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

TL;DR

This paper proves that for large $d$-dimensional boxes, the energy levels of the spin-$1/2$ quantum Heisenberg ferromagnet are asymptotically ordered by total spin, with minimal energy achieved at specific total spin values.

Contribution

It establishes an asymptotic ordering of energy levels for the Heisenberg ferromagnet on large boxes, a significant step in understanding quantum spin systems.

Findings

Energy levels are asymptotically ordered by total spin.

Minimal energy states correspond to specific total spin values.

Results hold for sufficiently large box sizes.

Abstract

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 $(L^d/2)-n$.