Let $\eta_i=+1$ and $-1$ on the two sublattices of the square lattice. A unit-vector decomposition that separates staggered and uniform magnetization is
$$
\boxed{
\mathbf n_i
=\eta_i\widetilde{\mathbf n}(\mathbf x_i)
\sqrt{1-|\mathbf m(\mathbf x_i)|^2}
+\mathbf m(\mathbf x_i),}
$$
where
$$
|\widetilde{\mathbf n}|^2=1,
\qquad
\mathbf m\mathbin\cdot\widetilde{\mathbf n}=0,
\qquad
|\mathbf m|\ll1.
$$
The alternating part is the <Néel order parameter>, while $\mathbf m$ is the slowly varying <uniform magnetization> generated by canting the two sublattices.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 337 / 2 / a / Solution

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The Néel order parameter is the slowly varying staggered magnetization of an antiferromagnet. On a bipartite lattice it changes sign between neighboring sublattices before coarse graining.


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