Moments and Lanczos Study of the Anisotropic One Dimensional $XY$ Model in a Skewed Magnetic Field

Here we wish to study the ground-state and energy gap of the one dimensional spin $\frac{1}{2}$ anisotropic antiferromagnetic $XY$ Heisenberg model given by \begin{equation} H=\sum_{l=1}^{N}\left[ \left( 1+\gamma\right) S_{l}^{x}S_{l+1}^{x}+\left( 1-\gamma\right) S^{y}_{l}S_{l+1}^{y}-h\left( -1\right) ^{l}S_{l}^{z}\right] \end{equation} where $\gamma$ is the anisotropy parameter and $h$ is an external magnetic field. We shall investigate the ground-state energy as well as the energy gap as a function of both the anisotropy parameter as well as the number of holes as a function of the external magnetic field.