A $\mathrm{\Gamma }$-valley Moiré Platform for Tunable Square Lattice Hubbard Model

Moiré superlattices have emerged as a premier platform for simulating the Hubbard model, yet achieving high tunability in square-lattice systems remains a key challenge. We demonstrate that $\mathrm{\Gamma }$-valley twisted square homobilayers provide a faithful and highly tunable realization of $t-t^{\mathrm{\prime }}-U$ Hubbard model, extending the recent proposal in M-valley systems. We show that at small twist angles, an emergent layer-exchange symmetry decouples electronic states into flat bands residing on two nested square sublattices. An interlayer displacement field breaks this symmetry to induce controllable inter-sublattice hybridization, enabling wide-range experimental tuning of the effective hopping ratio $t^{\mathrm{\prime }}\mathrm{/}t$. By establishing a direct correspondence between $\mathrm{\Gamma }$- and M-valley systems, we provide a unified framework for understanding displacement-field tunability in square moiré physics. These findings establish $\mathrm{\Gamma }$-valley twisted bilayers as a versatile platform for simulating the square-lattice Hubbard model and exploring its rich landscape of correlated phenomena.