 Solution

At zero field the finite-volume law is invariant under the global rotation ${\theta }_z\mapsto {\theta }_z+\alpha$. Averaging $\cos (2{\theta }_x)$ over $\alpha$ gives zero. Reflection ${\theta }_z\mapsto -{\theta }_z$ likewise gives $⟨\sin {\theta }_y⟩=0$. Thusfor every finite $\Lambda$ containing $x,y$, so its infinite-volume limit exists and equals zero.