We propose twisted homobilayer of a carbon allotrope, ${\mathrm{C}}_{568}$, to be a promising platform to realize a controllable square lattice single-band extended Hubbard model. This setup has the advantage of a widely tunable $t^{\prime }/t$ ratio without adding external fields, and the intermediate temperature $t\ll T\ll U$ regime can be easily achieved. We first analyze the continuum model obtained from symmetry analysis and first-principle calculations, and calculate the band structures. Subsequently, we derive the corresponding tight-binding models and fit the hopping parameters as well as the Coulomb interactions. When a displacement field is applied, anisotropic nearest-neighbor hoppings can further be achieved. If successfully fabricated, the device could be an important stepping stone towards understanding high-temperature superconductivity.

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• Square lattice
• Twisted heterostructures
• Extended Hubbard model
• First-principles calculations
• Hubbard model
• Tight-binding model