Fractals, Anderson localization, and the Aubry-André model intersect in the realm of condensed matter physics, offering profound insights into the behavior of quantum systems. This thesis embarks on a multifaceted exploration of these intriguing phenomena. The introductory chapters lay the groundwork, elucidating Anderson localization's fundamentals and its critical role in condensed matter physics. Applications span various domains, including topological insulators and quantum spin Hall systems. Central to this study is the concept of the delocalized-localized phase transition, impacting wavefunction extension and localization. The subsequent sections delve into fractal analysis—a powerful tool for quantifying self-similarity and complexity within complex materials. The Participation Ratio, framed in terms of \(\psi^{2q}\), quantifies eigenstate localization, with \(q = 1.5\) chosen strategically to balance fine details and dominant characteristics. This choice enhances our understanding of Anderson localization's multifaceted nature, with emphasis on both moderate and high-amplitude regions. Numerical calculations harness Python to unravel the relationship between the number of neighboring couplings (\(k\)) and fermion count, shedding light on electronic occupancy within quasiperiodic lattices. The analysis zooms in on the eigenstate corresponding to half filling, a critical point where localization-delocalization transitions often occur. Fractal properties emerge as a focal point, offering a holistic perspective on complex systems. The Aubry-André model's behavior is dissected, revealing how fractal dimensions, wavefunction characteristics, and critical phenomena intertwine. This thesis unveils a comprehensive tapestry of quantum behavior, localization, and fractal intricacies, offering fresh perspectives on condensed matter physics. The synthesis of these topics enhances our understanding of complex material systems and their multifaceted behavior, illuminating avenues for future exploration.
