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The Hubbard and closely related t-J models are exciting platforms forunconventional superconductivity. Through state-of-the-art density matrixrenormalization group calculations using the grand canonical ensembles, weaddress open issues regarding the ground-state phase diagram of the extendedt-J model on a square lattice in the parameter regime relevant to cupratesuperconductors. On large 8-leg cylinders, we demonstrate that the pure t-Jmodel with only nearest-neighbor hoppings and superexchange interactions, for awide range of hole doping ($\delta = 0.1 - 0.2$), hosts an exotic SC statewhere the d-wave and unidirectional pair density wave SC co-exist with spin andcharge bond orders. Furthermore, a small next nearest neighbor hopping $t_2$suppresses the pair density wave and other co-coexisting orders, leading to ad-wave SC phase in both electron- and hole-doped cuprate model systems. Wereveal a simple mechanism for SC in the t-J model, in which the nearestneighbor hopping plays an essential role in driving the formation of Cooperpairs with real-space sign oscillations balancing the competition between thekinetic and exchange energies. Our work validates the t-J model as a properminimum model for describing fundamental physics of cuprate superconductors

D-Wave and Pair-Density-Wave Superconductivity in the Square-Lattice t-J Model