Competing $s$-wave pairing in overdoped $t$-$J$ model

The $d$-wave pairing symmetry has long been considered a defining feature of high-temperature superconductivity in cuprates. In this work, we reveal that $s$-wave pairing states exhibit variational energies comparable to the $d$-wave state in a square $t$-$J$ model, particularly at high doping levels ($\delta\gtrsim 15\%$) by using the state-of-the-art tensor network simulation. This surprising result suggests that $s$-wave pairing may play an important role in the cuprate phase diagram, especially for the overdoped region. Our findings provide a potential resolution to discrepancies in recent Josephson tunneling experiments on twisted bilayer cuprates and offer new insights into the evolution of pairing symmetry with doping.