Extended t−J-type model on square lattice: role of three-site terms, superconductivity and competing orders

Recent advances in twisted square-bilayer systems allow realization of tunable square-lattice $t−t′−U$ Hubbard models, where the effective hopping ratio $t'/t$ can be varied via twist angle or displacement field. Motivated by these developments, we investigate the strong-coupling regime near half-filling using a self-consistent slave-boson mean-field theory. Beyond the conventional two-site hopping processes (yielding antiferromagnetic $J$, $J′$ couplings) at quadratic order, we incorporate the three-site hopping processes, which in moiré platforms can be parametrically larger than the two-site terms. We find these terms fundamentally alter the pairing landscape: they actively enhance $s$-wave pairing while suppressing $d$-wave, resulting in the stabilization of an extended chiral $s+id$-wave superconducting phase over a broad parameter regime —among other interesting pairing states. We also analyze competition with charge- and spin-density-wave orders and show that superconductivity remains robust across a large window of coupling strengths. Our results thus reveal novel moiré-intrinsic physics in designer square-lattice quantum simulators.