Quantum Hall Effect and Chern Phases in the 1/5-Depleted Square Lattice

We investigate the fractional energy spectrum and quantum Hall response of a two-dimensional 1/5-depleted square lattice subjected to a perpendicular magnetic field. Using a tight-binding model that includes both nearest-neighbor (t_1) and next-nearest-neighbor (t_2) hopping, we compute the Hofstadter butterfly and extract quantized Hall conductivities via Chern number calculations. In the absence of diagonal hopping (t_2 =0), the spectrum exhibits exact particle-hole and flux-inversion symmetries, and the total Chern number across all bands vanishes. When t_2 is introduced, these symmetries are broken, the butterfly becomes deformed, new gaps open, and -remarkably-a nonzero total Chern sum can emerge, signaling unconventional topological phases. By systematically varying t_1 and t_2, we identify regimes with large individual Chern indices and parameter windows where gap stability and Hall plateaus are optimized. Our results demonstrated that lattice depletion combined with diagonal hopping provides a tunable route to engineer robust Chern insulators in both artificial and oxide-based square-lattice systems.