(\mathbf{x})$, where $\hat{n}^2=1$. Eliminating $\mathbf{M}(\mathbf{x})$ from Eq. , we obtain

$$\label{nlsm1}
       {\partial^2\hat{n}(\mathbf{x},t)\over\partial t^2}={z(SJa)^2\over2}\nabla^2\hat{n}(\mathbf{x},t),$$

corresponding to the following classical action for the vector field $\hat{n}$: $$\label{nlsm2}
      S_{\sigma}={1\over2}\int dt\int d^dx \left({1\over J}\left({\partial\hat{n}\over\partial t}\right)^2-{zJ(Sa)^2\over2}(\nabla\hat{n})^2\right),$$ with the constraint $\hat{n}^2=1$.

$S_{\sigma}$ is the non-linear-$\sigma$ model (NL$\sigma$M) for the unit vector field $\hat{n}(\mathbf{x})$.

Comparing Eqs. and , we see that the NL$\sigma$M can be viewed as a continuum model describing coupled rotors 