  Recent observations have revealed the emergence of an unusual normal phase
when a magnetic flux threads an ultra-thin superconducting cylinder. Moreover,
with increasing temperature, the resistance rises in a series of abrupt jumps.
These phenomena are explained using a novel approach, which allows calculation
of the resistance in the presence of amplitude and phase fluctuations of the
superconducting order parameter, and at the same time introduces a local probe
of the current and chemical potential. It is demonstrated that phase
fluctuations lead to the sequential breakdown of local superconducting phase
correlations, resulting in the formation of normal weak links, which give rise
to the emergence of the normal phase is a stepwise manner. Finally, specific
predictions are made on how the experimental observations change with the
cylinder geometry.
