Euler - Heisenberg effective action and magnetoelectric effect in multilayer graphene

The low energy effective field model for the multilayer graphene (at ABC stacking) is considered. We calculate the effective action in the presence of constant external magnetic field $B$ (normal to the graphene sheet). We also calculate the first two corrections to this effective action caused by the in-plane electric field $E$ at $E\mathrm{/}B\ll 1$ and discuss the magnetoelectric effect. In addition, we calculate the imaginary part of the effective action in the presence of constant electric field $E$ and the lowest order correction to it due to the magnetic field ($B\mathrm{/}E\ll 1$).