Interaction Stability of the Chiral Anomaly in 1+1d

The chiral anomaly is one of the counter-intuitive quantum effects in field theories with chiral symmetry. In many contexts, the chiral anomaly is a consequence of a topological term in the field theory and therefore not renormalized. However, based on some calculations on relativistic models, recent works have proposed that the chiral anomaly could be modified by interactions in Weyl systems. In this paper, we investigate the interaction stability of the chiral anomaly in different one-dimensional condensed matter systems by interactions. We show that in interacting 1DEG at incommensurate fillings, the chiral charge is defined as the total momentum divided by k_F, and the anomaly term originated from the non-gauge-invariance of the stress-energy tensor. With this definition, we find the chiral anomaly to be unrenormalized by interactions. The chiral charge in relativistic field theory is defined as being equal to the conventional current. We find that the chiral anomaly with this definition of chiral charge to be renormalized. We argue that this definition of chiral charge is the only meaningful definition if a back-scattering pinning potential is used to open a weak gap at the Fermi surface. In the presence of repulsive interactions, the chiral charge so defined can be associated with a conserved solitonic chiral charge, which is the total number of solitons and anti-solitons. This renormalized chiral anomaly can be measured by applying an electric field larger than the gap induced by pinning potential.