Directional Localization in Disordered 2D Tight-Binding Systems: Insights from Single Particle Entanglement Measures

We investigate the directional localization properties of wave-functions in a two-dimensional tight-binding model with uniform hopping and correlated random on-site energies. By controlling the disorder correlation strength with a parameter $\alpha$, we explore the effects of disorder on wave-function localization using Single Particle Entanglement Entropy (SPEE) and Single Particle R\'enyi Entropy (SPRE) at different values of $q$. Our analysis includes two distinct randomness structures: row-wise and fully correlated disorder. We find that row-wise disorder maintains maximal entanglement for horizontal cuts while enhancing horizontal spread for vertical cuts as $\alpha$ increases. In contrast, fully correlated disorder leads to reduced vertical entanglement for horizontal cuts and increased horizontal entanglement for vertical cuts with rising $\alpha$. Additionally, our results show that the difference between SPEE and SPRE provides valuable insights into localization behavior. These findings highlight the significance of directional properties in understanding localization transitions in disordered systems.