From Momentum to Real Space: Investigating the Hofstadter Butterfly and Landau Levels in Hexagonal Lattices

Abstract

This research examines Landau Levels (LLs) in real space within hexagonal lattice systems, utilizing the tight-binding model to elucidate band structures under strong magnetic fields, leading to the Hofstadter Butterfly pattern. The introduction of a real-space periodic tight-binding model provides a comprehensive perspective on the Density of States (DOS) in both periodic and open boundary conditions. A key contribution of this work is the innovative approach to decompose Landau Level wavefunctions into angular momentum eigenstates, revealing the inherent angular momentum attributes of the loop-like zones. While the study offers significant insights into the behavior of LLs in real space, it underscores the vast data spectrum associated with each system and highlights the potential for future research on variations in sample shape, the influence of disorder, impurities, and the integration of artificial intelligence for data interpretation.