  Recent works have provided evidence that an axial anomaly can arise in Weyl
semimetals. If this is the case, then the electromagnetic response of Weyl
semimetals should be governed by the equations of axion electrodynamics. These
equations capture both the chiral magnetic and anomalous Hall effects in the
limit of linear response, while at higher orders their solutions can provide
detectable electromagnetic signatures of the anomaly. In this work, we consider
three versions of axion electrodynamics that have been proposed in the Weyl
semimetal literature. These versions differ in the form of the chiral magnetic
term and in whether or not the axion is treated as a dynamical field. In each
case, we look for solutions to these equations for simple sample geometries
subject to applied external fields. We find that in the case of a linear chiral
magnetic term generated by a non-dynamical axion, self-consistent solutions can
generally be obtained. In this case, the magnetic field inside of the Weyl
semimetal can be magnified significantly, providing a testable signature for
experiments. Self-consistent solutions can also be obtained for dynamical
axions, but only in cases where the chiral magnetic term vanishes identically.
Finally, for a nonlinear form of the chiral magnetic term frequently considered
in the literature, we find that there are no self-consistent solutions aside
from a few special cases.
