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We study two Heisenberg-Kitaev t-J-like models on a honeycomb lattice,focusing on the zigzag magnetic phase of Na$_2$IrO$_3$, and investigate holemotion by exact diagonalization and variational methods. The spectral functionsare quite distinct from those of cuprates and are dominated by large incoherentspectral weight at high energy, almost independent of the microscopicparameters --- a universal and generic feature for zigzag magneticcorrelations. We explain why quasiparticles at low energy are stronglysuppressed in the photoemission spectra and determine an analog of a pseudogap.We point out that the qualitative features of the predominantly incoherentspectra obtained within the two different models for the zigzag phase aresimilar, and they have remarkable similarity to recently reported angularresolved photoemission spectra for Na$_2$IrO$_3$.

Hidden Quasiparticles and Incoherent Photoemission Spectra in Na2IrO3