  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.
