Superconductor Insulator Transition

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, Ψ = Δ exp ⁡ ( i θ ) {displaystyle Psi =Delta exp(i heta )} , changes. Here Δ {displaystyle Delta } is the amplitude of the order parameter, and θ {displaystyle heta } is the phase. Most theories involve either the destruction of the amplitude of the order parameter - by a reduction in the density of states at the Fermi surface, or by destruction of the phase coherence; which results from the proliferation of vortices. 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, Ψ = Δ exp ⁡ ( i θ ) {displaystyle Psi =Delta exp(i heta )} , changes. Here Δ {displaystyle Delta } is the amplitude of the order parameter, and θ {displaystyle heta } is the phase. Most theories involve either the destruction of the amplitude of the order parameter - by a reduction in the density of states at the Fermi surface, or by destruction of the phase coherence; which results from the proliferation of vortices. 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 ⟨ Ψ ( 0 ) Ψ ( r ) ⟩ ∝ r − γ {displaystyle langle Psi (0)Psi (r) angle propto r^{-gamma }} it decays algebraically).