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

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.

Figure 1

One out of the 210 solutions with $\kappa =6$ to Eq. (22) for a system with $M=N=10$ and ${\omega }_p=p{\omega }_0$, calculated numerically using the modified Newton-Raphson method with homotopy continuation. Observe that, as $\mu \to 0$, six out of the ten ${\mathrm{\Lambda }}_p$ values approach $-\infty$ (as $-1/\mu$), as expected for a $\kappa =6$ solution.

Figure 2

Energy eigenvalues calculated from Eq. (24) after obtaining the values of each ${\mathrm{\Lambda }}_p$. System parameters are identical to those in Fig. 1. The solutions for $\kappa >5$ are omitted, as each of these plots are identical to their $M-\kappa$ counterparts, up to a constant vertical offset.

Figure 3

In the region of small $\mu$, we observe numerous level crossings between different energy states of the Hamiltonian in Eq. (11). (a) Energy eigenvalues of all the $\kappa =5$ solutions for the case $N=M=10$, such as in the case of $\kappa =5$ with the same conditions as in Fig. 2. (b) Energy eigenvalues for all the solutions of a system with $N=M=3$, with eigenvalues corresponding to different $\kappa$ coded as follows: $\kappa =0$ (dotted line), 1 (solid lines), 2 (dashed lines), and 3 (dot-dashed line).

Figure 4

Overlaps, $|⟨{\nu }_{i_1},\dots ,{\nu }_{i_N}|{\psi }_n⟩{|}^2$, of an excited state $|{\psi }_n⟩$ with the mass basis states (with $i_1,\dots ,i_N=1,2$), for systems with different sizes $N$. Here, the $n$th energy eigenstate $|{\psi }_n⟩$ ($n=0,\dots ,2^N-1)$ is connected to the mass eigenstate $|{\nu }_{j_1},\dots ,{\nu }_{j_N}⟩$ in the limit $\mu \to 0$, where $j_{N-k+1}=1+$ ($k$th digit of $n$ in binary representation). Observe that a state with a given $\kappa$ has ${{}^{N}C}_{\kappa }$ nontrivial components for $\mu >0$. (a) Mass-basis decomposition for the state $|{\psi }_2⟩$ in the $N=M=3$ system, as a function of $\mu$; this state has $\kappa =1$. (b) Mass-basis decomposition of the state $|{\psi }_3⟩$ in the $N=M=4$ system, as a function of $\mu$; this state has $\kappa =2$. Two of the six nontrivial overlaps are numerically indistinguishable in (b).