  The Heisenberg ferromagnet has symmetry group ${\rm SU}(2)$. The property
known as ferromagnetic ordering of energy levels (FOEL) states that the minimum
energy eigenvalue among eigenvectors with total spin $s$ is monotone decreasing
as a function of $s$. While this property holds for certain graphs such as open
chains, in this note we demonstrate some counterexamples. We consider the spin
1/2 model on rings of length $2n$ for $n=2,3,...,8$, and show that the minimum
energy among all spin singlets is less than or equal to the minimum energy
among all spin triplets, which violates FOEL. This also shows some
counterexamples to the "Aldous ordering" for the symmetric exclusion process.
We also review some of the literature related to these examples.
