An Exactly Solvable Model of Phase-Fluctuating Superconductivity in Cuprates: The Role of Partially Flat Bands

Utilizing an exactly solvable Hubbard-like model that exhibits a pseudogap (PG) phase and a partially flat band, we perform self-consistent microscopic calculations of the superconductivity (SC) in cuprates, incorporating both thermal and zero-point superconducting phase fluctuations in the presence of long-range Coulomb interactions. The results reveal an important role of the partially flat band in determining phase-fluctuating SC as well as several key features that are consistent with experimental observations. Specifically, we find a dome-shaped $d$-wave superconducting region in the temperature-doping phase diagram with the optimal doping point located near the quantum critical point between the PG and the metallic phases. Near the optimal doping, the partially flat band suppresses and amplifies the fluctuation-induced destructing effect on the $d_{x^2-y^2}$- and $d_{xy}$-wave pairing, respectively, ensuring an absolutely dominant $d_{x^2-y^2}$-wave SC. While the phase fluctuations of the $d_{x^2-y^2}$-wave SC are relatively weak around optimal doping, they become significant in both underdoped and overdoped regimes. We then identify a discontinuity on the superconducting dome in underdoped regime that results from a transition from a strong- to a weak-phase-fluctuating state as doping approaches the optimal point.