In-medium polarization tensor in strong magnetic fields (II): Axial Ward identity at finite temperature and density

We investigate the axial Ward identity (AWI) for massive fermions in strong magnetic fields. The divergence of the axial-vector current is computed at finite temperature and/or density with the help of a relation between the polarization and anomaly diagrams in the effective (1+1) dimensions realized in the lowest Landau level (LLL). We discuss delicate interplay between the vacuum and medium contributions that determines patterns of the spectral flow in the adiabatic limit and, more generally, the diabatic chirality production rate. We also establish an explicit relation between the AWIs from the LLL approximation and from the familiar triangle diagrams in the naive perturbative series with respect to the coupling constant.