Quantum Hall effect and Chern phases in the 1/5-depleted square lattice

We investigate the fractal 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 demonstrate 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.