  We explore the propagation of a single hole in the quantum compass model,
whose nematic ground state is given by mutually decoupled antiferromagnetic
chains. The compass model can be seen as the strong-coupling limit of a
spinless two-band Hubbard model, which we study here using mean field theory
and the variational cluster approach. Due to the symmetries of the compass
model, the inherent disorder along one lattice direction turns out not to
affect hole motion and doping a hole consequently does not lift the
subextensive degeneracy of the nematic phase. In order to broaden and deepen
understanding, we derive a generalized itinerant model and address the
transition to two-dimensional Ising order. We observe coherent hole motion in
both the nematic and the antiferromagnetic phases, also in the presence of
quantum fluctuations away from pure Ising exchange. In addition to quantum
fluctuations and interorbital hopping, three-site hopping is found to play an
important role and to dominate propagation in the two-dimensional Ising limit
as well as along the antiferromagnetic chains in the nematic order which forms
in the compass model.
