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We explore with exact diagonalization the propagation of a single hole infour magnetic phases of the t-J-like Kitaev-Heisenberg model on a honeycomblattice: the Neel antiferromagnetic, stripe, zigzag and Kitaev spin-liquidphase. We find coherent propagation of spin-polaron quasiparticles in theantiferromagnetic phase by a similar mechanism as in the $t$-$J$ model forhigh-$T_c$ cuprates. In the stripe and zigzag phases clear quasiparticlesfeatures appear in spectral functions of those propagators where holes arecreated and annihilated on one sublattice, while they remain largely {\ithidden} in those spectral functions that correspond to photoemissionexperiments. As the most surprising result, we find a totally incoherentspectral weight distribution for the spectral function of a hole moving in theKitaev spin-liquid phase in the strong coupling regime relevant for iridates.At intermediate coupling the finite systems calculation reveals a well definedquasiparticle at the $\Gamma$ point, however, we find that the gapless spinexcitations wipe out quasiparticles at finite momenta. Also for this moresubtle case we conclude that in the thermodynamic limit the lightly dopedKitaev liquid phase does not support quasiparticle states in the neighborhoodof $\Gamma$, and therefore yields a {\it non-Fermi liquid}, contrary to earliersuggestions based on slave-boson studies. These observations are supported bythe presented study of the dynamic spin-structure factor for the Kitaev spinliquid regime.

Hole propagation in the Kitaev-Heisenberg model: From quasiparticles in quantum Neel states to non-Fermi liquid in the Kitaev phase