  We examine the dynamics of a single hole in the gapless phase of the Kitaev
honeycomb model, focusing on the slow-hole regime where the bare hopping
amplitude $t$ is much less than the Kitaev exchange energy $J$. In this regime,
the hole does not generate gapped flux excitations and is dressed only by the
gapless fermion excitations. Investigating the single-hole spectral function,
we find that the hole propagates coherently with a quasiparticle weight that is
finite but approaches zero as $t/J \to 0$. This conclusion follows from two
approximate treatments, which capture the same physics in complementary ways.
Both treatments use the stationary limit as an exactly solvable starting point
to study the spectral function approximately (i) by employing a variational
approach in terms of a trial state that interpolates between the limits of a
stationary hole and an infinitely fast hole and (ii) by considering a special
point in the gapless phase that corresponds to a simplified one-dimensional
problem.
