  We present a controlled microscopic study of mobile holes in the spatially
anisotropic (Abelian) gapped phase of the Kitaev honeycomb model. We address
the properties of (i) a single hole [its internal degrees of freedom as well as
its hopping properties]; (ii) a pair of holes [their (relative) particle
statistics and interactions]; (iii) the collective state for a finite density
of holes. We find that each hole in the doped model has an eight-dimensional
internal space, characterized by three internal quantum numbers: the first two
"fractional" quantum numbers describe the binding to the hole of the fractional
excitations (fluxes and fermions) of the undoped model, while the third "spin"
quantum number determines the local magnetization around the hole. The
fractional quantum numbers also encode fundamentally distinct particle
properties, topologically robust against small local perturbations: some holes
are free to hop in two dimensions, while others are confined to hop in one
dimension only; distinct hole types have different particle statistics, and in
particular, some of them exhibit non-trivial (anyonic) relative statistics.
These particle properties in turn determine the physical properties of the
multi-hole ground state at finite doping, and we identify two distinct ground
states with different hole types that are stable for different model
parameters. The respective hopping dimensionalities manifest themselves in an
electrical conductivity approximately isotropic in one ground state and
extremely anisotropic in the other one. We also compare our microscopic study
with related mean-field treatments, and discuss the main discrepancies between
the two approaches, which in particular involve the possibility of binding
fractional excitations as well as the particle statistics of the holes.
