 Solution

The spin-length sum rule and Fourier inversion giveThe zero mode contains the square of the field-directed magnetization, while the infrared bound controls all nonzero modes. After $L\to \infty$ and then $h\downarrow 0$,Near zero, $\epsilon (k)\asymp |k{|}^2$, so the integral is finite exactly when $d\ge 3$. Choose ${\beta }_0$ larger than the resulting finite constant. Then for $\beta >{\beta }_0$ the right side is positive, and $m(\beta )\ge c>0$.

If $h\downarrow 0$ before $L\to \infty$, every finite torus retains global $O(n)$ symmetry and its magnetization is zero. The reversed iterated limit is therefore zero. The order of limits is what permits spontaneous magnetization.