Preformed Cooper pairing and the uncondensed normal-state component in phase-fluctuating monolayer cuprate superconductivity

We develop a self-consistent microscopic framework beyond mean-field theory for monolayer cuprate superconductivity. It couples fermionic quasiparticles with collective phase dynamics to treat the gap and superfluid stiffness. The phase sector explicitly incorporates both smooth bosonic Nambu-Goldstone phase fluctuations, renormalized by long-range Coulomb interactions, and topological BKT-type vortex-antivortex fluctuations. The required input is the correlated single-particle spectral function, enabling direct interfacing with Hubbard-type models. The theory provides access to key superconducting observables, including $T$-dependent gap and phase stiffness, gap-closing temperature $T_{\rm os}$, and transition temperature $T_c$, across wide ranges of doping. Using a solvable interaction model as input, our simulations reveal several important features consistent with experimental observations in cuprate superconductors: a $d$-wave superconducting dome in $T$-$p$ phase diagram with a shoulder-like anomaly in underdoped regime, a pronounced separation between $T_c$ and $T_{\rm os}$ signaling preformed Cooper pairing, a finite uncondensed normal component persisting even at $T=0$, and the onset temperature $T_{\rm on,vortex}$ of vortex signals, offering a consistent understanding of how strong correlations and phase fluctuations cooperate to shape high-$T_c$ superconductivity.