Survival of superconducting correlations across the two-dimensional superconductor-insulator transition : A finite-frequency study

The complex ac conductivity of thin highly disordered $\mathrm{In}{\mathrm{O}}_{x}$ films was studied as a function of magnetic field through the nominal two-dimensional superconductor-insulator transition. We have resolved a significant finite-frequency superfluid stiffness well into the insulating regime, giving direct evidence for quantum superconducting fluctuations around an insulating ground state and a state of matter with localized Cooper pairs. A phase diagram is established that includes the superconducting state, a transition to a ``Bose'' insulator, and an eventual crossover to a ``Fermi'' insulating state at high fields. We speculate on the consequences of these observations, their impact on our understanding of the insulating state, and its relevance as a prototype for other insulating states of matter that derive from superconductors.