Effects of Next-Nearest-Neighbor Hopping on the Hole Motion in an Antiferromagnetic Background

In this paper we study the effect of next-nearest-neighbor hopping on the dynamics of a single hole in an antiferromagnetic (N\'{e}el) background. In the framework of large dimensions the Green function of a hole can be obtained exactly. The exact density of states of a hole is thus calculated in large dimensions and on a Bethe lattice with large coordination number. We suggest a physically motivated generalization to finite dimensions (e.g., 2 and 3). In $d=2$ we present also the momentum dependent spectral function. With varying degree, depending on the underlying lattice involved, the discrete spectrum for holes is replaced by a continuum background and a few resonances at the low energy end. The latter are the remanents of the bound states of the $t-J$ model. Their behavior is still largely governed by the parameters $t$ and $J$. The continuum excitations are more sensitive to the energy scales $t$ and $t_1$.