  We consider a class of Heisenberg all-to-all coupled spin models inspired by
neutrino interactions in a supernova with $N$ degrees of freedom. These models
are characterized by a coupling matrix that is relatively simple in the sense
that there are only a few, relative to $N$, non-trivial eigenvalues, in
distinction to the classic Heisenberg spin-glass models, leading to distinct
behavior in both the high-temperature and low-temperature regimes. When the
momenta of the neutrinos are uniform and random in directions, we can calculate
the large-$N$ partition function for the quantum Heisenberg model. In
particular, the high-temperature partition function predicts a non-Gaussian
density of states, providing interesting counter-examples showing the limits of
general theorems on the density of states for quantum spin models. We can
repeat the same argument for classical Heisenberg models, also known as rotor
models, and we find the high-temperature expansion is completely controlled by
the eigenvalues of the coupling matrix, and again predicts non-Gaussian
behavior for the density of states as long as the number of eigenvalues does
not scale linearly with $N$. Indeed, we derive the amusing fact that these
\emph{thermodynamic} partition functions are essentially the generating
function for counting permutations in the high-temperature regime. Finally, for
the case relevant to neutrinos in a supernova, we identify the low-temperature
phase as a unique state with the direction of the momenta of the neutrino
dictating its coherent state in flavor-space, a state we dub the
"flavor-momentum-locked" state.
