Scattering Neutrinos, Spin Models, and Permutations

TL;DR

This paper explores all-to-all coupled Heisenberg spin models inspired by neutrino interactions, revealing non-Gaussian density of states, permutation counting connections, and a unique low-temperature flavor-momentum-locked phase.

Contribution

It introduces a class of simplified Heisenberg models with few eigenvalues, derives their partition functions, and uncovers links to permutation counting and a novel low-temperature phase.

Findings

High-temperature partition functions predict non-Gaussian density of states.

Thermodynamic partition functions relate to permutation counting.

Identifies a unique low-temperature 'flavor-momentum-locked' phase.

Abstract

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.