Spin-hole polaron of the t-$J_z$ model

We present analytic and numerical results for the ground state of a dynamical hole in a quantum Ising antiferromagnet (the t-$J_z$ model) on a two-dimensional square lattice. We find that the ground state has a localized region of reduced Néel order associated with the hole, with an energy that is well described by ${\mathrm{\epsilon }}_{\mathit{h}}$/t≊-3.63+2.93(${\mathit{J}}_{\mathit{z}}$/t${)}^{23}$ for 6×${10}^{\mathrm{-}3}$≲${\mathit{J}}_{\mathit{z}}$/t≲1. This resolves a disagreement in the literature regarding the coefficient of ($J_z$/t${)}^{2/3}$. A simple string picture gives a reasonable description of the extent of the ground-state polaron in this intermediate-coupling regime. Disruption of long-range antiferromagnetic order in the bulk limit is estimated to occur at $J_z$/t≊0.3 for a hole-doping fraction of x=(1/16.