is summarized in section VI.

From semi-classical to non-linear-$\sigma$ model approaches for quantum antiferromagnets
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Here, we consider simple Heisenberg antiferromagnets on bipartite lattices (with sublattices $A$ and $B$) with the Hamiltonian $$\label{hm1}
  H=J\sum_{\langle i,j\rangle}\mathbf{S}_i\cdot\mathbf{S}_j,$$ where $J>0$ and $\langle i,j\rangle$ describes a pair of nearest neighbor sites in the bipartite lattice. In a bipartite lattice, any two nearest neighbor sites always belong to different sublattices. $\mathbf{S}$ is a quantum spin with magnitude $S=n/2$, where $n=$ positive integer. Examples of bipartite lattices include 1D spin chains, 2D square or honeycomb lattices, and 3D cubic lattices.

Two-spin problem
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The semi-classical approach begins with the assumption that the quantum spins are “close" to classical spins, and it is helpful to start by first analyzing the corresponding classical spin problem. For simplicity, we start by considering only two classical spins coupled by the Heisenberg interaction $$H=J\mathbf{S}_A\cdot\mathbf{S}_B. \\\\\ (J>0).$$ The classical spins obey Euler’s equation of motion: $$\label{eqm1}
   {\partial\mathbf{S}_{A(B)}\over\partial t}=J\mathbf{S}_{B(A)}\times\mathbf{S}_{A(B)}.$$

This equation can be solved most easily by introducing the magnetization and staggered magnetism vectors $\mathbf{M}(\mathbf{N})=\mathbf{S}_A+(-)\mathbf{S}_B$, where it is easy to show from Eq.  (\[eqm1\]) that $$\begin{aligned}
   \label{eqm3}
   {\partial \mathbf{M}\over\partial t} & = & 0,  \\ \nonumber
   {\partial\mathbf{N}\over\partial t} & = & J\mathbf{M}\times\mathbf{N},
   \end{aligned}$$ 