  In this paper Chiral Magnetic Effect (CME) in Dirac semimetals is studied by
means of lattice Monte Carlo simulation. We measure conductivity of Dirac
semimetals as a function of external magnetic field in parallel
$\sigma_{\parallel}$ and perpendicular $\sigma_{\perp}$ to the external field
directions. The simulations are carried out in three regimes: semimetal phase,
onset of the insulator phase and deep in the insulator phase. In the semimetal
phase $\sigma_{\parallel}$ grows whereas $\sigma_{\perp}$ drops with magnetic
field. Similar behaviour was observed in the onset of the insulator phase but
conductivity is smaller and its dependence on magnetic field is weaker. Finally
in the insulator phase conductivities $\sigma_{\parallel, \perp}$ are close to
zero and do not depend on magnetic field. In other words, we observe
manifestation of the CME current in the semimetal phase, weaker manifestation
of the CME in the onset of the insulator phase. We do not observe signatures of
CME in the insulator phase. We believe that the suppression of the CME current
in the insulator phase is connected to chiral symmetry breaking and generation
of dynamical fermion mass which take place in this phase.
