$${s_i^x}^2+{s_i^y}^2=1$$

$${\cal H}=\displaystyle\sum_{\langle i,j\rangle}J_{ij}\left(s_i^x s_j^x+s_i^y s_j^y\right)$$

$$\vec{s}_i=\left(s_i^x, s_i^y\right)$$

$$\left|\vec{s}_i\right|=1$$

$${\cal H}=\displaystyle\sum_{\langle i,j\rangle}J_{ij}\vec{s}_i\cdot\vec{s}_j$$

$${\cal H}=\displaystyle\sum_{\langle i,j\rangle}J_{ij}cos\left(\varphi_i-\varphi_j\right)$$