Heisenberg antiferromagnet

A Heisenberg antiferromagnet has positive exchange coupling in $H=J{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$, so neighboring spins energetically prefer antiparallel alignment.

With $H={\sum }_{ij\mathrm{bonds}}{J}_{ij}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$ counting bonds once, type-I and type-II classical collinear energies per spin are $S^2(-4J_1+3J_2)$ and $-3J_2{S}^2$. Their boundary is $J_2/J_1=2/3$. The corresponding ordering reflections have integer indices with odd sum, and three half-integer indices, respectively, in units of $2\pi /a$. This compares classical order, not an exact quantum product-state ground state.

A bipartite spin rotation rotates the spin on one sublattice by $\pi$, converting an alternating classical Néel state to an all-up reference state. Rotation about $x$ sends $S^z\mapsto -S^z$ and $S^{\pm }\mapsto S^{\mp }$ on that sublattice. For an isotropic nearest-neighbor Heisenberg antiferromagnet, a bond becomes $-S_i^z{S}_j^z+(S_i^{+}{S}_j^{+}+S_i^{-}{S}_j^{-})/2$. The transformation is unitary and preserves the spin commutation relations; it changes the reference frame, not the model's spectrum.