Past exam of the condensed matter physics major option course of the University of Oxford / 2019 / 5 / d / Solution

The canted pattern has opposite $x$ components but a common $z$ component. Write neighboring spins as ${\mathbf{S}}_A=S(\cos \theta ,0,\sin \theta )$ and ${\mathbf{S}}_B=S(-\cos \theta ,0,\sin \theta )$. Their scalar product is $-S^2\cos 2\theta$, and the cross-product magnitude is $S^2|\sin 2\theta |$.

There are three exchange bonds per spin, two eligible $xy$ bonds per spin for the $D$ term, and one single-ion term per spin. HenceFor the branch $\theta \ge 0$ close to the collinear state,Minimization is equivalent to maximizing $(3J+K/2)\cos 2\theta +2D\sin 2\theta$, which givesThe chosen branch is a minimum of the energy because it maximizes that positive-amplitude sinusoid. The resulting state is a canted antiferromagnet with a small common $z$ moment. The source's absolute-value interaction selects a canting magnitude but is symmetric between the two signs of the weak moment; it is not an oriented vector Dzyaloshinskii-Moriya interaction specifying a unique handedness.