we generalize the rotor approach to the many-spin systems described by the antiferromagnetic (AFM) Heisenberg model, keeping in mind the above two elements.

Following Haldane [@Haldane83PLA; @Haldane83PRL], we here consider Heisenberg antiferromagnets on a bipartite lattice described by the Hamiltonian given in Eq. . As in the two-spin problem, we introduce the magnetization vectors $\mathbf{M}(\mathbf{x}_i)$ and the staggered magnetization vectors $\mathbf{N}(\mathbf{x}_i)$ such that $$\begin{aligned}
        \label{sab}
        \mathbf{S}^A_i & = &  \mathbf{M}(\mathbf{x}_i)+\mathbf{N}(\mathbf{x}_i), \\  \nonumber
        \mathbf{S}^B_i & = &  \mathbf{M}(\mathbf{x}_i)-\mathbf{N}(\mathbf{x}_i),
        \end{aligned}$$ where $\mathbf{S}^{A(B)}$ denote spins on the $A(B)$ sublattices of the bipartite lattice. We assume that the ground state of the quantum system is “classical-like" with nearly anti-parallel spins on two nearest neighboring sites such that $\mathbf{M}(\mathbf{x}_i)\ll
        \mathbf{N}(\mathbf{x}_i)$, where both $\mathbf{M}(\mathbf{x})$ and $\mathbf{N}(\mathbf{x})$ are very slowly varying functions in space. (We show that this assumption can be justified in the following section.) The classical equation of motion for the spin at lattice site $i$ is $$\label{eqma}
       {\partial\mathbf{S}^{A(B)}_i\over\partial t}=J\left(\sum_{j=\text{NN}(i)}\mathbf{S}^{B(A)}_j\right)\times\mathbf{S}^{A(B)}_i,$$ where $j=\text{NN}(i)$ means that $j$ represents the nearest neighbor sites of $i$.

Using Eq. , after some straightforward algebra and taking the continuum limit, we obtain $$\begin{aligned}
    \label{eqmb}
     {\partial\mathbf{N}(\mathbf{x})\over\partial
     t} & \sim & Jz\mathbf{M}(\mathbf{x})\times\mathbf{N}(\mathbf{x}),  \\ \nonumber
     {\partial\mathbf{M}(\mathbf{x})\over\partial t} & \sim &
     -{Ja^2\over2}(\nabla^2\mathbf{N}(\mathbf{x}))\times\mathbf{N}(\mathbf{x}),
     \end{aligned}$$ where $a$ 