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We prove a result for the spin-$1/2$ quantum Heisenberg ferromagnet on$d$-dimensional boxes $\{1,\dots,L\}^d \subset \mathbb{Z}^d$. For any $n$, if$L$ is large enough, the Hamiltonian satisfies: among all vectors whose totalspin is at most $(L^d/2)-n$, the minimum energy is attained by a vector whosetotal spin is exactly $(L^d/2)-n$.

Asymptotic Ferromagnetic Ordering of Energy Levels for the Heisenberg Model on Large Boxes