Slow holes in the triangular Ising antiferromagnet

We consider the problem of the doped Ising antiferromagnet on the frustrated triangular lattice in the limit where the hole kinetic energy is much smaller than the Ising exchange. For a single hole we prove a "frustrated Nagaoka theorem" Showing that the ground state is magnetized and breaks translational symmetry, in contrast to the parent insulating state that is unmagnetized and spatially homogeneous. The extension of this physics to finite dopings depends on the strength of a density-density coupling that is inevitably present - we find either phase separation of the holes, or a superconducting state that is also magnetized and breaks translational symmetry in a feat of spatial self-organization. Finally, we derive an effective interaction between dilute holes at temperatures in excess of the hopping and find an oscillatory, long-ranged form reflective of the correlations in the underlying classical magnet which presages the breaking of translational symmetry at zero temperature.