  A half-century after the discovery of the superconductor-insulator transition
(SIT), one of the fundamental predictions of the theory, the charge
Berezinskii-Kosterlitz-Thouless (BKT) transition that is expected to occur at
the insulating side of the SIT, has remained unobserved. The charge BKT
transition is a phenomenon dual to the vortex BKT transition, which is at the
heart of the very existence of two-dimensional superconductivity as a
zero-resistance state appearing at finite temperatures. The dual picture points
to the possibility of the existence of a superinsulating state endowed with
zero conductance at finite temperature. Here, we report the observation of the
charge BKT transition on the insulating side of the SIT, identified by the
critical behavior of the resistance. We find that the critical temperature of
the charge BKT transition depends on the magnetic field exhibiting first the
fast growth and then passing through the maximum at fields much less than the
upper critical field. Finally, we ascertain the effects of the finite
electrostatic screening length and its divergence at the magnetic field-tuned
approach to the superconductor-insulator transition.
