Quasiparticle structure and coherent propagation in the $t-J_{z}-J_{\perp}$ model

Numerical studies, from variational calculation to exact diagonalization, all indicate that the quasiparticle generated by introducing one hole into a two-dimensional quantum antiferromagnet has the same nature as a string state in the $t-J_{z}$ model. Based on this observation, we attempt to visualize the quasiparticle formation and subsequent coherent propagation at low energy by studying the generalized $t-J_{z}-J_{\perp}$ model in which we first diagonalize the $t-J_{z}$ model and then perform a {\em degenerate} perturbation in $J_{\perp}$. We construct the quasiparticle state and derive an effective Hamiltonian describing the coherent propagation of the quasiparticle and its interaction with the spin wave excitations in the presence of the N\'{e}el order. We expect that qualitative properties of the quasiparticle remain intact when analytically continuing $J_{\perp}$ from the anisotropic $J_{\perp} < J_{z}$ to the isotropic $J_{\perp} = J_{z}$ limit, despite the fact that the spin wave excitations change from gapful to gapless. Extrapolating to $J_{\perp}=J_{z}$, our quasiparticle dispersion and spectral weight compare well with the exact numerical results for small clusters.