The Superconductor Insulator Transition is an example of a quantum phase transition, whereupon tuning some parameter in the Hamiltonian, a dramatic change in the behavior of the electrons occurs. The nature of how this transition occurs is disputed, and many studies seek to understand how the order parameter,
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Destruction of superconductivity
In two dimensions, the subject of superconductivity becomes very interesting because the existence of true long-range order is not possible. How then is superconductivity obtained? In the 70's, Kosterlitz and Thouless (along with 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
This picture changes if disorder is included. 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. At high temperatures, the system is in the normal state. As the system is cooled towards its transition temperature, superconducting grains begin to fluctuate in and out of existence. When one of these grains "pops" into existence, it is accelerated without dissipation for a time
Finite magnetic field
Cooling the system to
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.