  We explore the implications of Berezinskii-Kosterlitz-Thouless (BKT) critical
behavior and variable-range hopping on the two dimensional (2D) quantum
superconductor-insulator (QSI) transition driven by tuning the gate voltage. To
illustrate the potential and the implications of this scenario we analyze sheet
resistance data of Parendo et al. taken on a gate voltage tuned ultrathin
amorphous bismuth film. The finite size scaling analysis of the BKT-transition
uncovers a limiting length preventing the correlation length to diverge and to
enter the critical regime deeply. Nevertheless the attained BKT critical regime
reveals consistency with two parameter quantum scaling and an explicit quantum
scaling function determined by the BKT correlation length. The two parameter
scaling yields for the zero temperature critical exponents of the
QSI-transition the estimates zn = 3/2, z = 3, and n = 1/2, revealing that
hyperscaling is violated and in contrast to finite temperature disorder is
relevant at zero temperature. Furthermore, zn = 3/2 is also consistent with the
two variable quantum scaling form associated with a variable-range hopping
controlled insulating ground state.
