  We study a relativistic plasma containing charged chiral fermions in an
external electric field. We show that with the presence of both vector and
axial charge densities, the electric field can induce an axial current along
its direction and thus cause chirality separation. We call it the Chiral
Electric Separation Effect (CESE). On very general basis, we argue that the
strength of CESE is proportional to $\mu_V\mu_A$ with $\mu_V$ and $\mu_A$ the
chemical potentials for vector charge and axial charge. We then explicitly
calculate this CESE conductivity coefficient in thermal QED at leading-log
order. The CESE can manifest a new gapless wave mode propagating along the
electric field. Potential observable of CESE in heavy-ion collisions is also
discussed.
