Eigenvalues and eigenstates of the many-body collective neutrino oscillation problem

Abstract

We demonstrate a method to systematically obtain eigenvalues and eigenstates of a many-body Hamiltonian describing collective neutrino oscillations. The method is derived from the Richardson-Gaudin framework, which involves casting the eigenproblem as a set of coupled nonlinear "Bethe Ansatz equations", the solutions of which can then be used to parametrize the eigenvalues and eigenvectors. The specific approach outlined in this paper consists of defining auxiliary variables that are related to the Bethe-Ansatz parameters, thereby transforming the Bethe-Ansatz equations into a different set of equations that are numerically better behaved and more tractable. We show that it is possible to express not only the eigenvalues, but also the eigenstates, directly in terms of these auxiliary variables without involving the Bethe Ansatz parameters themselves. In this paper, we limit ourselves to a two-flavor, single-angle neutrino system.

Entanglement and collective flavor oscillations in a dense neutrino gas

“…A more detailed exposition can be found in our previous study, Ref. [73]-however, here we point out, at the end of our summary, a symmetry in the eigenvalue equations that can be used to reduce the computational time. Though arguments in this appendix can be generalized to the higher dimensional (i.e., j p > 1/2) representations, in this discussion we mainly focus on the case of j p = 1/2 for all p, which corresponds to having a single neutrino at each ω p .…”

“…(A8) Here, A is a diagonal matrix of Gaudin raising (lowering) operators, as defined in Ref. [73]. One can determine the corresponding m for a solution using either Λ or , as p µΛ p = ±κ and p p = −m.…”

“…. , 1/µ) [73,89]. Furthermore, by selecting either the raising or the lowering formalism for determining an eigenstate so that κ = min{N/2 + m, N/2 − m}, considerable computation time may be reduced in calculating these states in the mass basis.…”

“…It is worth pointing out, as observed in Refs. [42,73], that energy eigenvalues of the Hamiltonian in Eq. (4) exhibit numerous energy level crossings.…”

“…II. Adiabatic eigenvalues and eigenstates of this many-neutrino Hamiltonian are obtained [42,72,73] using the Richardson-Gaudin algebraic approach [20,84,85]. This solution is outlined briefly in Sec.…”