Abstract
The complex ac conductivity of thin highly disordered InOx 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.
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