Past exam of the condensed matter physics major option course of the University of Oxford / 2015 / 4 / b / ii / Solution

For collinear antiferromagnetic order in the $xy$ plane, take ${\mathbf{S}}_n=(-1{)}^{n}S\hat{\mathbf{x}}$. Nearest neighbors are antiparallel, giving $-S^2$, whereas second neighbors lie on the same sublattice and remain parallel, giving $+S^2$. ThusThe result follows from substituting into the specified XY Hamiltonian operator, regardless of whether that trial state is the global minimum for the chosen exchanges. Comparing trial-state energies does not eliminate the possibility of a lower helical state.