The nature of how this transition occurs is disputed, and many studies seek to understand how the order parameter,
In two dimensions, the subject of superconductivity becomes very interesting because the existence of true long-range order is not possible.
In the 1970s, J. Michael Kosterlitz and David J. Thouless (along with Vadim Berezinski) showed that a different kind of long-range order could exist - topological order - which showed power law correlations (meaning that by measuring the two-point correlation function
Kosterlitz-Thouless behavior can be obtained, but the fluctuations of the order parameter are greatly enhanced, and the transition temperature is suppressed.
The model to keep in mind in the understanding of how superconductivity occurs in a two-dimensional disordered superconductor is the following.
As the system is cooled towards its transition temperature, superconducting grains begin to fluctuate in and out of existence.
This has the effect of increasing the conductivity even before the system has condensed into the superconducting state.
is referred to as paraconductivity, or fluctuation conductivity, and was first correctly described by Lev G. Aslamazov and Anatoly Larkin.
As the system is cooled further, the lifetime of these fluctuations increase, and becomes comparable to the Ginzburg-Landau time Eventually, the amplitude
of the order parameter becomes well defined (it is non-zero wherever there are superconducting patches), and it can begin to support phase fluctuations.
These phase fluctuations set in at a lower temperature, and are caused by vortices - which are topological defects in the order parameter.
, all of the free vortices become bound into vortex-antivortex pairs, and the systems attains a state with zero resistance.
Increasing the field further leads to a very interesting possibility - in two-dimensions where the fluctuations are enhanced - that the vortices may condense into a Bose-condensate, which localizes the superconducting pairs.