1. Material Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
2. Institute of Semiconductor Physics, 630090, Novosibirsk, Russia
3. Institut für experimentelle und angewandte Physik, Universität Regensburg, D-93040 Regensburg, Germany
4. Theoretische Physik III, Ruhr-Universität Bochum, D-44801 Bochum, Germany
5. IMEC, Kapeldreef 75, B-3001 Leuven, Belgium

Correspondence to: Valerii M. Vinokur¹ Correspondence and requests for materials should be addressed to V.M.V. (Email: vinokour@anl.gov).

Synchronized oscillators are ubiquitous in nature¹, and synchronization plays a key part in various classical and quantum phenomena. Several experiments^{2, 3, 4} have shown that in thin superconducting films, disorder enforces the droplet-like electronic texture—superconducting islands immersed into a normal matrix—and that tuning disorder drives the system from superconducting to insulating behaviour. In the vicinity of the transition, a distinct state⁴ forms: a Cooper-pair insulator, with thermally activated conductivity. It results from synchronization of the phase of the superconducting order parameter at the islands across the whole system⁵. Here we show that at a certain finite temperature, a Cooper-pair insulator undergoes a transition to a superinsulating state with infinite resistance. We present experimental evidence of this transition in titanium nitride films and show that the superinsulating state is dual to the superconducting state: it is destroyed by a sufficiently strong critical magnetic field, and breaks down at some critical voltage that is analogous to the critical current in superconductors.