We report a comprehensive study of the complex ac conductivity of thin effectively two-dimensional amorphous superconducting InO x films at zero applied field. Below a temperature scale T c0 where the superconducting order parameter amplitude becomes well defined, there is a temperature where both the generalized superfluid stiffness acquires a frequency dependence and the dc magnetoresistance becomes linear in field. We associate this with a transition of the Kosterlitz-Thouless-Berezinskii ͑KTB͒ type. At our measurement frequencies the superfluid stiffness at T KTB is found to be larger than the universal value. Although this may be understood with a vortex dielectric constant of ⑀ v Ϸ 1.9 within the usual KTB theory, this is a relatively large value and indicates that such a system may be out of the domain of applicability of the low-fugacity ͑lowvortex-density͒ KTB treatment. This opens up the possibility that at least some of the discrepancy from a nonuniversal magnitude is intrinsic. Our finite-frequency measurements allow us access to a number of other phenomena concerning the charge dynamics in superconducting thin films, including an enhanced conductivity near the amplitude fluctuation temperature T c0 and a finite dissipation at low temperature which appears to be a universal aspect of highly disordered superconducting films.

The complex AC conductivity of thin highly disordered InOx films was studied as a function of magnetic field through the nominal 2D 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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