  Motivated by the existence of mobile low-energy excitations like domain walls
in one dimension or gauge-charged fractionalized particles in higher
dimensions, we compare quantum dynamics in the presence of weak Markovian
dephasing for a particle hopping on a chain and for an Ising domain wall whose
motion leaves behind a string of flipped spins. Exact solutions show that the
two models have near identical transport responses in the bulk. On the other
hand, in finite-length chains, the broadening of discrete spectral lines is
much more noticeable in the case of a domain wall. These results may be of
relevance to a broad class of systems including quasi-1D antiferromagnets,
polymer chains, and even retinal systems.
