  The two-dimensional (2D) logarithmic character of Coulomb interaction between
charges and the resulting logarithmic confinement is a remarkable inherent
property of high dielectric constant (high-$\kappa$) thin films with far
reaching implications. Most and foremost, this is the charge
Berezinskii-Kosterlitz-Thouless transition with the notable manifestation,
low-temperature superinsulating topological phase. Here we show that the range
of the confinement can be tuned by the external gate electrode and unravel a
variety of electrostatic interactions in high-$\kappa$ films. We find that by
reducing the distance from the gate to the film, we decrease the spatial range
of the 2D long-range logarithmic interaction, changing it to predominantly
dipolar or even to exponential one at lateral distances exceeding the dimension
of the film-gate separation. Our findings offer a unique laboratory for the
in-depth study of topological phase transitions and related phenomena that
range from criticality of quantum metal- and superconductor-insulator
transitions to the effects of charge-trapping and Coulomb scalability in memory
nanodevices.
