Can baryogenesis occur on the lattice?

We examine the question of how baryogenesis can occur in lattice models of the Standard Model where there is a global $U(1)$ symmetry which is accompanied by an exactly conserved fermion number. We demonstrate that fermion creation and annihilation can occur in these models {\em despite} this exact fermion number conservation, by explicitly computing the spectral flow of the hamiltonian in the two dimensional U(1) axial model with Wilson fermions. For comparison we also study the closely related Schwinger model where a similar mechanism gives rise to anomalous particle creation and annihilation.