the Kosterlitz-Thouless temperature , all of the free vortices become bound into vortex-antivortex pairs, and the systems attains a state with zero resistance. Finite magnetic field Cooling the system to and turning on a magnetic field has certain effects. For very small fields () the magnetic field is shielded from the interior of the sample. Above however, the energy cost to keep out the external field becomes too great, and the superconductor allows the field to penetrate in quantized fluxons. Now the superconductor has transitioned into the "mixed state", in which there is a superfluid along with vortices - which now have only one circulation. Increasing the field adds vortices to the system. Eventually the density of vortices becomes so large that they overlap. The core of the vortex contains normal electrons (i.e. the amplitude of the superconducting order parameter is zero), so when they overlap, superconductivity is killed by destroying the amplitude of the order parameter. 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. See also Metal–insulator transition Anderson's theorem References Superconductivity Quantum phases