The nature of the "pseudogap" in the insulating phase of highly disordered superconductors
Disordered thin films undergoing a superconductor-insulator transition provide a controlled setting for studying pseudogap physics in the absence of competing electronic orders. Although theory predicts that local Cooper pairing can survive deep into the insulating phase, direct spectroscopic confirmation has remained experimentally inaccessible because tunneling measurements in highly insulating films require ultra-high-resistance junctions and picoampere current sensitivity. Here we combine ultra-high-resistance planar tunneling spectroscopy on amorphous indium oxide films with quantum Monte Carlo simulations of the attractive Hubbard model to probe the single-particle excitation spectrum deep in the insulating regime. We find striking agreement between experiment and theory: the single-particle gap not only survives across the superconductor-insulator transition, but increases substantially with disorder, reaching values more than twice those observed on the superconducting side. With increasing temperature, the gap fills rather than closes, while coherence peaks are suppressed and spectral weight redistributes to energies far exceeding the gap scale. Our results provide direct quantitative experimental confirmation of the theoretically predicted Cooper-pair insulating state with localized Cooper pairs and establish a unified connection between the pseudogap above $T_c$ and the insulating gap as manifestations of pairing without global phase coherence.