Chern-Simons terms in the 3D Weyl semi-metals

Abstract: Based on some theoretical arguments, it has been suggested that electromagnetic response of 3D Weyl semi-metals with non-zero chiral- chemical potential may have a Chern-Simons term, $\frac{1}{2}k_\mu \epsilon^{\mu\nu\rho\sigma}F_{\nu\rho}A_\sigma$, in their effective action for the gauge field. An independent numerical study has shown that such a term is absent in a similar system. In this paper, we investigate the non-equilibrium and equilibrium response of 3D Weyl semi-metals. We argue that the controversy in literature stems from the difference in response of these two distinct states. We then develop a method to deal with well-known ambiguities in quantum electrodynamics in 3D (QED$_{3+1}$) with non-zero chiral-chemical potential and calculate the Chern-Simons term unambiguously. We find that time-like Chern-Simons term can exist in non-equilibrium conditions. We observe that there does not exist any chiral-magnetic effect in equilibrium and anomalous Hall effect replaces it.