Chiral Magnetic Effect out of equilibrium

We consider relativistic fermionic systems in lattice regularization out of equilibrium. The chiral magnetic conductivity $\sigma_{CME}$ is calculated in spatially infinite system for the case when the chiral chemical potential depends on time while the system initially was in thermal equilibrium at small but nonzero temperature. We find that the frequency dependent $\sigma_{CME}(\omega)$ for any nonzero $\omega$ both in the limits $\omega \ll T$ and $\omega \gg T$ is equal to its conventional value $1$ when the lattice model approaches continuum limit. Notice that $\sigma_{CME} = 0$ for the case when the chiral chemical potential does not depend on time at all. We therefore confirm that the limit of vanishing $\omega$ is not regular for the spatially infinite systems of massless fermions.