  Massless chiral excitations can arise from the interactions between a fermion
and an axion string, propagating along the string and allowing it to
superconduct. The properties of these excitations, or zero modes, dictate how
the string interacts with light and can thus have important phenomenological
consequences. In this paper, we add a nowhere-vanishing Dirac mass for the
fermion in the usual model of axion electrodynamics. We find that the zero
modes exhibit an interesting phase structure in which they delocalize from the
string's core as the mass increases, up until a critical value past which they
disappear. We study this structure from an analytic perspective, with explicit
numerical solutions, and via anomaly inflow arguments. Finally, we derive the
two-dimensional effective theory of the zero mode and its interactions with the
four-dimensional gauge field and show how this effective theory breaks down as
the zero modes delocalize.
