  The magnetic properties of 4d and 5d transition-metal insulating compounds
with the honeycomb structure are believed to be described by the
Heisenberg-Kitaev model, which contains both the isotropic Heisenberg
interaction J and anisotropic Kitaev interaction K. In this paper, to
investigate the charge dynamics in these materials, we study the single-hole
propagation of the t-J-K model in various magnetically ordered phases by the
self-consistent Born approximation. We find that there are low-energy coherent
quasiparticle (QP) excitations in all of these phases which appear firstly
around the K point in the Brillouin zone (BZ), but the band-widths of these QPs
are very small due to the hole-magnon coupling. Interestingly, in the zigzag
phase relevant to recent experiments, though the QP weights are largely
suppressed in the physical spectra in the first BZ, we find that they recover
in the extended BZs. Moreover, our results reveal that the low-energy QP
spectra are reduced with the increase of K.
