  We derive the long-time dynamics of a tracer immersed in a one-dimensional
active bath. In contrast to previous studies, we find that the damping and
noise correlations possess long-time tails with exponents that depend on the
tracer symmetry. For generic tracers, shape asymmetry induces ratchet effects
that alter fluctuations and lead to superdiffusion and friction that grows with
time when the tracer is dragged at a constant speed. In the singular limit of a
completely symmetric tracer, we recover normal diffusion and finite friction.
Furthermore, for small symmetric tracers, the active contribution to the
friction becomes negative: active particles enhance motion rather than oppose
it. These results show that, in low-dimensional systems, the motion of a
passive tracer in an active bath cannot be modeled as a persistent random
walker with a finite correlation time.
