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

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 total spin is at most $(L^d/2)-n$, the minimum energy is attained by a vector whose total spin is exactly $(L^d/2)-n$.