Recent scanning tunneling spectroscopy (STS) experiments revealed a remarkable role of a supercell consisting of $4\times 4 {\mathrm{CuO}}_2$ unit cells in the emergence of local nematic state and preformed local Cooper pairs and phase coherent cuprate superconductivity. By employing the numerically exact determinant Quantum Monte Carlo simulations, we mimic the effects of an experimental Ca vacancy by an external local potential to investigate the charge and spectral properties of the system hosting two-doped holes. The model numerically supports the role of the $4\times 4$ supercell as the building block of hole-doped cuprates via the hole density distribution and local spectra around the local potential. Our results might provide a theoretical support for the experimental observations and a platform for investigating local charge order and local Cooper pairs on the $4\times 4$ supercell as the plausible route to understanding unconventional cuprate superconductivity.

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• Cuprates
• High-temperature superconductors
• Strongly correlated systems
• Hubbard model
• Quantum Monte Carlo
• Superconductivity