Intrinsic constraint on $T_c$ for unconventional superconductivity

Can room temperature superconductivity be achieved in correlated materials under ambient pressure? Our answer to this billion-dollar question is probably no, at least for realistic models within the current theoretical framework. This is shown by our systematic simulations on the pairing instability of some effective models for two-dimensional superconductivity. For a square lattice model with nearest-neighbour pairing, we find a plaquette state formed of weakly-connected $2\times2$ blocks for sufficiently large pairing interaction. The superconductivity is suppressed on both sides away from its melting quantum critical point. Thus, the magnitude of $T_c$ is constrained by the plaquette state for the $d$-wave superconductivity, in resemblance of other competing orders. We then extend our simulations to a variety of effective models covering nearest-neighbour or onsite pairings, single layer or two-layer structures, intralayer or interlayer pairings, and find an intrinsic maximum of the ratio $T_c/J\approx 0.04-0.07$, where $J$ is the pairing interaction. Comparison with existing experiments supports this constraint in cuprate, iron-based, nickelate, and heavy fermion superconductors, despite that these compounds are so complicated well beyond our simplified models. As a result, the known families of unconventional superconductivity, possibly except the infinite-layer nickelates, seem to almost exhaust their potentials in reaching the maximal $T_c$ allowed by their respective $J$, while achieving room temperature superconductor would require a much larger $J$ beyond 400-700 meV, which seems unrealistic in existing correlated materials and hence demands novel pairing mechanisms. The agreement also implies some deep underlying principles of the constraint that urge for a more rigorous theoretical understanding.