  We show that the backreaction of left-handed neutrinos out of equilibrium on
the matter sector induces an electric current proportional to a magnetic field
even without a chiral imbalance for electrons in core-collapse supernovae. We
derive the transport coefficient of this effect based on the recently
formulated chiral radiation transport theory for neutrinos. This chiral
electric current generates a strong magnetic field via the so-called chiral
plasma instability, which could provide a new mechanism for the strong and
stable magnetic field of magnetars. We also numerically study the physical
origin of the inverse cascade of the magnetic energy in the
magnetohydrodynamics including this current. Our results indicate that
incorporating the chiral effects of neutrinos would drastically modify the
hydrodynamic evolutions of supernovae, which may also be relevant to the
explosion dynamics.
