Instability of Dynamical Axion Mode Towards Finite-Momentum

The chiral anomaly is a striking signature of quantum effects which leads to violations of the continuity equation for a classically conserved current. In the case of a Weyl semimetal, this leads to a breakdown in the conservation of the chiral-charge current, which is conserved by the classical dynamics. In condensed matter systems this leads to a signature magnetoelectric response associated to anomaly due to the separation of the Weyl points in momentum space. In the presence of strong interactions however, a Weyl semimetal phase become unstable towards spontaneous symmetry breaking. If the chiral symmetry is spontaneously broken this then leads to a Goldstone mode which couples to the chiral anomaly leading to a dynamical magnetoelectric response -- a situation known as a dynamical axion insulator.<br/><br/>Here we consider a simple model of a charge-density wave in a Weyl semimetal and calculate the equations of motion for the Goldstone mode. Surprisingly, we find that the Goldstone mode appears to exhibit a negative phase stiffness, signalling instability of the mean-field towards finite momentum condensation. This is expected to lead to very strong fluctuations in the dynamical anomaly. We suggest a suitable long-wavelength theory which may govern the new axion dynamics in this system and comment on possible signatures.