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We explore the implications of Berezinskii-Kosterlitz-Thouless (BKT) criticalbehavior and variable-range hopping on the two dimensional (2D) quantumsuperconductor-insulator (QSI) transition driven by tuning the gate voltage. Toillustrate the potential and the implications of this scenario we analyze sheetresistance data of Parendo et al. taken on a gate voltage tuned ultrathinamorphous bismuth film. The finite size scaling analysis of the BKT-transitionuncovers a limiting length preventing the correlation length to diverge and toenter the critical regime deeply. Nevertheless the attained BKT critical regimereveals consistency with two parameter quantum scaling and an explicit quantumscaling function determined by the BKT correlation length. The two parameterscaling yields for the zero temperature critical exponents of theQSI-transition the estimates zn = 3/2, z = 3, and n = 1/2, revealing thathyperscaling is violated and in contrast to finite temperature disorder isrelevant at zero temperature. Furthermore, zn = 3/2 is also consistent with thetwo variable quantum scaling form associated with a variable-range hoppingcontrolled insulating ground state.

Gate voltage tuned quantum superconductor to insulator transition in an ultrathin bismuth film revisited