Moments and Lancszos Study of the Anisotropic One Dimensional $t-J$ Model with Holes

In this work we wish to study the ground-state energy as well as the energy gap of the one-dimensional $t-J$ model \[ H=\sum_{l}^{N}\left[ S_{l+1}^{z}S_{i}+\frac{1}{2}\alpha\left( S_{l+1}% ^{+}S_{l}^{-}+S_{l+1}^{-}S_{l}^{+}\right) \right] \] where $\alpha$ is the anisotropy parameter. We will investigate both the ground-state energy as well as the energy gap as the number of holes is increased for lattices of length 16 sites.