Incoherent Cooper pairing and pseudogap behavior in single-layer FeSe/SrTiO$_3$

In many unconventional superconductors, the presence of a pseudogap - a suppression in the electronic density of states extending above the critical temperature - has been a long-standing mystery. Here, we employ combined \textit{in situ} electrical transport and angle-resolved photoemission spectroscopy (ARPES) measurements to reveal an unprecedentedly large pseudogap regime in single-layer FeSe/SrTiO$_3$, an interfacial superconductor where incoherent Cooper pairs are initially formed above $T_{\Delta}$ $\approx$ 60 K, but where a zero resistance state is only achieved below $T_{0}$ $<$ 30 K. We show that this behavior is accompanied by distinct transport signatures of two-dimensional phase fluctuating superconductivity, suggesting a mixed vortex state hosting incoherent Cooper pairs which persist well above the maximum clean limit $T_{c}$ of $\approx$ 40 K. Our work establishes the critical role of reduced dimensionality in driving the complex interplay between Cooper pairing and phase coherence in two-dimensional high-$T_c$ superconductors, providing a paradigm for understanding and engineering higher-$T_{c}$ interfacial superconductors.