The nature of the magnetic-field driven superconductor-to-insulator quantum-phase transition in two-dimensional systems at zero temperature has been under debate since the 1980s, and became even more controversial after the observation of a quantum-Griffiths singularity. Whether it is induced by quantum fluctuations of the superconducting phase and the localization of Cooper pairs, or is directly driven by depairing of these pairs, remains an open question. We herein experimentally demonstrate that in weakly-pinning systems and in the limit of infinitely wide films, a sequential superconductor-to-Bose insulator-to-Fermi insulator quantum-phase transition takes place. By limiting their size smaller than the effective penetration depth, however, the vortex interaction alters, and the superconducting state re-enters the Bose-insulating state. As a consequence, one observes a direct superconductor-to-Fermi insulator in the zero-temperature limit. In narrow films, the associated critical-exponent products diverge along the corresponding phase boundaries with increasing magnetic field, which is a hallmark of the quantum-Griffiths singularity.
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