  Rings are a model system for studying phase coherence in one dimension.
Superconducting rings have states with uniform phase windings that are integer
multiples of 2$\pi$ called fluxoid states. When the energy difference between
these fluxoid states is of order the temperature so that phase slips are
energetically accessible, several states contribute to the ring's magnetic
response to a flux threading the ring in thermal equilibrium and cause a
suppression or downturn in the ring's magnetic susceptibility as a function of
temperature. We review the theoretical framework for superconducting
fluctuations in rings including a model developed by Koshnick$^1$ which
includes only fluctuations in the ring's phase winding number called fluxoid
fluctuations and a complete model by von Oppen and Riedel$^2$ that includes all
thermal fluctuations in the Ginzburg-Landau framework. We show that for
sufficiently narrow and dirty rings the two models predict a similar
susceptibility response with a slightly shifted Tc indicating that fluxoid
fluctuations are dominant. Finally we present magnetic susceptibility data for
rings with different physical parameters which demonstrate the applicability of
our models. The susceptibility data spans a region in temperature where the
ring transitions from a hysteretic to a non hysteretic response to a periodic
applied magnetic field. The magnetic susceptibility data, taken where
transitions between fluxoid states are slow compared to the measurement time
scale and the ring response was hysteretic, decreases linearly with increasing
temperature resembling a mean field response with no fluctuations. At higher
temperatures where fluctuations begin to play a larger role a crossover occurs
and the non-hysteretic data shows a fluxoid fluctuation induced suppression of
diamagnetism below the mean field response that agrees well with the models.
