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Energy source for the magnetic field growth in magnetars driven by the electron-nucleon interaction

We study the magnetic field generation in a neutron star within the model based on the magnetic field instability in the nuclear matter owing to the electron-nucleon parity violating interaction. We suggest that the growing magnetic field takes the energy from thermal background fermions in the neutron star matter. The system of kinetic equations for the spectra of the magnetic helicity density and magnetic energy density as well as the chiral imbalance are solved numerically accounting for this energy source. We obtain that, for the initial conditions corresponding to a typical neutron star, the large scale magnetic field $\sim 10^{15}\thinspace\text{G}$ is generated during $(10^4-10^5)\thinspace\text{yr}$. We suggest that the proposed model describes strong magnetic fields observed in magnetars.