  We perform a detailed finite-size scaling analysis of the sheet resistance in
Bi-films and the LaAlO$_{3}$/SrTiO$_{3}$ interface in the presence and absence
of a magnetic field applied perpendicular to the system. Our main aim is to
explore the occurrence of Berezinskii-Kosterlitz-Thouless (BKT) and quantum
phase transition behavior in the presence of limited size, stemming from the
finite extent of the homogeneous domains or the magnetic field. Moreover we
explore the implications thereof. Above an extrapolated BKT transition
temperature, modulated by the thickness $d$, gate voltage $%V_{g}$ or magnetic
field $H$, we identify a temperature range where BKT behavior occurs. Its range
is controlled by the relevant limiting lengths,which are set by the extent of
the homogeneous domains or the magnetic field. The extrapolated BKT transition
lines $T_{c}\left(d,V_{g},H\right) $ uncover compatibility with the occurrence
of a quantum phase transition where $T_{c}\left( d_{c},V_{gc},H_{c}\right) =0$.
However, an essential implication of the respective limiting length is that the
extrapolated phase transition lines $T_{c}\left( d,V_{g},H\right) $ are
unattainable. Consequently, given a finite limiting length, BKT and quantum
phase transitions do not occur. Nevertheless, BKT and quantum critical behavior
is observable, controlled by the extent of the relevant limiting length.
Additional results and implications include: the magnetic field induced finite
size effect generates a flattening out of the sheet resistance in the
$T\rightarrow 0$ limit, while in zero field it exhibits a characteristic
temperature dependence and vanishes at $T=0$ only. The former prediction is
confirmed in both, the Bi-films and the LaAlO$_{3}$/SrTiO$_{3}$ interface, as
well as in previous studies. The latter is consistent with the
LaAlO$_{3}$/SrTiO$_{3}$ interface data, while the Bi-films exhibit a flattening
out.
