  If cosmic background neutrinos interact very weakly with each other, through
spin-spin interactions, then they may have experienced a phase transition,
leading to a ferromagnetic ordering. The small magnetic field resulting from
ferromagnetic ordering -- if present before galaxy formation -- could act as a
primordial seed of the magnetic fields observed in several galaxies. Our
findings suggest that the magnetization could occur in the right epoch, if the
exchange boson of neutrino-neutrino interaction is a massless boson beyond the
Standard Model, with a coupling constant of $2.2\times 10^{-13}
\left(\frac{m_\nu}{10^{-4}\,\rm{eV}}\right)^2<g<2.3\times 10^{-7}$. The
estimation of the magnetic seed is $2.3 \times 10^{-27}\,\rm{G}\lesssim B_{\rm
CNB}\lesssim 6.8\times 10^{-10}G$.
