Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 75 / 2 / a

On a connected bipartite graph, every exchange bond can attain its lower classical energy $-J{S}^2$. Set ${\mathbf{S}}_i=S\mathbf{n}$ on one sublattice and ${\mathbf{S}}_j=-S\mathbf{n}$ on the other, for any unit vector $\mathbf{n}$. This Néel state minimizes all bonds simultaneously and hasThe direction $\mathbf{n}\in S^2$ parametrizes the continuous orientation degeneracy; disconnected components can choose their orientations independently.

A triangular lattice illustrates geometric frustration. For three fixed-length spins on a triangle,The minimum requires their sum to vanish, giving coplanar $120$-degree spins rather than antiparallel alignment on every bond. A three-sublattice pattern realizes this condition on the nearest-neighbour triangular lattice. Its energy per bond is $-J{S}^2/2$, and global rotations produce equivalent classical states.