  We analyze the gapped phase of the Kitaev honeycomb model perturbatively in
the isolated-dimer limit. Our analysis is based on the continuous unitary
transformations method which allows one to compute the spectrum as well as
matrix elements of operators between eigenstates, at high order. The starting
point of our study consists in an exact mapping of the original honeycomb spin
system onto a square-lattice model involving an effective spin and a hardcore
boson. We then derive the low-energy effective Hamiltonian up to order 10 which
is found to describe an interacting-anyon system, contrary to the order 4
result which predicts a free theory. These results give the ground-state energy
in any vortex sector and thus also the vortex gap, which is relevant for
experiments. Furthermore, we show that the elementary excitations are emerging
free fermions composed of a hardcore boson with an attached spin- and phase-
operator string. We also focus on observables and compute, in particular, the
spin-spin correlation functions. We show that they admit a multi-plaquette
expansion that we derive up to order 6. Finally, we study the creation and
manipulation of anyons with local operators, show that they also create
fermions, and discuss the relevance of our findings for experiments in optical
lattices.
