Chiral separation effect in lattice regularization

We consider the chiral separation effect (CSE) in the lattice-regularized quantum field theory. We discuss two types of regularization: with and without exact chiral symmetry. In the latter case this effect is described by its conventional expression for the massless fermions. This is illustrated by the two particular cases of Wilson fermions and the conventional overlap fermions. At the same time, in the presence of the exact chiral symmetry the CSE disappears. This is illustrated by the naive lattice fermions, when the contributions of the fermion doublers cancel each other. Another example is the modified version of the overlap regularization proposed recently, where there is exact chiral symmetry, but as a price for this the fermion doublers become zeros of the Green function. In this case the contributions to the CSE of zeros and poles of the Green function cancel each other.